Assignment: Factor a Sum of Cubes
Follow the directions to find the missing information in each problem. Be sure to show all work
leading to your answers and simplify your answers whenever possible.
1. A shipping company offers various sized shipping boxes to its customers. Some of these
boxes are cube-shaped, with equal height, width, and depth. As part of an upcoming
sales promotion, the company will offer two cube-shaped boxes for the price of one.
a. Write an expression to represent the total volume of two different sized boxes as a
sum of cubes if one of the boxes has sides with a length of 1 foot and the other has
sides with a length of x feet.
x^3 + 1
b. Factor the sum of cubes.
x^3 + 1 = (x + 1)(x^2 - x * 1 + 1^2)
x^3 + 1 = (x + 1)(x^2 - x + 1)
c.
Calculate the total volume of the two boxes if x = 3 feet.
3^3 + 1
27 + 1
Total Volume = 28 feet
2. A toy manufacturer is preparing to manufacture a puzzle cube in two different sizes. One
of the puzzle cubes will have a side length of 4 centimeters, and the size of the other is
yet to be determined.
a. Write an expression to represent the total volume of two different sized puzzle cubes
as a sum of cubes, using m centimeters as the side lengths of the second puzzle.
m^3 + 4^3
m^3 + 64
b. Factor the sum of cubes.
m^3 + 64 = (m + 4)(m^2 - m * 4 + 4^2)
m^3 + 64 = (m + 4)(m^2 - 4m + 16)
c.
Calculate the total volume of the two puzzle cubes if m = 5 centimeters.
5^3 + 4^3
125 + 64
Total Volume = 189 centimeters
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3. A pet supply store is planning to offer cube-shaped fish tanks. The store manager
decides to order one tank with a 12-inch side length but has not yet decided on the size
of the other.
a. Write an expression to represent the total volume of two different sized tanks as a
sum of cubes, using 2t centimeters as the length of the sides in the second tank.
2t^3 + 12^3
2t^3 + 1,728
b. Factor the sum of cubes.
2t^3 + 1,728 = (2t + 12)(2t^2 - 2t * 12 + 12^2)
2t^3 + 1,728 = (2t + 12)(2t^2 - 24t + 144)
c.
Calculate the total volume of the two fish tanks if t = 7 inches.
2 * 7^3 + 12^3
2 * 343 + 1,728
686 + 1,728
Total Volume = 2,414 inches
4. Think of another real-world situation involving two different sized cube-shaped items.
a. Write a sentence to describe what the cubes represent and then write an expression
to represent the total volume of the two items as a sum of cubes, using 2a to
represent the side lengths in one cube and 3b to represent the side lengths in the
other cube.
A builder is planning to make two building that are cube shaped. The builder has not
yet decided on how long the side lengths will be.
Write an expression to represent the total volume of two different sized buildings as a
sum ofthe
cubes,
2a centimeters as the length of the sides in one building and 3b in
b. Factor
sumusing
of cubes.
the second building.
2a^3 + 3b^3 = (2a + 3b)(2a^2 - 2a * 3b + 3b^2)
2a^3 + 3b^3 = (2a + 3b)(2a^2 - 6ab + 3b^2)
2a^3 + 3b^3
c.
Choose two different values for a and b and use these values to calculate a specific
total volume for the two items.
Calculate the total volume of the two buildings if a = 3 and b = 5.
2 * 3^3 + 3 * 5^3
2 * 27 + 3 * 125
54 + 375
Total Volume = 429 feet
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