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Chapter 0: Problem 23

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box can be made. If the cardboard is 15 in. long and 8 in. wide and the square cutaways have dimensions of \(x\) in. by \(x\) in., find a function that gives the volume of the resulting box.

Short Answer

Expert verified
The volume function of the resulting open box after cutting identical squares of dimensions x by x inches from each corner of the cardboard and folding the flaps is given by: \[V(x) = 4x^3 - 30x^2 + 120x\]

Step by step solution

01

Identify the dimensions of the open box after folding

After cutting the squares of side x from each corner of the cardboard, the resulting open box will have a length of (15 - 2x), a width of (8 - 2x), and a height of x inches. This is because we remove an x by x square from both ends of the length and width, therefore reducing the length and width by 2x.
02

Write a formula for the volume of the open box

The volume of a rectangular box is given by the product of its length, width, and height. In this case, the length, width, and height of the open box are given by (15 - 2x), (8 - 2x), and x, respectively. We can multiply these three expressions to get the volume of the open box. So the volume function V(x) is given by: \[V(x) = (15-2x)(8-2x)x\]
03

Simplify the expression for the volume function

To simplify the expression for V(x), we will first multiply the expressions (15-2x) and (8-2x) using the distributive property, and then multiply the resulting expression by x: \[(15-2x)(8-2x) = (15 \cdot 8) + (-2x \cdot 8) + (15 \cdot -2x) + (-2x \cdot -2x)\] This simplifies to: \[(15-2x)(8-2x) = 120 - 30x + 4x^2\] Now, multiply the resulting expression by x: \[V(x) = x(120 - 30x + 4x^2) = 4x^3 - 30x^2 + 120x\] So, the volume function of the resulting open box is: \[V(x) = 4x^3 - 30x^2 + 120x\]

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