If 600 cm2 of material is available to make a box with a square base and a closed top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter without commas. For example, if the answer is 2,000 write 2000.

Question

dumbcow · Answer

let the sides of square base be "x" and height be "h"
first come up with expressions for surface area and volume
$$SA = 2x^2 + 4 hx = 600$$
$$V = h x^2$$
Next get h in terms of x and sub it into Volume equation
$$h = \frac{300 -x^2}{2x}$$
$$\rightarrow V = (\frac{300-x^2}{2x}) x^2$$
$$V = 150 x - \frac{1}{2} x^3$$
maximize volume by setting derivative equal to 0
$$\frac{dV}{dx} = 150 - \frac{3}{2}x^2 = 0$$
solve for x
$$x = 10$$
which means h =10
max volume occurs when box is a cube
10^3 = 1000 cm^3

anonymous · Answer

thank you!