Four small squares of side x are cut out of a square of side 12 cm to make a tray by folding the edges. What is the value of x so that the tray has the maximum volume?

Concept:

• For finding maxima or minima of a function f(x),
• Put f'(x) = 0 to find the critical point.
• If f''(x) > 0, then function is minima and if f''(x) < 0, function has maxima.

Calculation:

Given side of the square = 12 cm.

Four small square of side x are cut out of a square.

∴ Figure looks like:

Dimensions of the tray is:

Side length of base of the tray = 12 - 2x cm.

Height of the tray is x cm.

∴ Volume of the tray is V = (12 - 2x)²x cm³.

V = (4x² - 48x + 144)x

V = 4x³ - 48x² + 144x.

\(\rm\frac{dV}{dx}=12x^2-96x+144\)

For maximum volume , put \(\rm\frac{dV}{dx}=0\)

⇒ 12x² - 96x + 144 = 0.

⇒ x² - 8x + 12 = 0

⇒ (x - 2)(x - 6) = 0

⇒ x = 2 or x = 6.

Now \(\rm\frac{d^2V}{dx^2}\) = 24x - 96

\(\rm\frac{d^2V}{dx^2}\) at x = 2 is - 48

⇒ V(x) at x = 2 is maxima.

Volume is maximum when length of x is 2 cm.