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area of the target is equal to pi * r^2

radius of the target equals 122/2 = 61 cm.

radius of the bullseye = 12.2/2 = 6.1 cm.

outside of the bullseye there are 9 evenly spaced concentric rings.

this means that the width of each of those rings would equal (61 - 6.1)/9 = 6.1 cm each.

the target forms 10 concentric circles.

the bullseye is one of these circles and there are an additional 9 circles surrounding the bullseye.

the area of each of these circles is from the center of the target to the line forming that circle.

the circle formed by the bullseye has a radius of 6.1 cm.
the circle formed by the 1st ring has a radius of 12.2 cm.
the circle formed by the 2d ring has a radius of 18.3 cm.
the circle formed by the 3d ring has a radius of 24.4 cm.
the circle formed by the 4th ring has a radius of 30.5 cm.
the circle formed by the 5th ring has a radius of 36.6 cm.
the circle formed by the 6th ring has a radius of 42.7 cm.
the circle formed by the 7th ring has a radius of 48.8 cm.
the circle formed by the 8th ring has a radius of 54.9 cm.
the circle formed by the 9th ring has a radius of 61 cm.

the total area of the target is equal to pi * 61^2 = 11689.86626 cm^2.

this is the same as the area of the circle formed by the 9th ring.

the probability of hitting anywhere on the target is equal to 1.0 (100%). the assumption is that you will never miss the target.

the area of the bullseye = pi * 6.1^2 = 116.8986626

the probability of hitting the bullseye = 116.8986626 / 11689.86626 = .01

the probability of landing in the outmost 4 rings is calculated as follows:

4 outer rings are the 6th, 7th, 8th, and 9th rings.

the radius of the circle formed by the 9th ring is 61 cm.

the radius of the circle formed by the 5th ring is 36.6 cm.

the area from the center of the circle to a circle with a radius of 36.6 cm = pi * (36.6)^2 = 4208.251855 cm^2.

the area from the center of the circle to a circle with a radius of 61 cm = pi * 61^2 = 11689.86626 cm^2 (this is the area of the target we calculated earlier).

the area for the outer 4 circles is equal to the area of the circle formed by the 9th ring minus the area of the circle formed by the 5th ring.

the area for the outer 4 circles is therefore equal to:

11689.86626 - 4208.251855 = 7481.514409 cm^2.

the probability of hitting this area is equal to the area of the 4 outer rings divided by the total area of the target = 7481.514409 / 11689.86626 = .64

the area of the 5th ring only is the area of the circle formed by the 5th ring minus the area of the circle formed by the 4th ring.

that would be pi * (36.6)^cm^2 minus pi * (30.5)^2.

that equals 4208.351855 minus 2922.466566 = 1285.885289 cm^2

the probability of hitting this area would be 1258.885289 / 11689.86626 = .11

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formula for the surface area of a sphere is:

s = 4 * pi * r^2

formula for the volume of a sphere is:

v = (4/3) * pi * r^3

since s = 4 * pi * r^2, and since s = 3600 * pi, then we get:

3600 * pi = 4 * pi * r^2

divide both sides of this equation by pi to get:

3600 = 4 * r^2

divide both sides of this equation by 4 to get:

900 = r^2

take square root of both sides of this equation to get:

r = 30

We want to find what v/pi equals.

since v = (4/3) * pi * r^3, and since r = 30, we get:

v = (4/3) * pi * (30)^3 which becomes:

v = (4/3) * pi * 27000

divide both sides of this equation by pi to get:

v/pi = (4/3) * 27000 = 36000.

our answer is:

v/pi = 36000

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f = number of liters of fruit juice.
b = nume of liters of bubbly water.

ratio f/b = 3/5

total liters = 24.

your equation is f + b = 24.

since f/b = 3/5, thie means that:

f = 3b/5

equation of f + b = 24 becomes:

3b/5 + b = 24

multiply both sides of this equation by 5 to get:

3b + 5b = 120 which becomes:

8b = 120 which becomes:

b = 120/8 = 15.

since f + b = 24, this means that f = 24-15 = 9.

you have f = 9 and b = 15

f + b = 9 + 15 = 24

f/b = 9/15 = 3/5

ratio holds and the sum is correct.

if you used a strip diagram, you might do the following:

let 1 strip represent 3 liters of fruit juice.
this strip would be divided into 3 equal sections.

let another strip represent 5 liters of bubbly water.
this strip would be divided into 5 equal sections.

each section would represent 1 liter.

first strip would look like fff
second strip would look like bbbbb

since the ratio has to remain constant, then combine these strips into one long strip that is composed of 3 sections of f and 5 sections of b.

that would look like:

fffbbbbb = 8 liters.

since you want 24 liters, how many times does this strip divide into 24.

answer is 3.

you get:

fffbbbbb + fffbbbbb + fffbbbbb = 24 liters.

add up the f's and you get 9.
add up the b's and you get 15.

you have 9 liters of fruite juice and 15 liters of bubbly water to make a total of 24 liters of mixture.

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looks like you started off ok.

your formula is:

S = P * (1+i)^n

you used 10,000 for P and 1.21 * 10,000 = 12100 for S.

that's correct but not necessary because you could just as easily have used 1 for P and 1.21 for S.

I don't know why you filled in 4 for n.

if compounding is annually, then n should have been 2 to represent 2 years.

using your numbers for S and P, your formula should have become:

12100 = 10000 * (1+x)^2

x is the annual interest rate.

divide both sides of this equation by 10000 to get:

1.21 = (1+x)^2

take the square root of both sides of this equation to get:



subtract 1 from both sidesof this equation to get:



solve for x to get:

x = .1

looks like you are looking for 10% a year growth compounded annually for 2 years.

to confirm, substitute in your original equation to get:

12100 = 10000 * (1+x)^2 becomes:

12100 = 10000 * (1.1)^2

simplify to get:

12100 = 12100 confirming the answer is good.

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if we assume that you can hit anywhere in the scoring area of the dartboard 100% of the time, then the total scoring area of the dartboard represents 100% probability.

any area less than that on the dartboard means the probability of hitting that area = that area divided by the total area of the dartboard.

total area of the dartboard is equal to the area of the circle from the center of the circle to the outer double ring line.

the area of a circle is equal to pi*r^2 where r is the radius.

singe the diameter of a circle is equal to 2 times the radius of a circle, then the area of a circle can be shown as pi*(d/2)^2 where d is the diameter.

the scoring area of the dartboard is equal to pi * (41/2)^2 = pi * 20.5^2 = 1320.254313 cm^2.

if you look at the picture in the attached link, that's the area from the center of the dartboard to the double ring line.

that's the total scoring area of the dartboard and we are assuming you will hit in the scoring area of the dartboard 100% of the time.

-------------------------------------------------------------------

if you want to hit in the double scoring area, that's the area between the outer double ring line and the inner double ring line.

the diameter of the outer double ring line is 41 cm.
the diameter of the innter duble ring line is 36 cm.

the double ring area is the area from the center of the circle to the outer double ring line minus the area from the center of the circle to the inner double ring line.

the area between those lines is equal to pi*(41/2)^2 - pi*(36/2)^2 = pi*20.5^2 - pi*18^2 = 1320.254313 - 1017.87602 = 302.3782929 cm^2.

the probability of hitting in the double scoring area is therefore equal to:

302.3782929 cm^2. / 1320.254313 cm^2 = .229030339

-----------------------------------------------------------------------

the probability of hitting the single bullseye depends on what you call the single bullseye.

you have an outer bullseye line and an inner bullseye line.

the very center is the double bullseye which means the single bullseye area must be between the outer bullseye line and the inner bullseye line.

this equals the area from the center of the circle to the outer bullseye line minus the area from the center of the circle to the inner bullseye line.

the area of the circle from the center to the outer bullseye line is equal to:

pi*(4.4/2)^2 = pi*2.2^2 = 15.20530844 cm^2.

the area of the circle from the center to the inner bullseye line is equal to:

pi*(1.4/2)^2 = pi*.7^2 = 1.5393804 cm^2.

the area between the outer bullseye line and the inner bullseye line is equal to 15.20530844 cm^2 - 1.5393804 cm^2 = 13.66592804 cm^2

the probability of hitting in the single bullseye area is therefore equal to:

13.66592804 cm^2 / 1320.254313 = .010350982

-----------------------------------------------------------------

the area of the triple 20 is equal to 1/20th of the triple area.

the triple area is the area between the outer triple line and the inner triple line.

the area from the center of the circle to the outer triple line is equal to:

pi * (25.6/2)^2 = pi * 12.8^2 = 514.7185404

the area from the center of the circle to the inner triple line is equal to:

pi * (21.2/2)^2 = pi * 10.6^2 = 352.9893506

the triple area is the area between these 2 lines which equals 514.7185404 minus 352.9893506 = 161.7291898 cm^2

the triple 20 area would be 1/20th of this which would equal 8.086459493

the probability of hitting into the triple 20 area is therefore equal to:

8.086459493 / 1320.254313 = .006124926

the whole thing depends on the folloowing:

*** you will hit the scoring area of the dartboard 100% of the time.

*** the probability of hitting one area of the scoring area of the dartboard is the same as the probability of hitting any other area of the scoring area of the dartboard.

*** the probability of hitting any area within the scoring area of the dartboard is based on the size of that area divided by the size of the scoring area of the dartboard.

any change in these assumptions changes the outcome.

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let s = number of women members of parliament in sweden.
let n = number of women members of parliament in norway.

s + n = 206

2s + 3n = 477

solve for s in the first equation to get s = 206 - n

substitute in the second equation to get 2*(206-n) + 3n = 477

simplify to get 412 - 2n + 3n = 477

combine like terms to get 412 + n = 477

subtract 412 from both sides of equation to get n = 477 - 412 = 65

if n = 65, then s = 206 - 65 = 141

equation of s + n = 206 becomes 141 + 65 = 206 which is true.

equation of 2s + 3n = 477 becomes 2*141 + 3*65 = 477 becomes 282 = 195 = 477 which is also true.

numbers look good.

s = 141 = number of women members of parliament in sweden.
n = 65 = number of women members of parliament in norway.

total members of parliament in sweden is 349.
total women members of parliament in sweden is 141.
141/349 = .404011461 = 40% to the nearest whole percent.

total members of parliament in norway is 165.
total women members of parliament in norway is 65.
65/165 = .393939394 = 39% to the nearest whole percent.

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13/11 = 1+2
26/11 = 2+4
39/11 = 3+6
52/11 = 4+8
65/11 = 5+10
78/11 = 7+1 *****************************8

looks like the number is 6

6*13 = 78

78/11 = 7 + a remainder of 1.

7*11 = 77 + 1 = 78

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7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649
7^7 = 823543
7^8 = 5764801
7^9 = 40353607
7^10 = 282475249
7^11 = 1977326743

the last digit is repeating in a pattern of:

7,9,3,1,
7,9,3,1,
7,9,3,1,

to find out what the last digit is, you need to make a formula.

the formula would be taking the larger exponent and breaking it down into the smaller exponent

example:

suppose your number is 7^6.

6/4 = 2 and 7^2 = 49 so the last digit is 9.

look at 7^6 above and you can see that the last digit is 9.

suppose your number is 7^11.

11/4 = 2 with a remainder of 3.

7^3 = 343 so the last digit is 3.

look up 7^11 above and you can see that the last digit is 3.

you can use this formula for larger numbers even though you can't check them because the calculator doesn't carry enough digits.

I used excel and the highest I could go was 7^17 as shown below:

base	exponent	number	           last digit
7	1	        7	           7
7	2	       49	           9
7	3	       343	           3
7	4	       2401	           1
7	5	       16807	           7
7	6	       117649	           9
7	7	       823543	           3
7	8	       5764801	           1
7	9	       40353607	           7
7	10	       282475249	   9
7	11	       1977326743	   3
7	12	       13841287201	   1
7	13	       96889010407	   7
7	14	       678223072849	   9
7	15	       4747561509943	   3
7	16	       33232930569601	   1
7	17	       232630513987207	   7
7       18                                 9
7       19                                 3
7       20                                 1 
7       21			           etc.....        

for a final check, try 7^16.

16/4 = 4 with a remainder of 0.

7^0 = 1 so the remainder should be 1.

it is, as shown in the table from excel above.

note that 16 / 4 = 4 with a remainder of 0. it also equals 3 with a remainder of 4 which confirms that the last digit is 1.

In your problem, the number is 7^2005

divide 2005 by 4 to get 501 with a remainder of 1.

7^1 = 7, so the last digit of 7^2005 = 1

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let x = selling price of house.

she must pay .07*x to the real estate agent.

she wants to make $140,000 after she had paid the real estate agent.

equation is:

x - .07*x = 140000

this becomes x * (1-.07) = 140000

this becomes x * .93 = 140000

divide both sides of this equation by .93 to get:

x = 140000 / .93 = 150537.6344

if she sells the house for that price, then the broker gets:

.07*150537.6344 = 10537.63441

selling price is 150537.6344
broker's commission is 10537.63441

150537.6344 - 10537.63441 = 140000.

your answer is:

she must sell the house for $150,537 in order to make $140,000, assuming the real estate agent gets 7% of the selling price.

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your equations are:

7r - 6s = -22
6r + 7s = 54

if you let:

y = s and x = r, your equations become:

7x - 6y = -22
6x + 7y = 54

now you can graph them on an xy plane where x is the horizontal axis and y is the vertical axis.

in order to graph them, you have to solve for y in both equations.

first equation becomes y = (7/6)x + (22/6)

second equation becomes y = -(6/7)x + (54/7)

graph of these equations looks like this:



to graph them manually, you would need coordinates, but you would only need 2 for each equation since they are straight lines.

if you let x = 0, one of the coordinates pops up easily.

then just pick any other value for x, like 5 or 10, and you get your second coordinate.

then just draw a straight line passing through those 2 points.

these equations are perpendicular to each other.

after you solved for y, they are in slope intercept form.

slope intercept form is y = mx + b where m is the slope and b is the y-intercept.

y-intercept is the value of y when x = 0.

slope for the first equation is (7/6).

slope for he second equation is -(6/7).

negative reciprocal of (7/6) is -(6/7) telling you that these lines are [perpendicular to each other.

the form that they started off in is the standard form of the equation of a straight line which is ax + by = c

if you graph them manually, you could keep them as r and s.

just make r the horizontal axis and s the vertical axis.

I had to convert them to x and y because the graphing function required them to be x and y.

keeping them as r and s, then the slope intercept form of the equations is:

s = (7/6)r + (22/6)
s = -(6/7)r + (54/7)

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x = 17.

17 - 14 = 3
17 - 13 = 4
17 - 12 = 5

345 triangle is a right triangle.

solve by use of pythatogrean formula.

a^2 + b^2 = c^2

x is the side of the equilateral triangle.

(x-12) = largest side which would be equal to c.

(x-13) = midsize side which would be equal to b.

(x-14) = smallest side which would be equal to a.

a^2 + b^2 = c^2 becomes:

(x-14)^2 + (x-13)^2 = (x-12)^2

this becomes

x^2 - 28x + 196 + x^2 - 26x + 169 = x^2 - 24x + 144

combine like terms to get:

2x^2 - 54x + 365 = x^2 - 24x + 144

subtract x^2 from both sides of equation to get:

x^2 - 54x + 365 = -24x + 144

add 24x to both sides of equation to get:

x^2 - 30x + 365 = 144

subtract 144 from both sides of equation to get:

x^2 - 30x + 221 = 0

factor this quadratic equation to get:

(x-17) * (x-13) = 0

this makes x = 17 or x = 13.

x can't be 13 because then one of the sides of the triangle will be 0.

x has to be 17.

when x is 17, you get:

x-12 = 17-12 = 5
x-13 = 17-13 = 4
x-14 = 17-14 = 3

that's your 345 triangle which is a right triangle.

side of the equilateral triangle is 17.

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bill = $239.25

let x = number of students.

let y = amount each student paid.

if all students paid the bill, then:

(first equation)

if all but 4 paid the bill, then:

(second equation)

substitute for y in second equation by the value for y in first equation to get:



you had one equation in 2 unknowns. the substitution allows you to have 1 equation in 1 unknown.

multiply both sides of this equation by (x-4) to get:



multiply the factors to get:



subtract 239.25 from both sides of the equation to get:



remove parentheses to get:



multiply both sides of equation by x to get:



rearrange terms to get:



use quadratic formula to solve for x to get:

x =

this becomes:

x = which becomes:

x = which becomes:

x = which becomes:



or:



since x has to be positive, then the solution narrows down to:

.

when 33 students pay the bill, the bill is $239.25 / 33 = $7.25 apiece.

when 29 students pay the bill (33-4 = 29), the bill is $239.25 / 29 = $8.25.

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1 quarter = 25 cents
1 dime = 10 cents
1 nickel = 5 cents
1 penny = 1 cent

it would make sense to make a table of this working our way down from the highest denomination to the lowest denomination.

the total value of each row is equal to 50 cents.

           quarters       dimes      nickels      pennies
              2             0          0            0 
              1             2          1            0           
              1             2          0            5           
              1             1          3            0 
              1             1          2            5
              1             1          1            10   *** example 1
              1             1          0            15
              1             0          5            0
              1             0          4            5
              1             0          3            10
              1             0          2            15
              1             0          1            20
              1             0          0            25
              0             5          0            0
              0             4          2            0
              0             4          1            5
              0             4          0            10
              0             3          4            0
              0             3          3            5   *** example 2
              0             3          2            10
              0             3          1            15
              0             3          0            20
              0             2          6            0
              0             2          5            5
              0             2          4            10
              0             2          3            15
              0             2          2            20
              0             2          1            25
              0             2          0            30
              0             1          8            0
              0             1          7            5
              0             1          6            10
              0             1          5            15
              0             1          4            20
              0             1          3            25
              0             1          2            30
              0             1          1            35
              0             1          0            40
              0             0          10           0
              0             0          9            5
              0             0          8            10
              0             0          7            15
              0             0          6            20
              0             0          5            25   *** example 3
              0             0          4            30
              0             0          3            35
              0             0          2            40
              0             0          1            45
              0             0          0            50

this counts out to be 49 possible combinations.

I couldn't figure out a formula to use so I just tabulated all the possible combinations I could think of.

The total is 49.

I believe I have them all covered.

Any combination you can think of should be in the table.

Example 1:

1 quarter 1 dime 1 nickel 10 pennies.

start with the quarter column and scan down until you see 1.
shift to the dime column and scan further down until you find 1.
shift to the nickel column and scan further down until you find 1.
shift to the penny column and you should see 10.

Example 2:

3 dimes 3 nickels 5 pennies.

start with the quarter column and scan down until you see 0.
shift to the dime column and scan further down until you find 3.
shift to the nickel column and scan further down until you find 3.
shift to the penny column ahd you should see 5.

Example 3:

5 nickels 25 pennies.

start with the quarter column and scan down until you see 0.
shift to the dime column and scan further down until you find 0.
shift to the nickel column and scan further down until you find 5.
shift to the penny column and you should see 25.

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should be equal to (4!)/(2!*2!) = (4*3*2*1)/(2*1*2*1) = 6

let's see what they would be:

1199
9911
1919
9191
1991
9119

this winds up being a permutation with some of the elements being the same.

formula for a permutation is n!

if x of the n are the same, then the formula becomes n!/x!

if, in addition to x being the same, y are also the same, then the formula becomes n!/(x!*y!)

in your problem, n was equal to 4.
x was equal to 2 because you had 2 ones.
y was also equal to 2 because you had 2 nines.

n!/(x!*y^1) became 4!/(2!*2!)

to see what the difference is, look at 3 letters (a,b,c)

how many ways can they be formed?

3! = 3*2*1 = 6 ways because each of the letters is different.

they are:

abc
acb
bac
bca
cab
cba

take the same 3 letters and make 2 of them the same.

let's assume a and b become the same letter d.

you would then have ddc

how many ways can they be formed.

we have n = 3 and x = 2 and our formula is n!/x! = 3!/2! = 3

the number of ways they can be formed is 3 as follows:

ddc
dcd
cdd

please keep in mind that we are talking about permutations here, and not combinations.

with permutations, order is important. the same elements in a different order are a different set.

with combinations, order is not important. each set has to have different elements in it.

take the set of abc.

with combinations, this forms one set. the same 3 elements can only be used once regardless of what order they are in the set.

with permutations, this forms 6 sets. the same 3 elements can be used 6 times in a different order each time.

the number of combinations possible would be 1.
the numer of permutations possible would be 6.

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I = 700*W/H2
W = weight in pounds
H = height in inches
< 25 = low risk

I presume when you says H2, you mean .

Jerome is 67 inches tall.

the formula to work with is:



substitute 67 for h to get:



In order for his index to be between 23 and 33, this formula needs to be modified as follows:

and

for greater than or equal to 23, the formula is:



multiply both sides of this equation by (67)^2 = 4489 to get:



divide both sides of this equation by 700 to get



Solve for W to get:

W >= 147.4957143

Similarly, the second equation of:

becomes:



Solve for W to get:

W <= 211.6242857

To the nearest tenth of a pound, Jerome needs to keep his weight between 147.5 and 211.6 pounds in order to keep his index between 23 and 33.

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a diagram of your switch could look like this:

                      A1                             A2
                      xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

battery xxxxxxxxxxxxxxx  C1                      C2  xxxx light xxxxx ground

                      xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
                      B1                             B2

you have battery coming in from the left going through the switches and then through the light and then to ground.

the following connections are possible through switch 1 and switch 2.

                       switch 1              switch 2           light

                       C1 to A1              C2 to A2             ON

                       C1 to A1              C2 to B2             OFF
                       
                       C1 to B1              C2 to B2             ON

                       C1 to B1              C2 to A2             OFF

if we let:

C1 to A1 = UP
C1 to B1 = DOWN
C2 to A2 = UP
C2 to B2 = DOWN

then the table becomes:

                       switch 1              switch 2           light

                       UP                    UP                   ON

                       UP                    DOWN                 OFF
                       
                       DOWN                  DOWN                 ON

                       DOWN                  UP                   OFF

this becomes the basis for your truth table.

in order to construct the truth tables, we will have 4 variables.

they will be U1, U2, D1, D2

U1 means switch 1 is up.
U2 means switch 2 is up.
U3 means switch 3 is down.
U4 means switch 4 is down.

T means True means the switch is as indicated.
F means False means the switch is not as indicated.

the 5th column will be L representing the light.

T means True means the light is on.
F means False means the light is off.

Here's what your truth table would look like if constructed this way.

                        U1  D1  U2  D2  L
                        T   F   T   F   T
                        T   F   F   T   F
                        F   T   F   T   T
                        F   T   T   F   F

if the implication is that when a switch is not in the up position, then it has to be in the down position, we can simplify the truth table to be as follows:

                        U1  U2  L
                        T   T   T
                        T   F   F
                        F   F   T
                        F   T   F

U1 being true means that switch 1 is in the up position.
U1 being false means that switch 1 is in the down position.
Similarly for U2

the logic statements for this truth table would look like:

U1 & U2 => L

this means that the state of U1 and the state of U2 both define the state of L.

Hopefully this is what you are looking for, or at least something similar to it.

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you want 2 of one kind, 2 of another kind, and any card for the 5th.

your hand would basically be 2 pairs plus a fifth card that is not the same as either of the pairs.

there are 4 suits in a deck and each suit has 13 cards making a total of 4 * 13 = 52.

That means you have 4 aces, 4 kings, 4 queens, 4 jacks, 4 tens, etc. down to 4 deuces.

since there are 4 suits of the same card, and you need to get 2 of them, then your possible combinations are C(4,2) = 4!/2!2! = 6 possible ways for each pair.

assume it's a pair of aces.

you can get them as follows:

ace of clubs and ace of spades
ace of clubs and ace of diamonds
ace of clubs and ace of hearts
= 3 +
ace of spades and ace of diamonds
ace of spades and ace of hearts
= 5 +
ace of diamonds and ace of hearts
= 6

so for each pair, you can do it in 6 ways.

since there are 13 sets of the same card, the total number of combinations of two different pairs you can make from this deck of card is C(13,2) = 13!/(11!*2!) = 13*12/2 = 13*6 = 78.

some examples would be:

kings and queens
kings and jacks
kings and tens
etc.

once you have your 2 pairs, the total number of ways you can draw the 5th card is (52-8) * 1 = 44

total possible combinations of different pairings would therefore be:

6*6*78*44 = 123552

that's 78 combinations of different pairs.
each pair can be derived in 6 ways.
each combination of pairs can therefore be derived in 6*6 = 36 ways.
you get 6*6*78 possible combinations of pairs.
for each of those possible combinations of pairs, there are 44 possible ways that the fifth card can happen.

-------------------------------------------------------------------------

to make this easier to understand in an example, let's assume your deck of cards had only kings, queens, and jacks in it.

let's also assume there are only 3 suits (clubs, spades, hearts).

that would be a total of 9 cards in the deck with 3 suits.

you want 2 of one kind, 2 of another kind, and any card for the 5th.

your hand would basically be 2 pairs plus a fifth card that is not the same as either of the pairs.

there are 3 suits in this deck and each suit has 3 cards making a total of 3 * 3 = 9 cards in the deck.

That means you have 3 kings 3 queens and 3 jacks.

since there are 3 of each kind, and you need to get 2 of them to make a pair, then your possible combinations for each pair are C(3,2) = 3!/1!2! = 3.

assume it's a pair of kings

you can get them as follows:

king of clubs and king of spades
king of clubs and king of hearts
king of spades and king of hearts

so for each pair, you can do it in 3 ways.

the total number of combinations of two different pairing you can make from this deck of card is C(3,2) = 3!/(1!*2!) = 3*2*1/1*2*1 = 6/2 = 3.

those possible combinations of pairings would be:

kings and queens
kings and jacks
queens and jacks

once you have your 2 pairs, the total number of ways you can draw the 5th card is (9-6) * 1 = 3

for kings and queens, the fifth card can be:

jack of clubs
jack of spades
jack of hearts

total possible number of combinations of pairing would therefore be:

3*3*3*3 = 81

you have 3 possible combinations of two pairs each.
each pair has 3 possible ways it can be achieved.
each combination of pairs can therefore happen in 3*3 = 9 ways.
total possible combinations of pairing is therefore 9*3 = 27 possible ways.
for each of these possible combinations of pairings, the fifth card can be picked in 3 ways.
that's a total of 27*3 = 81 possible ways.

here are the posible combinations for just 1 of the 3 possible combinations of pairings.

the 3 possible combination of pairings are:

kings and queens
kings and jacks
queens and jacks

this pairing will be for kings and queens.

let k = king
let q = queen
let j = jack

let 1 = clubs
let 2 = spades
let 3 = hearts

you get:

k1 = king of clubs, k2 = king of spades, k3 = king of hearts
q1 = queen of clubs, q2 = queen of spades, q3 = queen of hearts
j1 = jack of clubs, j2 = jack of spades, j3 = jack of hearts

for kings paired with queens, possible combinations of pairings are:

k1 k2 q1 q2 j1   k1 k2 q1 q3 j1   k1 k2 q2 q3 j1
k1 k2 q1 q2 j2   k1 k2 q1 q3 j2   k1 k2 q2 q3 j2
k1 k2 q1 q2 j3   k1 k2 q1 q3 j3   k1 k2 q2 q3 j3

k1 k3 q1 q2 j1   k1 k3 q1 q3 j1   k1 k3 q2 q3 j1
k1 k3 q1 q2 j2   k1 k3 q1 q3 j2   k1 k3 q2 q3 j2
k1 k3 q1 q2 j3   k1 k3 q1 q3 j3   k1 k3 q2 q3 j3

k2 k3 q1 q2 j1   k2 k3 q1 q3 j1   k2 k3 q2 q3 j1
k2 k3 q1 q2 j2   k2 k3 q1 q3 j2   k2 k3 q2 q3 j2
k2 k3 q1 q2 j3   k2 k3 q1 q3 j3   k2 k3 q2 q3 j3

for k1k2 you have 3 possible combinations with queens.
those would be q1q2, q1q3, q2q3.

for kik3 you have 3 more possible combinations with queens.
those would be q1q2, q1q3, q2q3.

for k2k3 you have 3 more possible combinations with queens.
those would be q1q2, q1q3, q2q3.

that's a total of 3*3 = 9 possible combinations with kings and queens.

for each one of those combinations you have 3 possible combinations of the fifth card.
it can be j1, j2, or j3.

that makes a total of 9*3 = 27 possible combinations for kings and queens alone.

you can see that for kings and queens there are 3*3*3 = 27 possible hands that can be dealt.

kings can also be paired with jacks which would yield another 27.

queens can also be paired with jacks which would yield another 27.

total possible pairings would be 81.

same principle applies to the larger deck of 4 suits with 13 cards in each suit.

with the larger deck (52 cards, 13 suits, 4 of the same card in each suit), you get the following:

you have a total of 6 possible ways to get each pairing.

each combination of pairing can happen in 6*6 = 36 ways.

there are a total of 78 possible combinations of pairings.

that makes a total of 78*36 = 2808 possible combinations of pairings.

for each of those possible combinations of pairings, you have a choice of 44 different cards to be the fifth card.

that makes a total of 2808 * 44 = 123552 possible ways you can get a hand with 2 pairs only.

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problem is:



since , then the reverse is also true, i.e.:



applying this rule of logarithms makes your equation become:



since if and only if , applying this rule makes your equation become:



multiply both sides of this equation by and you get:



multiply the factors out and you get:



subtract from both sides of this equation and combine like terms to get:



factor this equation to get:



solve for x to get:

or

confirm by substituting in original equation of:



to confirm using your calculator, you need to convert the base of the logarithm from 2 to 10 or e.

I used 10 because it is the LOG function of my calculator.

the conversion formula says:



if we let a = base of 2 and we let c = base of 10, this formula becomes:



your formula of becomes:



substituting x = (1/5) into this equation, we get:



this becomes:

which becomes:

confirming that x = (1/5) is good.

when x = -1, confirmation was not successful because:

original equation is:



substitute -1 for x to get:



this becomes:



since you can't take log of 0 to any base, x = -1 is not a valid answer.

graph of your original equation looks like this:



the answer to your problem is:

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12 foot ladder leaning against a wall.
painter is 2/3 of the way up when he drops his brush.
brush lands directly beneath him 1 foot away from the wall.
how far from the base of the ladder is the wall.

2/3 of the way up is 8 feet on the ladder.

diagram of the ladder against the wall is shown below.

                  A

                  x
                  x x
                  x   x
                  x     x       4 feet
                  x       x
                  x         x
                  x           x
               F  x x x x x x x x  D
                  x             x x
                  x             x   x
                  x             x     x
                  x             x       x
                  x             x         x
                  x             x           x
                  x             x             x     8 feet
                  x             x               x
                  x             x                 x
                  x             x                   x
                  x             x                     x
                  x x x x x x x x x x x x x x x x x x x x 

                  C             E                       B

A is the point where the ladder touches the wall.
B is the point where the ladder touches the ground (base of the ladder).
C is the point where the wall touches the bround (base of the wall).
D is where the painter is.
E is where the brush drops
F is where a line from D parallel to the ground touches the wall.

you have formed 2 similar triangles.

they are triangle ABC and triangle ADF

the distance from C to E on the ground is 1 foot.

the distance from F to D parallel to the ground is also 1 foot.

since the triangle are similar, then their corresponding sides are proportional.

this means that AD / FD = AB / CB

since AD is 4 feet and AB is 12 feet and FD is 1 foot, this formula becomes:

4 / 1 = 12 / CB

cross multiply to get:

4*CB = 1*12

divide both sides of this equation by 4 to get:

CB = 12/4 = 3

your answer is that the distance from the base of the ladder to the base of the wall is 3 feet.

--------------------------------------------------------------------

kite is 1000 feet long from the flyer of the kite to the kite along the length of the string.

angle kite makes to the ground is 42 degrees.

this forms a triangle as shown below:

                  A

                  x
                  x x
                  x   x
                  x     x
                  x       x
                  x         x
                  x           x
                  x             x  
                  x               x   1000 feet
                  x                 x
                  x                   x
                  x                     x
                  x                       x
                  x                         x
                  x                           x
                  x                             x
                  x                               x
                  x                    42 degrees   x
                  x                                   x
                  x x x x x x x x x x x x x x x x x x x x 

                  C                                     B

point A is the kite.
point B is the person flying the kite.
point C is a point directly beneath the kite perpendicular to the ground.

the ground is represented by line segment CB

length of line segment AB is 1000 feet.

the height of the kite is represwented by line segment AC.

tangent of 42 degrees = opposite / hypotenuse = AC / AB = AC / 1000

tan(42) = AC / 1000

multiply both sides of this equation by 1000 to get:

1000 * tan(42) = AC = 900.4040443.

height of the kite is 900.4040443 feet.

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I couldn't find any formulas that would let you find the minimum of that equation.

I graphed it instead and saw that the minimum point looks like it is at x = 1.



I took 2 points before and after 1 to test this out.

I chose x = .8 and x = 1.2

at x = .8, y = 5.0016

at x = 1.2, y = 5.0016

this kind of confirms that the axis of symmetry is at x = 1.

you can try other points like .9 and 1.1, and .7 and 1.3.

I suspect the minimum point is at x = 1.

I checked out a website that talked about minimum and maximum points of a polynomial, but I couldn't find anything in there that showed how it could be done other than graphically.

click here for that.

http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_poly_maxmin.xml




at x = 1.2, y =

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every 10 numbers, you have a last digit of 6.

300 - 100 = 200 numbers in between.

divide that by 10 and you get 20.

you will have 20 numbers that have a last digit of 6.

those number will be:
106
116
126
136
146
156
166
176
186
196
206
216
226
236
246
256
266
276
286
296

every 100 numbers, you will have 10 numbers that have a middle digit of 6.

those will be:

160 to 169
260 to 269

that makes a total of 20.

only half of them will be even.

those numbers will be:

160
162
164
166
168
260
262
264
266
268

this makes a total of 10 even number with a middle digit of 6.

within 2 of those numbers, however, you will have a number that has a middle digit of 6 and a last digit of 6.

those number will be:

166
266

we have to subtract these so they won't be counted twice.

the total number of numbers that have at least 1 digit of 6 would be 20 + 10 - 2 = 28

the total of even numbers between 100 and 300 = (300-100)/2 + 1 = 101

the total of even numbers between 100 and 300 that do not contain a digit of 6 either in the last digit or the middle digit would then be equal to:

101 - 28 = 73

that would be selection B.

here are the total even numbers that contain a digit of 6.

106
116
126
136
146
156
160
162
164
166
168
176
186
196
206
216
226
236
246
256
260
262
264
266
268
276
286
296

that's a total of 28

the total number of even digits is (300-100)/2 + 1 = 200/2 + 1 = 101

that 1 has to be added because the difference doesn't take it into account, and it is even.

example:

(30-20)/2 + 1 = 10/2 + 1 = 6

even numbers are:

20
22
24
26
28
30

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there are 12 boys.

3/4 * 12 are present which means that 1/4 * 12 are absent.

3 boys are absent.

there are 15 girls.

2/3 * 15 are present which means that 1/3 * 15 are absent.

5 girls are absent.

total student absent is 3 + 5 = 8.

that would be selection E.

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it depreciates at the rate of 14.3% per year.

this means that the value drops by 14.3% per year.

14.3% / 100% = .143

your equation increased the value by 14.3% per year.

you added .1453*6500 to 6500 to get:

.

when you factor out the 6500, you get:

= .

that does it for one year.

for n years, you then raised to the exponent of n to get:

.


your equation should have the decreased the value by 14.3% per year.

you needed to subtract .143*6500 from 6500 to get:

.

when you factor out the 6500, you get:

=

that does it for one year.

for n years, you then raised to the exponent of n to get:



where you used x, I used n.

if you prefer to use x, just replace n with x.

graph of looks like this.

the y values of the graph are in hundreds.

6000 = 60
7000 = 70
6500 = 65, etc.

the only x values you are interested are when x is positive.

those values are to the right of the y-axis which is the vertical line when x = 0.



you can see that as x increases, the value of y increases.

graph of looks like this.



you can see that as x increases, the value of y decreases.

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total red marbles are 4 * 8 = 32
total blue marbles are 4 * 11 = 44

a.

ratio of red marbles to blue marblesis 32/44 which is the same as 8/11.

note:

8/11 and 8:11 mean the same thing.
/ is the division sign.
: is the ratio sign.

a fraction is the ratio of the numerator to the denominator.

b.

ratio of blue marbles to red marbles is 44/32 which is the same as 11/8.

c.

ratio of red marbles to total marbles is 32/(44+32) = 32/76.

this is the same as 8/19.

d.

ratio of bags to marbles is 4/76 which is the same as 1/19.

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depends on the depth of the pool.

If the pool is circular, the formula for volume is

you get

v = 1000 so you get

you can solve for r or you can solve for h.

if you solve for r, you get =

take square root of both sides of equation and you get:

r = +/-

if you solve for h, you get

assuming the depth of the pool is 6 feet all around, the radius would be:

r = +/- feet.

graph of equation for the radius is shown below:

y value represents the radius of the pool.
x value represents the depth of the pool.

you can see that when the depth of the pool is 1 foot, the radius of the pool is:

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254123

bert gets 25 pieces of candy.
then he gets x - 17

karla gets 17 pieces of candy.
then she gets x - 25

bert has a total of 25 + x - 17 pieces of candy.

karla has a total of 17 + x - 25 pieces of candy.

to see how many more pieces of candy bert has, subtract the number of pieces of candy that karla has from the number of pieces of candy that bert has.

25 + x - 17 - (17 + x - 25) = 25 + x - 17 - 17 - x + 25 = 50 - 34 = 16.

bert has 16 more pieces of candy than karla has.

x can be any value.

as an example, assume x is 50 and 100

when x = 50, bert gets 25 and gives karla 25
karla gets 17 and gives bert 33
bert has a total of 25 + 33 = 58
karla has a total of 17 + 25 = 42
58-42 = 16

when x = 100, bert gets 25 and gives karla 75
karla gets 17 and gives bert 83
bert has a total of 25 + 83 = 108
karla has a total of 17 + 75 = 92
108-92 = 16

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i came up with 11 - 2i

here's how.

multiply (4-3i) * (2+i)

4 * 2 = 8
4 * i = 4i
-3i * 2 = -6i
-3i * i = -3i^2

add them up together and you get:


8 + 4i - 6i - 3i^2

combine like terms to get:

8 - 2i - 3i^2

since i^2 = -1, this becomes:

8 - 2i - 3*(-1) which becomes:

8 - 2i + 3

combine like terms to get:

11 - 2i

that would be selection 4.

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I suspect you want to simplify the expression.

without parentheses, you process left to right doing exponents and roots first, then multiplication and division, then addition and subtraction.

expression is:

5x/3x * x + 12/12

(5x/3x) = (5/3)

(5/3)*x = 5x/3

expression becomes:

5x/3 + 12/12

since 12/12 = 1, then expression becomes:

5x/3 + 1

multiply 1 by 3/3 to get 3/3.

expression becomes:

5x/3 + 3/3

combine numerators with common denominator to get:

(5x+3)/3

I don't believe it can be simplified any further.

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your equation to work with is:



The exponent in the equation is .





substituting in that equation, you get:



the exponent in the equation is

you want to isolate the exponential term on the right hand side of the equation so you need to divide both sides of the equation by 240 * 2 to get:



the exponent in the equation is

take the log of both sides of this equation to get:



since log(b^c) = c*log(b), your equation becomes:



multiply both sides of this equation by 13.8 to get:



divide both sides of this equation by -log(e) to get:



solve for t to get:

seconds.

substitute in original equation to see if this value for t is good.

original equation is:



exponent in the equation is




t = 57.39258655

equation becomes:




exponent in the equation is

simplify to get:



simplify further to get:



since the equation is true, the value for t is good.

t = 57.39258655 seconds

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measure of arc AD = measure of arc AB + measure of arc BC + measure of arc CD

this assumes all arcs are part of the same circle.

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assume the following as an example:

f(x) = x^2

g(x) = 3x+2

you want to find (f(g(-2))

start with g(x) = 3x+2

you want to find g(-2).

replace x with (-2) to get:

g(-2) = 3*(-2) + 2 = -6 + 2 = -4

you have g(-2) = -4

now you want to find f(g(-2)).

start with f(x) = x^2

replace x with g(-2) to get:

f(g(-2)) = (g(-2))^2

since g(-2) = -4, then replace g(-2) with -4 to get:

f(-4) = (-4)^2 = 16

you have f(g(-2)) = 16

----------------------------------------------------------------------

Here's another example.

let f(x) = sqrt(x)

let g(x) = x^3 + 3x^2 - 5

you want to find f(g(3))

first thing you want to do is find g(3).

since g(x) = x^3 + 3x^2 - 5, you replace x with 3 to get:

g(3) = (3)^3 + 3*(3)^2 - 5 = 27 + 27 - 5 = 49.

you now want to find f(g(3))

start with f(x) = sqrt(x).

replace x with g(3) to get:

f(g(3)) = sqrt(g(3))

since g(3) = 49, then replace g(3) with 49 to get:

f(49) = sqrt(49) = 7.

you have f(g(3)) = 7

--------------------------------------------

to keep this in the abstract, here's what's happening.

let f(x) = 5*x

let g(x) = 2*x

you want f(g(x)).

look at f(x) = 5*x

replace x with g(x) to get:

f(g(x)) = 5*(g(x))

since g(x) = 2*x, then replace g(x) with 2*x to get:

f(2*x) = 5*(2*x) = 10*x

if you are still having problems, send me your equation and I'll solve it for you to show you how it's done.

if you have any questions, send them to me and I'll answer them as best I can.

you also might want to look at the following link.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut30b_operations.htm