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.
"profit" = "amount paid for newspaper" - "cost of rental truck"
.
Let x = tons of newspaper collected for recycling
then
f(x) is the "profit" as a function of x
.
f(x) = 15x - 150

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a^2-8a+16
.
= (a-4)(a-4)
= (a-4)^2

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sophia used an 8-foot rope to secure a 6-foot tent pole as shown.
.
Let x = distance where rope is tied off
.
Because the rope, tent and ground forms a right-triangle, we can apply Pythagorean theorem:
.
8^2 = 6^2 + x^2
64 = 36 + x^2
28 = x^2
5.29 feet = x

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Log(x-3) + log(2x + 1) = 2logx
Log(x-3)(2x + 1) = logx^2
(x-3)(2x + 1) = x^2
2x^2 +x -6x - 3 = x^2
2x^2 - 5x - 3 = x^2
x^2 - 5x - 3 = 0
.
Using the quadratic equation to solve for x yields:
x = {5.541, -0.541}
We can toss out the negative solution leaving:
x = 5.541
.
Details of quadratic:

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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=37 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 5.54138126514911, -0.54138126514911. Here's your graph:

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A concrete driveway measures 16 yards by 6 yards. The driveway will be 0.25 yards thick. (a) How many cubic yards of concrete will be needed? (b) How much will it cost if concrete is $5.25 per cubic yard?
.
a)
16*6*0.25 = 24 cubic yards
.
b)
5.25*24 = $126

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log2(16)
= log2(2^4)
= 4

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Recall that total profit P is the difference between total revenue R and total cost C. Given R(x)=1000x-x^2 and C(x)=3000+20x, find the total profit, the maximum value of the total profit, and the value of x at which it occurs.
.
Profit = Revenue - Cost
f(x) = R(x) - C(x)
f(x) = (1000x-x^2) - (3000+20x)
f(x) = 1000x-x^2 - 3000 - 20x
f(x) = -x^2 + 980x - 3000
.
By looking at the coefficient associated with the x^2 term (-1), we know that it is a parabola that opens downward. Therefore, finding the vertex will provide the answer:
.
To do this, manipulate the equation into the "vertex form":
y= a(x-h)^2+k
.
f(x) = -x^2 + 980x - 3000
f(x) = (-x^2 + 980x) - 3000
f(x) = -(x^2 - 980x + __ ) - 3000
f(x) = -(x^2 - 980x + 240100) - 3000 + 240100
f(x) = -(x+490) + 237100
.
maximum value of the total profit: $237100
value of x at which it occurs: -490

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log base 2(5^2 + 7)
= log base 2(25 + 7)
= log base 2(32)
= log base 2(2^5)
= 5

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If you saw:
a^2 - b^2
This is a "difference of squares" -- which factors out to:
(a-b)(a+b)
.
So, if you started with:
(16X^4)-1
Notice we can rewrite it as:
(4x^2)^2 - 1^2
Now, you have a difference of squares resulting in:
(4x^2 - 1)(4x^2 + 1)
.
Oh wait, but (4x^2 - 1) is also a difference squares:
(2x-1)(2x+1)(4x^2 + 1)

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Write the equation of the line that passes through (-4,7) with a slope of a half.
.
Given:
point (-4,7)
slope, m, 1/2
.
Plug it all into the "point slope" form:
y-y1 = m(x-x1)
y-7 = (1/2)(x-(-4))
y-7 = (1/2)(x+4)
y-7 = (1/2)x + 2
y = (1/2)x + 9 (this is the slope-intercept form:what they're looking for)

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t=sqrt(h/16)
Plug in the height and solve for t:
t=sqrt(17000/16)
t=sqrt(1062.5)
t=32.6 seconds

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Given a circle with the radius of 2, which is the length of an arc measuring 75 degrees?
.
Circumference = 2(pi)r = 2(3.14)(2) = 12.56
.
Since the entire circle is 360 degrees:
the arc is then
(75/360)(12.56)= 2.62

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The equation:
x^2 + 6x + 8 = 0
is a quadratic
.
To solve, you can either "factor" or apply the "quadratic equation".
.
Always factor if you can, it is easier.
.
x^2 + 6x + 8 = 0
(x+4)(x+2) = 0
.
This would be true if either (x+4) or (x+2) was equal to zero.
.
To find the solutions, set each to zero and solve:
x+4=0
x = -4
and
x+2=0
x = -2
.
Therefore, two solutions for x:
x = {-4,-2}

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"A student missed 9 problems on a chemistry test and received a grade of 44%. If all the problems were of equal value, how many problems were on the test? Round off your answer to the nearest integer."
.
Let x = number of problems on the test
then
(x-9)/x = .44
x-9 = .44x
.56x = 9
x = 9/.56
x = 16.071
or, rounded to the nearest integer:
x = 16 problems on the test

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There are two dock workers, Alf and Bob. Alf can load 16 tons of steel per day, and Bob can load 20 tons per day. If they each work 8-hour days, how many tons of steel can the two of them load in one hour, assuming they maintain a steady rate?
.
Alf:
16 tons per day
or
16 tons per 8 hours
.
Bob:
20 tons per day
or
20 tons per 8 hours
.
In ONE hour, working together they can load:
16/8 + 20/8
= 36/8
= 4.5 tons

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the square of an integer is 30 more than the integer. find the integer.
.
Let x = the integer
then
x^2 = x + 30
x^2 -x -30 = 0
(x-6)(x+5) = 0
x = {-5,6}
.
Therefore, the integer could either be -5 or 6

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Apply "log rules":
log3x+log(x+2)=1
log(3x)(x+2)=1
(3x)(x+2)=10^1
3x^2+6x=10
3x^2+6x-10=0
Solve using the quadratic equation:
x = {1.082, -3.082}
We can toss out the negative solution leaving:
x = 1.082
.
Details of quadratic to follow:
.

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=156 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 1.08166599946613, -3.08166599946613. Here's your graph:

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Tashin Baker is planning an expansion of a square flower garden in a city park. If each side of the original square is increased by 7 meters and the new total area of the garden will be 144 square meters. Find the lengh of the original garden.
.
Let x = length of original garden
then
(x+7)(x+7) = 144
x^2 + 14x + 49 = 144
x^2 + 14x - 95 = 0
.
Factoring the left side:
(x+19)(x-5) = 0
x = {-19, 5}
We can toss out the negative solution leaving:
x = 5 meters (measure of one side of original garden)

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log(x+9)=log(2x-7)
(x+9)=(2x-7)
9 = x-7
16 = x

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Five, subtracted from three times a number.
.
Let n = a number
then
3n - 5 (this is what they're looking for)

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The first side of the triangle must be 8 feet shorter than the second side. the third side must be 4 times as long as the first side. Find the length of each side of the triangle if the perimeter is 26.
.
Let x = length of second side
then because "first side of the triangle must be 8 feet shorter than the second side" we get
x-8 = length of first side
and from "third side must be 4 times as long as the first side" we get
3(x-8) = length of third side
.
And, because the perimeter is 26:
26 = x + x-8 + 3(x-8)
26 = x + x-8 + 3x - 24
26 = 5x - 32
58 = 5x
11.6 feet = x (second side)
.
First side:
x-8 = 11.6-8 = 3.6 feet (first side)
.
Third side:
3(x-8) = 3(11.6-8) = 3(3.6) = 10.8 feet (third side)

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125t^3 + 8
.
What you're suppose to see is that this is the same as:
(5t)^3 + 2^3
Which is a special case -- "sum of cubes".
See this web site for additional info:
http://www.purplemath.com/modules/specfact2.htm
.
a^3 + b^3 = (a + b)(a^2 – ab + b^2)
In your case:
a is 5t
b is 2
.
Therefore:
125t^3 + 8
= (5t)^3 + 2^3
= (5t + 2)((5t)^2 - (5t)(2) + 2^2)
= (5t + 2)(25t^2 - 10t + 4)

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The height of a triangle is 4in greater than twice its base. The area of the trianlge is 168 square in. What is the base of the triangle?
.
We know that for any triangle:
area = (1/2)bh
where
b is base
h is height
.
Let b = length of the base
then
2b+4 = length of the height
.
168 = (1/2)b(2b+4)
336 = b(2b+4)
336 = 2b^2 + 4b
0 = 2b^2 + 4b - 336
0 = b^2 + 2b - 168
0 = (b+14)(b-12)
b = 12 inches (base)

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log(base3)(2x-1)=3
2x-1=3^3
2x-1=27
2x=28
x=14

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y^2+y-12
= (y+4)(y-3)

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y^2+12y-13
= (y+13)(y-1)

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log3x+log(x+2)=1
log3x(x+2)=1
3x(x+2)=10^1
3x^2+6x=10
3x^2+6x-10=0
Solving the above using the quadratic equation yields two solutions:
x = {1.082, -3.082}
.
Details of quadratic follows:
.

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=156 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 1.08166599946613, -3.08166599946613. Here's your graph:

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log(x+9)-logx=3
log(x+9)/x=3
(x+9)/x=10^3
(x+9)/x=1000
(x+9)=1000x
9=999x
1/111 = x

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3log2x=4
log(2x)^3=4
(2x)^3 = 10^4
8x^3 = 10000
x^3 = 1250
x = 10.772

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log(3x+7)=1
(3x+7)= 10^1
3x+7 = 10
3x = 3
x = 1

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2log4-log3+logx-4=0
log4^2-log3+logx-4=0
log(4^2/3)+logx-4=0
log(16/3)+logx-4=0
log(16(x-4)/3)=0
16(x-4)/3 = 10^0
16(x-4)/3 = 1
16(x-4) = 3
x-4 = 3/16
x = 3/16 + 4
x = 3/16 + 64/16
x = 67/16