Question

A box with a square base is to have an open top. The surface area of the box is 147 cm\(^2\). What should its dimensions be in order that the volume is largest?

Hint:

To maximize volume for a given surface area, express height as a function of the base side length and differentiate the volume function.

Answer 1

Step 1: Define the Variables
Let: - \( x \) be the side length of the square base, - \( h \) be the height of the box. The total surface area of the box is expressed as: \[ x^2 + 4(xh) = 147. \] Step 2: Solve for \( h \) in Terms of \( x \)
From the surface area equation: \[ h = \frac{147 - x^2}{4x}. \] Step 3: Maximize the Volume
The volume \( V \) of the box is given by: \[ V = x^2 h = x^2 \times \frac{147 - x^2}{4x}. \] \[ V = \frac{x(147 - x^2)}{4}. \] Differentiate the volume function with respect to \( x \): \[ \frac{dV}{dx} = \frac{147 - 3x^2}{4}. \] Set the derivative equal to zero to find the critical points: \[ 147 - 3x^2 = 0 \Rightarrow x^2 = 49 \Rightarrow x = 7. \] Step 4: Calculate \( h \)
Substitute \( x = 7 \) into the equation for \( h \): \[ h = \frac{147 - 49}{4(7)} = \frac{98}{28} = 3.5. \] Final Answer: \( x = 7 \) cm, \( h = 3.5 \) cm.