WAEC 2022 · Paper 2 · Q10

Applications of differentiation

A solid rectangular block has a base which measures $3x\text{ cm}$ by $2x\text{ cm}$ . The height of the block is $y\text{ cm}$ and its volume is $72\text{ cm}^3$ .

(a)
Express $y$ in terms of $x$ .
(b)(i)
Find an expression for the total surface area of the block in terms of $x$ only;
(b)(ii)
Find the value of $x$ for which the total surface area has a stationary value (2 d.p.).

Try it on a graph

Total surface area A(x) = 12x² + 120/x has its minimum at x = ∛5.

Worked solution (try it first)

(a)

The volume is $3x \times 2x \times y = 6x^2y = 72$ , so $y = \dfrac{12}{x^2}$ .

(b)(i)

Top and bottom: $2 \times 6x^2 = 12x^2$ .
The four sides: $2(3xy) + 2(2xy) = 10xy$ .
So $A = 12x^2 + 10xy$
$= 12x^2 + 10x \times \dfrac{12}{x^2}$
$= 12x^2 + \dfrac{120}{x}$ .

(ii)

$\dfrac{dA}{dx} = 24x - \dfrac{120}{x^2}$
$= 0$ , so $24x^3 = 120$ and $x^3 = 5$ .
So $x = \sqrt[3]5 \approx 1.71$ .

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