Chapter 10.4, Problem 30E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

*Minimum material*

(a) A box with an open top and a square base is to be constructed to contain 4000 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box.

(b) A box with an open top and a square base is to be constructed to contain *k* cubic inches. Show that the minimum amount of material is used to construct the box when each side of the base is

**(a)**

To determine

**To calculate:** The dimensions to minimize the material to construct the box.

Explanation

**Given Information:**

Volume of the box is

**Formula used:**

To find the minimum value, calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to

If the second derivative of the provided function is more than zero, then substituting the value of the independent variable will give the minimum value of the equation.

The power rule is used for a function in which the expression can be written as every term raised to a power (be it fractional, positive or negative). For the function

**Calculation:**

Consider the provided statement,

Volume of the box is

Assume side of the square base is *x* and other side of the box other than *x* is *y*

The surface area of the square base box, which has an open top is:

Use the equation of volume

Substitute

To find the stationary value for

Put

**(b)**

To determine

**To prove:** The dimensions are

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