You can put this solution on YOUR website!
3 {sqrt 3}/ {sqrt 6}
Factor the number inside the radical:
3 {sqrt 3}/ {sqrt 2*3}
ReWriting it as:
3 {sqrt 3}/[{sqrt 2}{sqrt 3}]
Canceling like-terms:
3/{sqrt 2}
Multiplying numerator and denominator by {sqrt 2}:
3{sqrt 2}/2
Rewriting:
(3/2){sqrt 2}

You can put this solution on YOUR website!
2 {sqrt x^3} + 5x{sqrt x} - 3{sqrt x^5}
You can rewrite as:
2 {sqrt x*x*x} + 5x{sqrt x} - 3{sqrt x*x*x*x*x}
Now, you can "pull out" pairs:
2x {sqrt x} + 5x{sqrt x} - 3x^2{sqrt x}
Factor out the sqrt x:
{sqrt x}[2x + 5x - 3x^2]
Combine like-terms:
{sqrt x}[7x - 3x^2]
Pull out the 'x':
x{sqrt x}[7-3x]

You can put this solution on YOUR website!
When "order" is important we refer to it as a "permutation".
When "order" is NOT important we refer to it as a "combination".
.
nPr = (n!)/(n-r)!
.
In our case:
n = 7
r = 7
.
nPr = (n!)/(n-r)!
7_Pr_7 = (7!)/(7-7)!
7_Pr_7 = (7!)/(0)!
7_Pr_7 = (7!)/1
7_Pr_7 = (7!)
7_Pr_7 = (1*2*3*4*5*7)
7_Pr_7 = 840

You can put this solution on YOUR website!
Remembering that:
log(100) = 2
or
100 = 10^2
.
Looking at your problem:
4logx = 4
logx = 1
x = 10^1
x = 10

You can put this solution on YOUR website!
If the police have 7 suspects, how many different ways can they select 5 for a lineup?
.
Since it doesn't matter which order the suspects line up, it is a "combination" as opposed to a "permutation".
.
Combination:
nCr = (n!)/[r!(n-r)!]
.
In our case:
n = 7
r = 5
.
nCr = (7!)/[5!(7-5)!]
nCr = (7!)/[5!(2)!]
nCr = (1*2*3*4*5*6*7)/[(1*2*3*4*5)(1*2)]
Canceling like-terms:
nCr = (6*7)/[(1*2)]
nCr = (42)/(2)
nCr = 21

You can put this solution on YOUR website!
You are given:
R = -0.037t + 50.1
where
R is the record
t is years after 1925
.
So, they want to know when the record will be 47.6 seconds or less...
So, just "plug it in" and solve for 't':
R = -0.037t + 50.1
47.6 = -0.037t + 50.1
47.6 + 0.037t = 50.1
0.037t = 2.5
t = 2.5/0.037
t = 67.568 years after 1925
.
So,
1925 + 67.568 = 1992.568
.
Conclusion:
1992 and AFTER the world record will be 47.6 seconds or less

You can put this solution on YOUR website!
.
Let x = pounds of lima beans added
then
.90x + .50(16) = .65(16+x)
.90x + 8 = 10.40 + .65x
.90x = 2.40 + .65x
.25x = 2.40
x = 9.6 pounds (of lima beans)

You can put this solution on YOUR website!
Area of a Triangle. Find the exact area of a triangle with a base of
sqrt(30)meters and a height of sqrt (6) meters.
.
Since the area of any triangle is
(1/2)bh
where
b is base
h is height
.
Plugging in our numbers:
(1/2)bh
(1/2)(sqrt(30))(sqrt(6))
(1/2)(sqrt(30*6))
(1/2)(sqrt(6*30))
(1/2)(sqrt(2*2*3*3*5))
Pulling out pairs:
(1/2)(2)(3)(sqrt(5))
3(sqrt(5))

You can put this solution on YOUR website!
"A man invests a certain sum of money at 14% and twice that amount at 15%. If his total interest earnings are $660 per year, how much has he invested at 14% and how much at 15%?"
.
Let x = amount invested at 14%
then
2x = amount invested at 15%
.
.14x + .15(2x) = 660
.14x + .30x = 660
.44x = 660
x = 660/.44
x = $1500 (amount invested at 14%
.
Amount invested at 15%:
2x = 2(1500) = $3000 (amount invested at 15%)

You can put this solution on YOUR website!
Domain is simply "the range" of values 'x' can take.
.
For your equation:
g(x)= (7x+4)/(x+4)
The ONLY values of x where it will be undefined is when the DENOMINATOR (x+4) is zero. So, set it to zero and solve for x:
(x+4) = 0
x = -4
.
So, the domain is ALL real numbers EXCEPT for -4.

You can put this solution on YOUR website!
Two trains started fromt he same station at the same time and traveled in opposite directions. After traveling 10 hours, they were 1,400 miles apart. The rate of the fast train exceeded the rate if the slow train by 5 miles per hour. Find the rate of each train.
.
Let x = speed of slow train
then
x+5 = speed of fast train
.
Using the "distance formula":
d = rt
where
d is distance
r is speed or rate
t is time
.
10x + 10(x+5) = 1400
10x + 10x + 50 = 1400
20x + 50 = 1400
20x = 1350
x = 67.5 mph (slow train)
.
Fast train:
x+5 = 67.5+5 = 72.5 mph

You can put this solution on YOUR website!
(4x + 1)(2x^2 + 5x + 1)
Using the "distributive property" :
(4x)(2x^2 + 5x + 1) + (1)(2x^2 + 5x + 1)
Expanding:
(8x^3 + 20x^2 + 4x) + (2x^2 + 5x + 1)
8x^3 + 20x^2 + 4x + 2x^2 + 5x + 1
8x^3 + 22x^2 + 9x + 1

You can put this solution on YOUR website!
Let x = final grade
.
The inequality looks like this:
70 <= (2/3)(48) + (1/3)x <= 79
70 <= (2)(16) + x/3 <= 79
70 <= 32 + x/3 <= 79
.
Looking at the left inequality first:
70 <= 32 + x/3
210 <= 96 + x
114 <= x
.
Looking at the right inequality:
32 + x/3 <= 79
96 + x <= 237
x <= 141
.
Putting it all together:
114 <= x <= 141
This means Wendy has to score between 114 and 141 on her final exam.

You can put this solution on YOUR website!
Find an angle such that 3 times the complement of the angle is 50 degrees greater than the supplement of the angle.
.
Two things you need to know:
If two angles are supplementary, the sum of the two angles = 180.
If two angles are complementary, the sum of the two angles = 90.
.
Let x = the angle
then
90-x = complement of angle
180-x = supplement of angle
.
3(90-x) = (180-x) + 50
270-3x = 180-x+50
270-3x = 230-x
270 = 230+2x
40 = 2x
20 degrees = x

You can put this solution on YOUR website!
s = vt + 16t^2
subtract vt from both sides:
s - vt = 16t^2
subtract s from both sides:
-vt = 16t^2 - s
multiply both sides by -1:
vt = s - 16t^2
finally divide both sides by t:
v = s/t - 16t

You can put this solution on YOUR website!
4x-5(3x+10)=126
First, distribute the -5 to terms inside the parenthesis:
4x-15x-50 = 126
combine like-terms:
-11x-50 = 126
add 50 to both sides:
-11x = 176
finally, divide both sides by 11
x = -176/11
x = -16

You can put this solution on YOUR website!
The problem gives you:
P(x)/(x-2) = 2x^2+x-2+-5/(x-2)
P(x)/(x-2) = 2x^2+x-2 + -5/(x-2)
.
If you multiply BOTH sides by (x-2), the (x-2) cancels from both sides leaving you with your answer:
P(x) = (x-2)(2x^2+x-2) + -5
.
At this point, you should "expand" the left term:
P(x) = (2x^3+x^2-2x)-(4x^2+2x-4) + -5
P(x) = 2x^3+x^2-2x-4x^2-2x+4 + -5
P(x) = 2x^3-3x^2-4x+4 + -5
P(x) = 2x^3-3x^2-4x-1

You can put this solution on YOUR website!
Let x = amount invested at 8%
then because "total invested was 18000"
18000-x = amount invested at 10%
.
.10(18000-x) = .08x + 360
1800 - .10x = .08x + 360
1800 = .18x + 360
1440 = .18x
1440/.18 = x
$8000 = x (amount invested at 8%)
.
Amount invested at 10%:
18000-x = 18000-8000 = $10000 (amount invested at 10%)

You can put this solution on YOUR website!
1. what is the approximate weight of oil in a container 3 feet high and 9 inches in diameter? (oil weighs approximately 55 pounds per cubic foot).
volume of container = (pi)r^2 * height
radius = 1/3(diameter) = 4.5 inches
Convert that into feet: 4.5 / 12 = 0.375 feet
.
volume of container = (pi)r^2 * height
volume of container = (3.14)(0.375)^2 * 3 = 1.3246875 cubic feet
.
"volume" * 55 = 1.3246875 * 55 = 72.86 pounds
.
2. find the area of a rectange which measures 12 feet, 3 inches by 3 feet, 9 inches. The correct answer expressed in square feet is?
.
convert 12 feet 3 inches to feet:
12 + 3/12 = 12 + 1/4 = 48/4 + 1/4 = 49/4 feet
.
convert 3 feet, 9 inches:
3 + 9/12 = 3 + 3/4 = 12/4 + 3/4 = 15/4
.
Area then is:
(49/4)(15/4) = 735/16 = 45.94 square feet
.
3.the diameter of a circle is 6 feet, 3 inches. what is the circumference of the circle? NOTE: to convert decimal fractions to inches in this problem, multiply by 12.
Circumference = (pi)d
where
pi is 3.14
d is diameter
.
Convert 6 feet, 3 inches to feet:
6 + 3/12 = 6 + 1/4 = 24/4 + 1/4 = 25/4 feet
.
Circumference = (pi)d = (3.14)(25/4) = 78.5/4 = 19.625 feet
or in inches:
19.625 * 12 = 235.5 inches

You can put this solution on YOUR website!
area of rectangle:
width*height
25*20 = 500 square feet
.
area of half circle:
area for a full circle = (pi)r^2
half circle would be half of the above:
(1/2)(pi)r^2
plugging in our values:
(1/2)(3.14)10^2
(1/2)(3.14)100
(3.14)50
157 square feet
.
Area of shaded region is:
"area of rectangle" - "area of half circle"
500 - 157 = 343 square feet
.
Perimeter:
Edge along the rectangle:
height + 2(width)
20 + 2(25)
20 + 50
70 feet
.
Edge along the half-circle:
"circumference of a full circle" is
(pi)(d)
we only want half of that so:
(1/2)(pi)(d)
(1/2)(3.14)(20)
(3.14)(10)
31.4 feet
.
Perimeter then is:
70 + 31.4 = 101.4 feet

You can put this solution on YOUR website!
Let x = length
then from "width of a rectangle is 3 feet less than its length" we have
x-3 = width
.
Perimeter = 2(length+width)
22 = 2(x + x-3)
22 = 2(2x-3)
22 = 4x-6
28 = 4x
7 feet = x (length)
.
width:
x-3 = 7-3 = 4 feet (width)

You can put this solution on YOUR website!
One number is four more than a second number. Two times the first number is 12 more than four times the second number.
.
Let x = second number
then from "One number is four more than a second number"
x+4 = first number
.
From "Two times the first number is 12 more than four times the second number" we get our equation:
2(x+4) = 12 + 4x
2x+8 = 12 + 4x
8 = 12 + 2x
-4 = 2x
-2 = x (second number)
.
First number:
x+4 = -2+4 = 2 (first number)

You can put this solution on YOUR website!
2x^2+x-6=0
Group terms first:
(2x^2+x)-6=0
(2x^2+x) = 6
2(x^2 + (1/2)x) = 6
(x^2 + (1/2)x) = 3
(x^2 + (1/2)x + 1/16) - 1/16 = 3
(x^2 + (1/2)x + 1/16) = 3 + 1/16
(x^2 + (1/2)x + 1/16) = 48/16 + 1/16
(x + 1/4)^2 = 49/16
Taking the square root of both sides:
x + 1/4 = (+-)7/4
x = -1/4 + (+-)7/4
.
Two possible solutions then are:
x = -1/4 + 7/4 = 6/4 = 1.5
or
x = -1/4 - 7/4 = -8/4 = -2

You can put this solution on YOUR website!
For log rules see:
http://www.purplemath.com/modules/logrules.htm
.
log(base 3)(x+8) + log(base 3)(x) = 2
.
Applying log rule: logb(mn) = logb(m) + logb(n)
log(base 3)(x^2+8x) = 2
.
Applying log rule: logb(m) = n =>> m = b^n
x^2+8x = 3^2
.
x^2+8x = 8
x^2+8x-8 = 0
.
Since we can't factor, use the quadratic equation. Doing so will yield:
x = {0.899, -8.899}
.
Details of quadratic:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=96 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 0.898979485566356, -8.89897948556636. Here's your graph:

= 0

You can put this solution on YOUR website!
s(t)= - 16t^2 + 48t
.
The idea is that the equation describes a parabola. If the coefficient 'a' is negative, the parabola opens downward. Thus the vertex will give you the "maximum".
.
The vertex form of a parabola is:
y= a(x-h)^2+k
.
The idea is to convert the given equation into the above form.
s(t)= - 16t^2 + 48t
s(t)= -16(t^2 - 4t)
Completing the square, we have:
s(t)= -16(t^2 - 4t + 4) + 64
s(t)= -16(t-2)^2 + 64
.
We now "see" that the vertex is at:
(h,k) = (2, 64)
.
Which says, that at
2 seconds, the ball will reach the maximum height of 64 feet.

You can put this solution on YOUR website!
Difficult to explain in words, but, here goes...
.
Given that you drew two axes -- where the vertical axis is the y-axis and the horizontal axis is the x-axis then:
.
y=5
is represented by a HORIZONTAL line (that is parallel to the x-axis) AND crosses the y-axis at +5.
.

You can put this solution on YOUR website!
Draw a diagram of the problem -- it'll help you see how to solve it.
.
If you do, the "inner rectangle" represents the garden while the "outer rectangle" (bordered by the outside edge of the gravel path).
.
"outer rectangle" - "inner rectangle" = "area of gravel path"
.
Let x = width of gravel path
.
"outer rectangle" then is:
(2x+18)(2x+13)
= 4x^2 + 26x + 36x + 234
= 4x^2 + 62x + 234
.
"Inner rectangle" is:
18 * 13 = 234
.
Now, instead of:
"outer rectangle" - "inner rectangle" = "area of gravel path"
we have:
(4x^2 + 62x + 234) - 234 = 516
4x^2 + 62x = 516
4x^2 + 62x - 516 = 0
2x^2 + 31x - 258 = 0
Since, it is difficult to factor, use the quadratic equation. It will yield:
x = {6, -21.5}
.
Since the negative solution does not make sense, throw it out leaving:
x = 6 feet (width of gravel path)
.
Details of quadratic:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=3025 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 6, -21.5. Here's your graph:

You can put this solution on YOUR website!
Let x = younger sister's age
and y = older sister's age
.
Since we have two unknowns, we'll need two equations.
.
Equation 1: "the sum of the ages of two sister's is 37"
x+y = 37
.
Equation 2: "the difference between four times the older sister's and three times the younger sister's age is 43"
4y - 3x = 43
.
Using the substitution method, solve equation 1 for y:
x+y = 37
y = 37-x
.
Substitute the above into equation 2 and solve for x:
4y - 3x = 43
4(37-x) - 3x = 43
148 - 4x - 3x = 43
148 - 7x = 43
148 = 7x+43
105 = 7x
15 years = x (younger sister's age)
.
Older sister's age, if found by plugging the above into equation 1 and solving for y:
x+y = 37
15+y = 37
y = 22 years (older sister's age)

You can put this solution on YOUR website!
Let x = son's age
then because "woman is three times as old as her son"
3x = woman's age
.
From:"sum of their ages id 52" we get
x + 3x = 52
4x = 52
x = 13 years (son's age)
.
woman's age:
3x = 3(13) = 39 years (mother's age)

You can put this solution on YOUR website!
Let x = Ms Ree's age
then because "Mr ree is 4 years older than ms ree"
x+4 = Mr Ree's age
.
From: "the sum of the squares of their ages is 3706"
x^2 + (x+4)^2 = 3706
x^2 + x^2+8x+16 = 3706
2x^2+8x+16 = 3706
2x^2+8x-3690 = 0
x^2+4x-1845 = 0
.
Factoring:
(x-41)(x+45) = 0
x = {41, -45}
.
We can toss out the negative answer so:
x = 41 (Ms Ree's age)
.
Mr. Ree's age:
x+4 = 41+4 = 45 years (Mr. Ree's age)

You can put this solution on YOUR website!
Let S = smaller number
and
L = larger number
.
Since we have two unknowns, we'll need two equations.
.
Equation 1: "five times the larger plus three times the smaller is 47"
5L + 3S = 47
.
Equation 2: "four times the larger minus twice the smaller is 20"
4L - 2S = 20
.
Using the "substitution method", solve equation 2 for S:
4L - 2S = 20
4L = 2S + 20
4L-20 = 2S
2L-10 = S
.
Substitute the above into equation 1 and solve for L:
5L + 3S = 47
5L + 3(2L-10) = 47
5L + 6L-30 = 47
11L - 30 = 47
11L = 77
L = 7
.
Finally, plug the above back into equation 2 and solve for S:
4L - 2S = 2
4(7) - 2S = 2
28 - S = 1
28 = S + 1
27 = S
.
Solution: The two numbers are 7 and 27