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.
(1^3 + x^3) = (1 + 3)(1^2 - 1x + x^2) = 0
b. Factor the sum of cubes.
(a^3 + b^3 = (a + b) (a^2 - ab = b^2)
c.
Calculate the total volume of the two boxes if x = 3 feet.
(1^3 + x^3) = (1 + 3)(1^2 - 1x + x^2) = 7
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^2 + 4^3
b. Factor the sum of cubes.
(m + 4) (m^2 - 2m + 4) (m + 5)
c.
Calculate the total volume of the two puzzle cubes if m = 5 centimeters.
(m + 4) (m^2 - 2m + 4) (m + 5) = 21
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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.
(12)^3 + (2t)^3 = 2t^2 + 1730 cm^3
b. Factor the sum of cubes.
(2t + 12^3) (2 x 7 - 2t + 12^2)
c.
Calculate the total volume of the two fish tanks if t = 7 inches.
1,274 cm
4. Think of