Answered: . A box with a square base and open top…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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. A box with a square base and open top must have a volume
of 32,000 cm cu . . Find the dimensions of the box that mini -
mize the amount of material used

Expert Solution

Step 1

To Determine: Find the dimensions of the box that minimize the amount of material used

Given: A box with a square base and open top must have a volume of 32,000 cm cube.

Explanation: let side of square bas be x and height of the box be h then the volume will be

$V = x^{2} h$

since volume is 32000 $c m^{3}$

then we have

$x^{2} h = 32000 h = \frac{32000}{x^{2}} - - - - - - - - - (1)$

surface area of box with open top $S = x^{2} + 4 x h$

so substitute $h = \frac{32000}{x^{2}}$ then we have

$\begin{array}{rcl} S (x) & = & x^{2} + \frac{4 x}{x^{2}} \times 32000 \\ = & x^{2} + \frac{128000}{x} \end{array}$

differentiate and equate to zero then we have

$S' (x) = 2 x - \frac{128000}{x^{2}} S' (x) = 0 2 x - \frac{128000}{x^{2}} = 0 2 x = \frac{128000}{x^{2}} x^{3} = \frac{128000}{2} x^{3} = 64000 x = 40$

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