The length and width of the cardboard are 20 ft and 15 ft. We are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give maximum volume.

Question

17-04-2024
Mathematics

High School

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The length and width of the cardboard are 20 ft and 15 ft. We are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give maximum volume.

Answer :

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prabaase87 prabaase87 · Answer 1 · 2023-12-16T16:41:15-05:00

Final answer:

To find the height that maximizes the volume of a box made from a 20ft x 15ft cardboard sheet, we set up a volume equation based on the lengths of the sides and solve for the height, 'x'. The maximum volume occurs when the height is approximately 2.5ft.

Explanation:

This question is about maximizing the volume of a box made with cardboard. Let's denote the length of the cardboard that we cut out as 'x', so the sides of the box, after cutting and folding, are (20-2x), (15-2x), and x. Therefore, the volume V of the box is expressed as V = x (20-2x)(15-2x).

We are asked to find the height 'x' that maximizes this volume. This is a problem in calculus: finding the maximum of a function. Differntiating the volume function and setting it to 0 gives the dimensions that maximize the volume. However, without calculus, the pragmatic way to solve is to use a graphing calculator or an online plotting tool to plot V = x(20 - 2x)(15 -2x) within the domain of x = 0 to x = 7.5 and identifying the peak of the function.

A common solution for this problem indicates that the height of the box that gives the maximum volume is approximately 2.5 ft.

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