Minimum and Maximum Values — Question 10

Question 10 - Solution

Let $x$ be the side length of the square base, and $h$ be the height of the box.

(a) Surface area as a function of one variable:

Volume constraint: $$x^2 h = 500 \Rightarrow h = \frac{500}{x^2}$$

Surface area (no top): $$A(x) = x^2 + 4xh = x^2 + 4x \cdot \frac{500}{x^2} = x^2 + \frac{2000}{x}$$

(b) Minimize the surface area:

Differentiate: $$A'(x) = 2x - \frac{2000}{x^2}$$

Set derivative to 0: $$2x - \frac{2000}{x^2} = 0 \Rightarrow 2x^3 = 2000 \Rightarrow x^3 = 1000 \Rightarrow x = 10$$

Find $h$: $$h = \frac{500}{10^2} = 5$$

Dimensions: $\boxed{x = 10 \, \text{cm}, \, h = 5 \, \text{cm}}$

(c) Minimum surface area: $$A = 10^2 + \frac{2000}{10} = 100 + 200 = \boxed{300 \, \text{cm}^2}$$

Graph of $A(x) = x^2 + \frac{2000}{x}$: