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Starting with:
s= 2(lw + hw + hl)
s/2 = lw + hw + hl
s/2 - lw = hw + hl
s/2 - lw = h(w + l)
(s/2-lw)/(w+l) = h

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Alfonzo' pizzeria bakes olive pieces in the outer crust of its 20 inch (diameter) pizza . there is at least one olive piece per inch of crust . how many olive pieces will you get in one slice of pizza ? assume the pizza is cut into eight slices
.
Area of pizza = (pi)r^2 = (3.14)10^2 = (3.14)(100) = 314 sq inches
.
Area of a slice of pizza: 314/8 = 39.25 square inches
.
Since there is one olive per square inch:
39.25 olives per slice

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Rationalizing denominator:

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The area of a rectangle is 30 square centimeters and its perimeter is 26 centimeters. What is the length and width of the rectangle?
Let x = width
and y = length
.
From:"The area of a rectangle is 30 square centimeters" (equation 1)
xy = 30
.
From: "perimeter is 26 centimeters" (equation 2)
26=2(x+y)
.
Solve equation 2 for y:
26=2(x+y)
13=x+y
13-x = y
.
Substitute above into equation 1 and solve for x:
xy = 30
x(13-x) = 30
13x-x^2 = 30
0 = x^2-13x+30
0 = (x-15)(x+2)
x = {-2, 15}
We can toss out the negative solution leaving
x = 15 centimeters
.
Substitute above into equation 1 and solve for y:
xy = 30
15y = 30
y = 2 centimeters
.
Dimensions of rectangle:
2 by 15 centimeters

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Starting with:

.
Multiply both sides by 12 to get rid of denominators:

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what is the measure of an angle that is 48 degrees greater than its complement
.
Let x = measure of our angle
then its
complement = 90-x
.
We get our equation from:"measure of an angle that is 48 degrees greater than its complement"
x = (90-x)-48
x = 90-x-48
x = 42-x
2x = 42
x = 21 degrees

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a certain 90-mile trip took 2 hours. exactally 1/3 of the distance traveledwas by raail, and this part of the trip took 1/5 of the travel time. what was the average rate, in miles per hour of the rail portion of the trip?
.
1/3 of the trip is then:
(1/3)(90) = 30 miles
.
Time it takes to travel that distance is:
(1/5)(2) = 2/5 of an hour
.
Average rate:
30/(2/5) = 150/2 = 75 mph

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what would you get if you divided a half of a half of a half by a half of a fourth of a half
.
Translate to:
[(1/2)(1/2)(1/2)]/[(1/2)(1/4)(1/2)]
[1/8]/[1/16]
multiplying by 16/16 we get:
16/8
which reduces to:
2

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Two boats leave the dock at 12:00 noon, one of them moving northward at 6 miles per hour, and the other moving westward at 8 mph. How far apart are the boats after 2hours?
A. 20 miles B. 28 miles C. 10 miles D. 14 miles
.
After two hours:
northward boat travels: 2*6 = 12 miles
westward boat travels: 2*8 = 16 miles
.
To find distance apart, apply Pythagorean theorem:
Let d = distance apart
then
d^2 = 12^2 + 16^2
d^2 = 144 + 256
d^2 = 400
d = 20 miles

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log AB^2
log A + log B^2
log A + 2log B

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The perimeter of a rectangle equals one and a half times its area. Express the length of the rectangle in terms of the width (use variable "w" for width)
.
Area of any rectangle = width * length
Perimeter of any rectangle = 2(width+length)
.
Let w = width
and temporarily
x = length
.
From the problem, perimeter is:
1.5wx
.
But,
Perimeter = 2(w+x)
So, then we have
1.5wx = 2(w+x)
Solving for x:
1.5wx = 2(w+x)
1.5wx = 2w+2x
1.5wx-2x = 2w
factoring the left:
x(1.5w-2) = 2w
x = 2w/(1.5w-2)
.
So, your answer would be
length is equal to:
2w/(1.5w-2)

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Tom wants to build a fence around his rectangular swimming pool. The dimensions of the pool are 18 feet by 22feet. He wants the border around the pool to be of uniform width. If he has 60 yards of fencing, find the width of the border.
.
Convert 60 yards to feet:
60 * 3 = 180 feet
.
Let x = width of border
then
18+2x = width of fence
22+2x = length of fence
.
perimeter = 2(width+length)
180 = 2(18+2x + 22+2x)
180 = 2(40+4x)
90 = 40+4x
50 = 4x
12.5 feet = x

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How many pints of a 10% cleaning solution must be mixed with 6 pints of an 8% cleaning solution to give a 9% solution?
.
Let x = pints of 10% solution added
then
.10x + .08(6) = .09(6 + x)
.10x + .48 = .54 + .09x
.01x + .48 = .54
.01x = .06
x = .06/.01
x = 6 pints

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9.6 to 6.8
multiplying both sides by 10:
96 to 68
dividing both sides by 4:
24 to 17

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I'm correcting my "absolute" conversion...
3.03+(-2.05)-(-9.25)+|-10|
3.03+(-2.05)-(-9.25)+10
3.03-2.05+9.25+10
0.985+9.25+10
10.23+10
20.23

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5 times the sum of a number and -1 is the same as 6 times the number.
.
Let x = the number
then
5(x+(-1)) = 6x
5(x-1) = 6x
5x-5 = 6x
-5 = x
.
Answer: the "number" is -5

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A rectangular garden is 40 ft wide. If its area is 3800 ft2, what is the length of the garden?
.
Let x = length
then
since width*length=area we have:
40x = 3800
x = 3800/40
x = 380/4
x = 95 feet (length)

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P=a+b+c solve for b
.
Starting with:
P=a+b+c
subtract 'a' from both sides:
P-a=a+b+c-a
P-a=b+c
subtract 'c' from both sides:
P-a-c=b+c-c
P-a-c=b

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in the equations y= -2x-4 the slope is
.
the equation is given in the "slope-intercept" form of:
y=mx+b
where
m is the slope
.
So, the slope, by inspection, is -2

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A planet orbiting the sun 42,000 miles from the center of the sun. It makes 1 revolution in 3 days. Find the speed of the planet in miles per hour.
.
circumference is (pi)d
where
d is diameter
.
circumference = (3.14)(2)42000 = 263760 miles
.
3 days = 3(24) = 72 hours
.
Speed = 263760/72 = 3663.33 mph

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A bakery sells single layer mini-cakes that are 3 inches in diameter for $4 each. They also have a 9-inch cake for $15. If both cakes are the same thickness, which option gives you more cake for the money, nine mini-cakes or one 9-inch cake?
.
Volume of cake = r^2(pi)
Let t = thickness
volume of mini-cake = (1.5)^2(3.14)t = 7.065t
Since there are nine mini-cake, total volume is: 9(7.065t) = 63.585t
.
volume of 9-inch cake = 3^2(3.14)t = 28.26t
.
Relate volume per cost:
mini-cake:63.585t/12 = 5.29875t (vol per dollar)
9-inch cake: 28.26t/15 = 1.884t (vol per dollar)
.
Answer:mini-cake gives MORE volume per dollar

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the length of a rectangle is 4 more than seven times the width. write an expression for the area a of the rectangle.
.
Let w = width
then from "length of a rectangle is 4 more than seven times the width" we have
7w+4 = length
.
since area=width*length:

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Car 1 leaves Miami and travels at a speed of 47mph. Car 2 leaves 2 hours later and travels at 58mph. How long will it take for car 2 to catch up to car 1.
.
Apply distance formula: d=rt
.
Let x = time (hours) to catch up
then
47x = 58(x-2)
47x = 58x-116
-11x = -116
x = 116/11
x = 10.55 hours
or
x = 10 hours and 33 minutes

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apply "order of operations": (PEMDAS)
.
Starting with:
-23-(-24)/-11-(-17) + |-22|
eliminate the absolute value:
-23-(-24)/-11-(-17) + 22
eliminate the double negatives:
-23+24/-11+17 + 22
perform division:
-23-2.182+17 + 22
-25.182+17 + 22
-8.182 + 22
13.182

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Put the equation of a line into slope-intercept form. 3x + 2y = 10
.
Starting with:
3x + 2y = 10
Subtract 3x from both sides:
2y = -3x + 10
Divide both sides by 2:
y = (-3/2)x + 5 (this is what they're looking for)
.
slope-intercept form:
y = mx + b
where
m is the slope
b is the y-intercept

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How many liters of water must be added to 30liters of a 20% alcohol solution to dilute it to a 15% solution?
.
Let x = liters of water to add
then
"amt of alcohol at start" = "amt of alcohol at end"
.20(30) = .15(30+x)
6 = 4.5+.15x
1.5 = .15x
1.5/.15 = x
10 liters = x

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Angle A and angle B are supplementary angles. If the measure of angle B is 6 degrees greater twice the measure of angle A, find the measure of each angle.
.
By definition of "supplementary angles" we have:
A + B = 180
.
Then, from:
"angle B is 6 degrees greater twice the measure of angle A" we have
B = 2A+6
.
Substitute the above into:
A + B = 180
A + 2A+6 = 180
3A+6 = 180
3A = 174
A = 58 degrees
.
Substitute above into the following to find B:
A + B = 180
58 + B = 180
B = 122 degrees

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Write a system of two equations using x and y to represent the following problem. A chemist is mixing two solutions together.
The chemist wants 100 gallons when complete. He is mixing a 24% solution with a 50% solution to get a 36% solution. How many gallons of each will he use?
.
Let x = amount of 24% solution
and y = amount of 50% solution
.
From: "The chemist wants 100 gallons when complete." we get equation 1:
x+y = 100
From: "He is mixing a 24% solution with a 50% solution to get a 36% solution." we get equation 2:
.24x + .50y = .36(100)
.
Solve equation 1 for y:
x+y = 100
y = 100-x
.
Substitute the above into equation 2:
.24x + .50y = .36(100)
.24x + .50(100-x) = .36(100)
.24x + 50-.50x = 36
50-.26x = 36
-.26x = -14
x = 53.85 gallons (of 24% solution)
.
to find 50% solution substitute above into equation 1:
x+y = 100
53.85+y = 100
y = 46.15 gallons

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Part of $18,000 is invested at 8% annual interest, and the rest is invested at 6%. After one year, the accounts paid $1,220 in interest. How much was invested at the lower rate? (Give your answer correct to the nearest cent.)
.
Let x = amount invested at lower rate of 6%
then
18000-x = amount invested at 8%
.
Our equation:
.06x + .08(18000-x) = 1220
.06x + 1440 - .08x = 1220
1440 - .02x = 1220
-.02x = -220
x = $11,000.00

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Starting with:

reduces to:

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Solve for x
tx-ux=3t
factoring the left:
x(t-u) = 3t
dividing both sides by (t-u):
x = 3t/(t-u)