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If you want to be paid $20,000 per year for the rest of your life without the balance of the fund going down, and assuming that the interest rate at which the account will increase each year is 4%, and you expect to start withdrawing the money immediately in year 20, then the amount of money that needs to be in the fund is equal to $520,000.

That's equivalent to $20,000 / .04 = $500,000 plus $20,000 to be withdrawn immediately to make a total of $520,000 required to be in the account in year 20.

If you have to figure in taxes, then the amount taken out each year will have to pay the going tax rate, and you will have to have more than $500,000 in the account to start with in order to cover the withdrawal plus taxes on the withdrawal.

We'll forget about taxes for sake of simplicity.

You will need to have $520,000 in the account 20 years from now.

If this is year 0, then year 20 is an additional 20 years from now.

If the investment rate is 4%, then you would need to invest $237,321.212 today in order to have $520,000 in the account in year 20.

The formula to use to calculate the present value of the future amount of $520,000 is shown below:

PRESENT VALUE OF A FUTURE AMOUNT

PV = Present Value
FA = future amount
i = Interest Rate per Time Period
n = Number of Time Periods

I used this formula to duplicate the result that I obtained through the use of a financial calculator.

You should be able to duplicate the result also.

i = .04
n = 20
FA = $520,000
PV is calculated to be

On a yearly cash flow analysis, this is what it would look like:

     	ACCOUNT	        ACCOUNT
YEAR    BALANCE         WITHDRAWAL
0	$237,321.21	$0.00
1	$246,814.06	$0.00
2	$256,686.62	$0.00
3	$266,954.09	$0.00
4	$277,632.25	$0.00
5	$288,737.54	$0.00
6	$300,287.04	$0.00
7	$312,298.52	$0.00
8	$324,790.47	$0.00
9	$337,782.08	$0.00
10	$351,293.37	$0.00
11	$365,345.10	$0.00
12	$379,958.91	$0.00
13	$395,157.26	$0.00
14	$410,963.55	$0.00
15	$427,402.10	$0.00
16	$444,498.18	$0.00
17	$462,278.11	$0.00
18	$480,769.23	$0.00
19	$500,000.00	$0.00
20	$520,000.00	$20,000.00
21	$520,000.00	$20,000.00
22	$520,000.00	$20,000.00
23	$520,000.00	$20,000.00
24	$520,000.00	$20,000.00
25	$520,000.00	$20,000.00

The money is invested in year 0 and grows at 4% per year until year 20 at which point 20,000 is withdrawn each year keeping the balance at a steady $520,000.

I went to year 25 to show you that the account balance remains the same, despite the $20,000 withdrawal each year.

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We'll assume you start from year 0.

The present value of 15 years worth of payments would be $6,810.864489.

This includes payments for years 3 and 5.

You need to subtract them.

The present value for a $1,000 payment in year 3 is equal to $711.7802478.

The present value for a $1,000 payment in year 5 is equal to $567.4268557.

Subtract both these payments from the present valueof $6,810.864489 and you have your answer.

The answer would be $5,531.657386

I used a financial calculator, but you'll get the same result by use of the present value of payments equation and the present value of a future value equation.

For the 15 years of payments, use the present value of payments equation.

For the present value of the missing payments, use the present value of a future value equation.

A cash flow analysis, using the same technique, would show the following result:

year	payment         present value of payment
0		
1	$1,000.00	$892.86
2	$1,000.00	$797.19
3	$1,000.00	$711.78
4	$1,000.00	$635.52
5	$1,000.00	$567.43
6	$1,000.00	$506.63
7	$1,000.00	$452.35
8	$1,000.00	$403.88
9	$1,000.00	$360.61
10	$1,000.00	$321.97
11	$1,000.00	$287.48
12	$1,000.00	$256.68
13	$1,000.00	$229.17
14	$1,000.00	$204.62
15	$1,000.00	$182.70
sum of payments and present value of payments from year 1 through year 15 are shown below.	
	$15,000.00	$6,810.86
missing payments and present value of apyments that need to be subtracted are shown below.		
3	$1,000.00	$711.78
5	$1,000.00	$567.43
total minus missing payments and present value of missing payments is shown below.		
	$13,000.00	$5,531.66

the formulas you would use are shown below:

PRESENT VALUE OF A PAYMENT



PV = present value
PMT = payment per time period
i = interest rate per time period
n = number of time periods

PRESENT VALUE OF A FUTURE AMOUNT

PV = Present Value
FA = Future Amount
i = Interest Rate per Time Period
n = Number of Time Periods

Using these formulas, I was able to duplicate the answer.

You should be able to also.

Write if you have questions.

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3 barrels
each contains equal number of r, b, g, y balls.
probability of getting all 3 of the same color is?

p of getting first ball is equal to 1
p of getting second ball the same color is .25
p of getting third ball the same color is .25

probability of getting 3 of the same color is 1 * .25 * .25 = .0625

p of all red = .25 * .25 * .25 = .015625
p of all blue = .25 * .25 * .25 = .015625
p of all green = .25 * .25 * .25 = .015625
p of all yellow = .25 * .25 * .25 = .015625

p of either red or blue or green or yellow = 4 * .015625 = .0625

the total different ways of choosing a ball from each barrel would be:

4 * 4 * 4 = 64

the possible ways to win would be 4 * 1 * 1 = 4

winning means either all red, or all green, or all blue, or all yellow.

the probability of getting a win would be 4/64 = .0625

the probability of winning on any one turn is .0625.

if the probability of winning on any one turn is .0625, then the probability of losing on any one turn is 1 - .0625 = .9375.

if the bet is 1 dollars, then every 16 times, you will win once and lose 15 times.

your total loss for the 16 times at bat is 15 dollars.

your total winning has to be 15 dollars for you to break even.

the pot, therefore, needs to be 15 dollars in order for you to break even.

mathematically that comes out to be .9375 / .0625 = 15.

in English, that means the pot should be equal to the probability of losing divided by the probability of winning in order to break even.

if the pot is more than that, then you have the advantage.

if the pot is less than that, then the person running the game has the advantage.

since, in this case, the pot is 20 dollars, you have the advantage and the person running the game has the disadvantage.

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let I = investment.
let L = .05 * I = 110

you get .05*I = 110

divide both sides of that equation by .05 to get:

I = 110/.05

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probability of a call resulting in a reservation = .15
probability of a call not resulting in a reservation = .85

15 calls are made.

what is the probability that at least 1 of them will be a reservation?

that equals 1 minus the probability that all of them do not result in a reservation.

since the probability of any one call not resulting in a reservation equals .85, then the probability that all 15 calls will not result in a reservation equals .85^15.

that number is .087354219.

the probability that at least 1 call will result in a reservation is 1 minus the probability that no calls will result in a reservation.

that probability = .912645781.

that's the probability that at least 1 call will result in a reservation.

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teacher has 2 solutions.
1 = 5% hydrochloric acid.
1 = 65% hydrochloric acid.
how much of each solution to get 42 mL of 45% hydrochloric acid?

let x = number of mL of first solution.
let y = number of mL of second solution.

amount of HCI in first solution is .05*x
amount of HCI in second solution is .65*y

teacher wants 42 mL of 45% HCI.

amount of HCI in the solution that the teacher wants is .45 * 42 mL = 18.9 mL.

first equation is .05*x + .65*y = 18.9 mL of HCI.

first equation tells teacher how much HCI in each solution is required to total the amount of HCI required in the final solution.

second equation is x + y = 42 mL of total solution.

second equation tells teach how much total of each solution is required to get the proper mix.

these 2 equations need to be solved simultaneously to get the right answer.

the 2 equations are:

.05*x + .65*y = 18.9
x + y = 42

solve for y in the second equation to get y = 42 - x.

substitute for y in the first equation to get:

.05*x + .65*(42-x) = 18.9

first equation now has 1 unknown and can be solved.

solve for x.

simplify by removing parentheses in second equation to get:

.05*x + .65*42 - .65*x = 18.9

simplify further to get:

.05*x + 27.3 - .65*X = 18.9

subtract 27.3 from both sides of the equation to get:

.05*x - .65*x = - 8.4

combine like terms to get:

-.6*x = -8.4

divide both sides of the equation to get:

x = -8.4/-.6 = 14.

since x + y = 42, then 14 + y = 42.

subtract 14 from both sides of this equation to get:

y = 42 - 14 = 28

teacher has x = 14 and y = 28 which means that:

teacher needs 14 mL of the 5% solution and 28 mL of the 65% solution.

if teacher does that, then teacher gets:

.05*14 + .65*28 = .7 + 18.2 = 18.9 mL of HCI.

Answer looks good.

Answer is:

teacher needs 14 mL of 5% solution and 28 mL of 65% solution to get 42mL of 45% solution.

14 + 28 = 42 mL total solution.

.05*14 + .65*28 = 18.9 mL of HCI.

18.9 / 42 = .45 = 45% of HCI in final solution.

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let x = hours spent tutoring.
let y = hours spend lawn mowing.

x + y < 12 is one equation.

x >= 4 is another equation.

y <= 6 is another equation.

y >= 0 is another equation (number of hours can't be negative).

All 4 are constraint equations.

The equation she is looking to maximize is the revenue equation.

That equation is R = 5*x + 6.5*yt

She needs to graph the constraint equations and then look at the intersection points to find the maximum revenue.

To graph the equations, make them an equality rather than an inequality.

you get:

x + y = 12               (E2 in the picture)
x = 4                    (E1 in the picture)
y = 6                    (E3 in the picture)
y = 0                    (E4 in the picture)

x + y = 12 is equivalent to y = 12 - x when you solve for y.

replace x + y = 12 with y = 12 - x and your equations are:

y = 12 - x               (E2 in the picture)
x = 4                    (E1 in the picture)
y = 6                    (E3 in the picture)
y = 0                    (E4 in the picture)

your intersection points for these 4 constraint equations are:

(x,y) = :
(4,0)
(4,6)
(6,6)
(12,0)

Solve the revenue equation at these intersection points.

revenue equation is:

R = 5*x + 6.5*y

at (4,0) this equation becomes 5*4 + 6.5*0 = 20
at (4,6) this equation becomes 5*4 + 6.5*6 = 59
at (6,6) this equation becomes 5*6 + 6.5*6 = 69
at (12,0) this equation becomes 5*12 + 6.5*0 = 60

maximum revenue occurs at (6,6).

6 hours of tutoring and 6 hours of mowing.

it meets the constraints since hours of tutoring is greater than or equal to 4 and hours of mowing is less than or equal to 6 and total hours is less than or equal to 12.

obviously, if hours of mowing was allowed to be more than 6, she would spend more hours mowing because mowing gives her 6.5 per hours while tutoring only gives her 5, but the constraints forced hours of mowing to be equal to or less than 6.

note that the maximum hours allowed for mowing is the maximum revenue solution.

this won't always be the case, but it is here because there are no other constraints that would not make it so.

the picture of the constraint equations is shown below.

the shaded area is the area in which all constraint are met.

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smaller wheel radius = 5.23 cm.
larger wheel radius = 8.16 cm.

circumference of smaller wheel = 2*pi*5.23 = 32.86105916 cm.
circumference of larger wheel = 2*pi*81.16 = 51.27079211 cm.

for each revolution of the smaller wheel, the larger wheel will go through .640931373 of a revolution.

when the smaller wheel rotates 60 degrees, it has rotated 1/6 of its circumference.

32.86105916 / 6 = 5.4767843193 cm.

the circumference of the larger wheel will have moved the same distance.

5.4767843193 / 51.27079211 = .106821895 which means that the larger wheel has moved through 1.06821895 of its circumference.

that would be equal to approximately 360 * .106821895 = 38.45588235 degrees.

since the number of degrees the wheel turns is equivalent to the proportion of the circumference that it moves, then you can shorten this equation to become the ratio of the circumference of the smaller wheel to the circumference of the larger wheel.

that ratio is equal to 32.86105916 / 51.27079211 = .640931373.

if the smaller wheel moves through a rotation of 60 degrees, then the larger wheel move through a rotation of .640931373 * 60 = 38.45588235 degrees.

60 degree rotation in the smaller wheel = 60 / 360 * 32.86105916 = 5.476843193.

38.45588235 degree rotation in the larger wheel = 38.45588235 / 360 * 51.27079211 = 5.476843193.

concepts:

circumference of circle = 2 * pi * r

length of arc of circle = degrees of rotation / 360 * 2 * pi * r

full rotation = 360 / 360 * 2 * pi * r = circumference of the circle.

wheels interlocking means a movement of the circumference of the smaller wheel is exactly equal to the movement of the circumference of the larger wheel.

the movement is the same in both wheels, but the proportion of movement in each wheel is different.

to see clearer, let the circumference in the smaller wheel be 6 and the circumference in the larger wheel = 12.

they are interlocking.

the smaller wheel rotates through 360 degrees.

circumference of smaller wheel moves 6 which is one complete rotation of the smaller wheel.

circumference of larger wheel also moves 6 only this is only 1/2 the complete rotation of the larger wheel.

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rate * time = distance.

divide both sides of this equation by rate to get:

time = distance / rate.

distance = 2.5 miles
rate = 3.7 miles per hour.
time = 2.5 / 3.7 = .675675676 hours

60 minutes per hour makes .675675676 hours = 40.54054054 minutes.

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you want to raise 4000.

let x = original number of members.

let y = amount required from each member.

you get:

y = 4000 / x

with an increase of 20 members, the amount each member needed to raise went down by 10.

you get:

y-10 = 4000 / (x + 20)

you have 2 equations that need to be solved simultaneously.

since y = 4000 / x, you can substitute for y in the second equation to get:

(4000 / x) - 10 = 4000 / (x + 20)

multiply both sides of this equation by (x + 20) * x to get:

4000 * (x + 20) - (10 * (x + 20) * x) = 4000 * x

Simplify by removing parentheses to get:

4000*x + 80,000 - 10*x^2 - 200*x = 4000 * x

subtract 4000 * x from both sides of this equation to get:

80,000 - 10*x^2 - 200*x = 0

re=order the terms to get:

-10*x^2 - 200*x + 80,000 = 0

multiply both sides of this equation by -1 to get:

10x^2 + 200*x - 80,000 = 0

you don't have to do that, but it's easier to factor the equation if the coefficient of the x^2 term is positive.

divide both sides of the equation by 10 to get:

x^2 + 20*x - 8000 = 0

factor the equation to get:

(x + 100) * (x - 80) = 0

your values of x can be either -100 or +80.

since x has to be positive, then your answer is x = 80.

substitute in your original equation to confirm this is a good solution.

y = 4000 / x becomes y = 4000 / 80 becomes y = 50

with the original number of members, each member is required to provide 50.

y - 10 would then equal 40.

your second equation is:

y-10 = 4000 / (x + 20)

y - 10 = 40
x + 20 = 100

you get:

40 = 4000 / 100 which becomes:

40 = 40 which is true.

when you add 20 members, each member now has to pay 40.

solution is confirmed as good.

there were 80 original members in the club.

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the graph will give you an approximate solution.

the algebra will give you an exact solution.

the reason for this is that the accuracy of the graph is not sufficient to home in on the exact solution.

an example:

x^2 + y^2 = 64
y = 4

solve this algebraically and you will see that the solution is:

x = +/- 6.92820323

solve this graphically and you will see that the solution lies somewhere around that value, but there's no way you can get it right on unless you zero in to a very minute detail which is not possible with most graphing software that might be available to you.

the exact solution is subject to the vagaries of the accuracy of the graphing software that you use.

here's the solution on the algebra.com graphing software.



you can see that the points of intersection are somewhere around x = +/- 7 but there's no way you can see that the actual solution is +/- 6.92820323.

The graphing solution is good to give you a general idea of where your solution lies, or if you even have a solution.

To get the exact solution, you need to go to the algebraic solution.

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you graph the equation of y = -2x - 1.

you shade the area of y < -2x - 1.

it will be the area of the graph that is underneath the line of the equation.

the graph of the equation of y = -2x - 1 will look like this.



you can see that, for every value of x, y = -2x-1 will be on the line and y < -2x-1 will be underneath the line.

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your equations are:

f(x) = 3x^2 - 1

g(x) = 2x + 5

you want to find f(g(-1))

first you have to find g(-1)

since g(x) = 2x + 5, then g(-1) = 2*(-1) + 5 = -2 + 5 = 3

f(g(-1)) becomes f(3) because g(-1) = 3

since f(x) = 3x^2 - 1, then f(3) = 3*(3^2) - 1 = 3*9 - 1 = 27 - 1 = 26

solve for the inner function first and then use that value to solve for the outer function.

you want to solve for g(f(-1))

since f(x) = 3x^2 - 1, then f(-1) = 3*(-1)^2 - 1 = 3 - 1 = 2

since g(x) = 2x + 5, then g(f(-1)) = g(2) = 2*2 + 4 = 4 + 5 = 9

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48 less than e would be written as:

e - 48

48 less a number e would be written as:

48 - e

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equation is:

51 = m - 40

add 40 to both sides of the equation to get:

91 = m

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70% of 8 = .7 * 8 = 5.6

5.6 people will like the hamburger.

this means that the expectation is that 5 or 6 people will like the hamburger, with a slight edge given to 6 people liking it rather than 5.

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one equation represents a circle and the other equation represents a line.

you can have 0, 1, or 2 solutions.

0 if the line and the circle never intersect
1 if the line is tangent to the circle.
2 if the line intersects the circle at any other angle.

the line can never intersect the circle at more than 2 points.

your equation is:

x^2 + y^2 = 64
x = 1

this is like solving equations simultaneously to get a common solution.

if x = 1, you can substitute in the first equation to get 1 + y^2 = 64 which give you y^2 = 63 which gives you y = +/- sqrt(63).

this implies 2 solutions to this equation.

The graph of the equation of the circle should confirm this.

To graph the equation of the circle, solve for y to get:

y = +/- sqrt(64-x^2)

that graph is shown below:



Draw a vertical line at x = 1 and you will see that the intersection of that vertical line and the circle is around y = +/- 7.937253933.

I drew 2 horizontal lines at y = +/- 7.937253933 so you can see that a little easier.

when x = 0, y = +/- sqrt(64-0) = +/- sqrt(64) = +/- 8.

That's the highest and lowest point of the circle.

when x = 1, y = +/- sqrt(64-1) = +/- sqrt(63) = +/- 7.937253933.

That's close to the highest point but hot exactly there (just a little below).

Solving the 2 equations simultaneously yielded the result.

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you need to solve for y.

then you can graph it.

2y-5x < 13

add 5x to both sides to get:

2y < 5x + 13

divide both sides by y to get:

y < (5x+13)/2

now you can graph the equation of:

y = (5x+13)/2.

any value of y below the the graph of that equation is good.

any value of y on or higher than the graph of that equation is not good.

you would shade the area of the graph below the line to indicate that is the good region.

your graph would look like this:



to prove your graph is good, take the value of x = 0 and substitute it in your equation to get:

y = (5x+13)/2 which becomes y = 13/2 which becomes y = 6.5 which is confirmed by the graph of the equation crossing the y-axis at x = 6.5.

to further prove your graph is good, take the value of y = 0 and substitute it in your equation to get:

0 = (5x+13)/2.

multiply both sides of the equation to get:

0 = 5x + 13

subtract 13 from both sides of the equation to get:

-13 = 5x

divide both sides of the equation by 5 to get:

-13/5 = x

-13/5 is equal to -2.6 which is the value of y when x = 0 and is confimred by the graph crossing the x-axis at approximately that point.

you don't graph the inequality directly.

you graph the equality and then apply logic so see where the values of y can lie.

in this case the values of y that lie below the equation are the values that are valid.

take any value of x.

find the value of y that lies on the graph of the equation of y = (5x+13)/2.

for that value of x, any value of y less than that will be below the line, and any value of y greater than that will be above the line.

the region that is valid for the equation y < (5x+13)/2 will be below the line.

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the diagonal forms the hypotenuse of a right triangle.

the length is one leg of the right triangle.

the width is the other leg.

if you let:

a = 1 leg of the right triangle
b = the other leg of the right triangle
c = the hypotenuse of the right triangle

then:

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

c^2 = 15^2 + 8^2 which becomes:

c^2 = 225 + 64 which becomes:

c^2 = 289.

Take the square root of 289 to get:

c = 17.

You got it right.

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I will solve it for you and then you can see where you went wrong.

problem is:

-7(5y – 2) > 6(6y-1) which can also be written as:

-7*(5*y – 2) > 6*(6*y-1).

The * indicates multiplication and can make things clearer if there is any ambiguity in how the formula is shown.

remove parentheses to get:

-35*y + 14 > 36*y - 6

add 6 to both sides of the equation to get:

-35*y + 20 > 36*y

add 35*y to both sides of the equation to get:

20 > 36*h + 35*y

combine like terms to get:

20 > 71*y

divide both sides of the equation by 71 to get:

20/71 > y

if 20/71 > y, then:

y < 20/71

that should be your answer.

you confirm by substituting in the original equation to see if the equation is true.

take 3 values for y.

pick values that are easy to calculate wherever possible.

take y = 0 which is smaller than 20/71

take y = 20/71 which is equal to 20/71

take y = 1 which is greater than 20/71

if y is smaller than 20/71, the original equation should be true.

if y is greater than or equal to 20/71, the original equation should be false.

the original equation is:

-7(5y – 2) > 6(6y-1)

substitute 0 for y to get:

14 > -6 which is true.

substitute 1 for y to get:

-21 > 30 which is false.

substitute 20/71 for y to get:

4.14084507 > 4.14084507 which is false.

confirmation of y < 20/71 is good since:

when y < 20/71, the original equation is true.

when y >= 20/71, the original equation is false.

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A common factor of both expressions would probably be w^3*x^6

Your first expression would be equivalent to (w^3*x^6)*(3*w*y) = 3*w^4*x^6*y

Your second expression would be equivalent to (w^3*x^6)*(16) = 16*w^3*x^6

To confirm this is true, then use some random numbers for w and x and y and see if the relationships are equivalent.

Let w = 3 and x = 4 and y = 5.

(w^3*x^6) = 110592

(w^3*x^6) * (3*w*y) becomes 110592 * 45 = 4976640

(w^3*x^6) * (16) becomes 110592 * 16 = 1769472

3*w^4*x^6*y = 4976640 which makes it equivalent to (w^3*x^6) * (3*w*y)

16*w^3*x^6 = 1769472 which makes it equivalent to (w^3*x^6) * (16)

The common multiple appears to be:

w^3*x^6

I couldn't find any other common multiples in these equations.

The applicable basic laws of multiplication and exponentiation appear to be:

a*(b*c) = (a*b)*c

a*b = b*a

a^b * a^c = a^(b+c)

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You can use brackets, or you can use parentheses.
parentheses are easier to write.
brackets are more formal.
i'll use parentheses.

formula is a(n) = S(n) - S(n-1) where () means sub.

In general, S(n) = a(1) + a(2) + ........ a(n-1) + a(n)

In general S(n-1) = a(1) + a(2) + ........a (n-1)

If you subtract one from the other, you are left with a(n) because all of the a's from a(0) to a(n-1) cancel out.

a(1) - a(1) = 0
a(2) - a(2) = 0
a(n-1) - a(n-1) = 0
a(n) - 0 = a(n)

nothing left but a(n).

there's a more formal proof that can be done, but it's more complicated to go through and not worth the effort unless you absolutely have to do it that way.

for your purposes, this shows that it's true and should be sufficient.

you can then cover with examples:

let a1 = 5 and let d = 7 (d = common difference in arithmetic sequence).

s(7) = 5 + 12 + 19 + 26 + 33 + 40 + 47 = 182

s(8) = 5 + 12 + 19 + 26 + 33 + 40 + 47 + 54 = 236

s(8) - s(7) = 236 - 182 = 54

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in 5 years you have 7800
in 9 years you have 8840
from the 5th year to the 9th year you have gained 1040.
since this is simple interest, then you have gained 1040 / 4 = 260 per year.
this can be seen as follows:
year 5 = 7800
year 6 = 8060
year 7 = 8320
year 8 = 8580
year 9 = 8840

since this is simple interest, then you gain the same interest per year and so you can work back from 7800 to get how much you started with.

since it took 5 years to get 7800, then subtract 5 * 260 from 7800 to get the principal amount in year 0.

since 5 * 260 = 1300, that would be equal to 7800 - 1300 = 6500.

your principal amount is equal to 6500.

your interest rate based on the principal amount would be equal to 260 / 6500 = .04 which is the same as 4% per year.

your yearly amounts come out to be:

year 0 = 6500
year 1 = 6760
year 2 = 7020
year 3 = 7280
year 4 = 7540
year 5 = 7800 ********
year 6 = 8060
year 7 = 8320
year 8 = 8580
year 9 = 8840 ********

you can graph the equation for simple interest as follows:

since this is simple interest, it would be a straight line graph.

let y = value
let x = number of years

your points are (x,y) = (5,7800) and (9,8840)

your slope is (8840 - 7800) / (9 - 5) = 1040 / 4 = 260

your straight line equation becomes:

y = 260*x + b

substitute one of the points in this equation to get the y-intercept.

8840 = 260*(9) + b

you get b = 6500.

your equation is therefore y = 260*x + 6500

graph of your equation looks like this:



horizontal lines at y = 7800 and y = 8840 show you that the intersection of these lines with the equation occur at x = 5 and x = 9 respectively.

the interest rate had to be calculated over the principal amount at year 0.

easiest way to get the answer was to get the annual increase year over year which had to be the same because you are dealing with simple interest.

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looks like your y-intercepts is at (0,-3).

looks like your x-intercepts are at (1,0) and (3,0).

looks like your axis of symmetry is at x = 2.

looks like your vertex is at (2,1).

coordinates are shown as (x,y) where x is the value along the x-axis and y is the value along the y-axis.

the point (2,1) is the intersection of a vertical line at x = 2 and a horizontal line at y = 1.

you find the x-intercepts by solving for the roots of the equation.

your equation is -x^2 + 4x - 3.

to find the roots you set the equation equal to 0.

you get -x^2 + 4x - 3 = 0

it's easier to factor if the coefficient of the x^2 term is positive so multiply both sides of the equation by -1 to get:

x^2 - 4x + 3 = 0

multiplying both sides of an equality by -1 keeps the equality so you are changing the equation, but not changing the equal relations between the 2 sides of the equation.

the equal relationship is what's important to solving for the roots, so changing the equation by multiplying both sides by -1 will give you the same answer.

down below, at the end, i'll show you how the factors worked out if you did not multiply both sides of the eqution by -1 before factoring.

your factors appear to be (x-3) * (x-1) = 0

if you multiply (x-3) * (x-1), you will get x^2 -x -3x + 3 which simplifies to x^2 - 4x + 3.

this makes the value of y equal to 0 then x = 3 or x = 1.

plug 3 into your original equation and you will see that it will equal to 0.

plus 1 into your original equation and you will see that it will also equal to 0.

the graph confirms that the values of x are 1 and 3 when the value of y is equal to 0.

since the coefficient of your x^2 term in the original equation is negative, the graph of the equation will point up and open down.

that means that the vertex of the equation will be a maximum point.

if the coefficient of your x^2 term was positive, the graph ofr the equation would have pointed down and open up.

that means that the vertex of the equation would have been a minimum point.

but it's not.

it's a maximum point because the coefficient of the x^2 term is negative.

the vertex of your equation is found by the equation:

x = -b/2a.

once your equation is in standard form, you can find the value of a, b, and c.

the standard form of a quadratic equation is ax^2 + bx + c = 0

your equation, in standard form, is -x^2 + 4x - 3 = 0.

that makes a = -1, b = 4, c = -3

a is the coefficient of the x^2 term.
b is the coefficient of the x term.
c is the coefficient of the constant term.

the formula for the vertex of a quadratic equation is x = -b/2a.

in your equation, that translates to x = - (-4) / 2 which becomes 4/2 which becomes 2.

the y value of the vertex is the value of y when x = 2.

that would be -x^2 + 4x - 3 = -(2^2) + 8 - 3 = -4 + 8 - 3 = 4 - 3 = 1.

that puts the vertex of your quadratic equation at (x,1) as confirmed by the graph.

the axis of symmetry of your graph is the x value of your vertex.

that makes the axis of symmetry of your graph at x = 2.

to prove that it is the axis of symmetry, you take a value of y and solve for x.

you can choose value of y = 0 to make it easy.

when y = 0, x = 1 and x = 3.

that makes the value of x equidistant from the axis of symmetry as it should be.

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

to find that, simply replace x with 0 in your equation to get y = -x^2 + 4x - 3 becomes y = 0 + 0 - 3 which becomes y = -3 as confirmed by the graph.

to plot the graph, you simply take values of x and solve for y and then plot the value of the (x,y) pairs on the graph.

a simple table below shows you what you will find.

 X	Y = -X^2 + 4X - 3

-2	-15
-1	-8
0	-3
1	0
2	1
3	0
4	-3
5	-8
6	-15

Since the axis of symmetry is at x = 2, you can see that:

when x = 0 and x = 4, the value of y is equal to -3.

that's because 0 is 2 units away from 2 and 4 is also 2 units away from 2.

since the graph is symmetric about the value of x = 2, the value of y at those points equi-distant from the axis of symmetry will be the same.

same goes for y = -15 when x = 6 and x = -2.

same goes for y = -8 when x = 5 and x = -1.

back to factoring.

if you did not multiply both sides of the equation by -1, you can still factor it, only it's a little harder to see the factors.

i'll do it here just to show you.

your original equation is -x^2 + 4x - 3.

since the coefficient of the x^2 term is negative, one of your x values has to be negative, so you will get factors that look like:

(-x + a) * (x + b) = -x^2 + 4x - 3

a and b are the constant terms that you want to find.

since your constant term has to be -3, then you will multiply by 3 * -1, or you will multiply by -3 * 1

let's try -3 * 1.

you will get (-x - 3) * (x + 1)

that results in -x^2 - x -3x - 3 which results in -x^2 -4x - 3 which is not what we want.

let's try (-x + 3) * (x - 1)

that results in -x^2 - x + 3x - 3 which results in -x^2 + 2x - 3 which is not what we want.

let's try (-x + 1) * (x - 3)

that results in -x^2 + 3x + x - 3 which results in -x^2 + 4x - 3 which is what we want so we can stop here.

our factors become:

(-x+1) * (x-3) = 0

from the factor of (x-3) = 0, we derive x = 3.

frm the factor of (-x+1) = 0, we derive -x = -1 which results in x = 3 when we multiply both sides of the equation by -1.

whether we multiplied both sides of the quadratic equation in standard form by -1 in the beginning or not, we still wind up with the same factors.

it's just easier to see what the factors are by doing that.

you are left to do that or not at your discretion.

i always found it easier to solve the quadratic equation if the coefficient of the x^2 term was positive.

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no.

The quotient (result of the division) can be positive in the following ways.

both numerator and denominator are positive.

both numerator and denominator are negative.

A minus times a minus is a plus.

A minus divided by a minus is also a plus.

to confirm, you can use your calculator.

take 25/5 and you'll get 5.

take -25 / -5 and you'll also get 5.

if you take -25 and divide it by 5, then you'll get -5.

if you take 25 and divide it by -5, then you'll get -5.

a minus divided by a plus is always a minus.

a plus divided by a minus is always a minus.

takes a while to get it, but once you have it down, it's a snap.

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(y^3)^(-5) is the same as 1 / (y^3)^5 which equals 1 / y^15

Looks like you got it.

If you let y = 2, then y^3 = 125 and 125 ^ -5 = .0000305118 by the calculator, and 1/(y^15) = .000030518 also.

They're equivalent.

In general, a^(-b) is equal to 1/(a^b)

In your equation, y^3 was equal to a, and 5 was equal to b.

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You divide the numerator by the denominator to get the decimal equivalent.

13/20 = .65

1/2 = .5

1/4 = .25

etc.

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non-member are charged $10 an hour to use the tennis courts.
Members are charged $300 per year plus $4 per hour.

If you let x be the number of hours you use the gym in a year, then:

You will pay x * 10 if you are a non-member.

You will pay 300 + 4 * x if you are a member.

You want to know when the membership fee plus the membership cost per hour becomes less than the non-membership cost per hour.

Your formula would be:

300 + 4 * x < 10 * x

Subtract 4 * x from both sides of this equation to get:

300 < 10 * x - 4 * x

Combine like terms to get:

300 < 6 * x

Divide both sides of this equation by 6 to get:

300 / 6 < x

Simplify to get:

50 < x

This is the same as:

x > 50

The number of hours you use the tennis courts in a year has to be greater than 50 in order for becoming a member to be cheaper than remaining a non-member.

Assume the hours are 49, 50, and 51 respectively.

Non-member cost would be:

49 * 10 = 490
50 * 10 = 500
51 * 10 = 510

Member costs would be:

300 + 49*4 = 496
300 + 50*4 = 500
300 + 51*4 = 504

At 49 hours, the non-member cost is cheaper.
At 50 hours, the non-member cost and the member cdost are the same.
At 51 hours, the member cost is cheaper.

The answer provided is confirmed.

The number of hours need to be greater than 50 for the member cost to be less than the non-member cost.

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x is the reciprocal of y if x*y = 1.

example:

The reciprocal of 3 is equal to 1/3 because 3 * 1/3 = 1

Let x equal the reciprocal of y.

You get 1/6 + 4*x = 21/10.

Multiply both sides of this equation by 60 to get:

10 + 240 * x = 126

Subtract 10 from both sides of this equation to get:

240*x = 116

Divide both sides of this equation by 240 to get:

x = 116 / 240

Since x is the reciprocal of y, this means that y is equal to 240 / 116

The number you are looking for should be 240 / 116.

To find out if that is true, substitute 240 / 116 for the number in your original problem statement.

The original problem statement is:

If 1/6 is added to 4 times the reciprocal of a certain number, the result is 21/10. Find the number.

If the number is 240 / 116, then the reciprocal of the number has to be 116 / 240 because 240 / 116 * 116 / 240 = 1.

So, the reciprocal we are looking for is 116 / 240.

4 * 116 / 240 = 464 / 240.

Add 1/6 to that and you get:

464 / 240 + 1/6 = 464 / 240 + 40 / 240 = 504 / 240

If this is correct, then 504 / 240 should be equal to 21 / 10

If you divide numerator and denominator of 504 / 240 by 24, then you get 21 / 10 which is the same as 21 / 10, so the answer is good.

The number you are looking for is 240 / 116.

The reciprocal of that number is 116 / 240.

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Your tree diagram would look something like this:

                     A               B                C

                     x               x                x
                     x               x                x
                     x               x                x
                     x               x                x
                =.26*T          =.38*T           =.36*T
                 xxxxxxxxx       xxxxxxxxx        xxxxxxxxx
                 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       x       x        x       x
                 x       x       x       x        x       x
            =.08*A  =.92*A  =.05*B  =.95*B   =.04*C  =.96*C

T = total number of parts in the bin.
A = total number of parts contributed by machine A.
B = total number of parts contributed by machine B.
C = total number of parts contributed by machine C.

Since A = .26 * T and B = .38 * T and C = .36 * T, then:

Total defective in the bin from machine A would be .26 * .08 = .0208 * T
Total defective in the bin from machine B would be .38 * .05 = .0190 * T
Total defective in the bin from machine C would be .36 * .04 = .0144 * T

Total defective parts in the bin is equal to (.0208 + .0190 + .0144) * T.

Total defective parts in the bin is therefore equal to .0542 * T.

Similarly,

Total good in the bin from machine A would be .26 * .92 = .2392 * T
Total good in the bin from machine B would be .38 * .95 = .3610 * T
Total good in the bin from machine C would be .36 * .96 = .3456 * T

Total good parts in the bin is equal to (.2392 + .3610 + .3456) * T.

Total good parts in the bin is therefore equal to .9458 * T.

The total good and bad in the bin should be equal to T.

.0542 * T + .9458 * T = 1 * T = T so this is ok.

Given that you pick a defective part out of the bin, what is the probability that it came from machine A?

The probability that it came from machine A would be .0208 / .0542 = .383763838.

Given that you pick a good part out of the bin, what is the probability that it came from machine B?

The probability that it came from machine B would be .3610 / .9458 = .38168746.

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The equation is:



Subtract 2 from both sides of this equation to get:



Multiply both sides of this equation by 4x to get:



Mulltiply both sides of this equation by 8 to get:



Simplify this equation to get:



Divide both sides of this equation by -60 to get:



Since 12 goes into 24 two times, and 12 goes into 60 five times, then this equation reduces to:



I believe this is the answer you are looking for.