Question

Assuming that the petrol burnt in a motor boat varies as the cube of its velocity, the most economical speed when going against a current of c km/hr is –

• A. (3c/2) km/hr
• B. (3c/4) km/hr
• C. (5c/2) km/hr
• D. (c/2) km/hr

Answer 1

Answer: (A)

(3c/2) km/hr

Answer 2

Suppose p is the quantity of petrol burnt in one hour.

Then p = ky³. Let the distance of the journey be s km.

Duration of the journey = $\frac{s}{v - c}$ hrs

\ total petrol burnt for the whole journey

= $\frac{sp}{v - c}$ = $\frac{skv^{3}}{v - c}$

Take u = $\frac{v^{3}}{v - c}$ (sk = constant),

$\frac{du}{dv}$ = $\frac{(v - c)3v^{2} - v^{3}}{(v - c)^{2}}$= $\frac{v^{2}(2v - 3c)}{(v - c)^{2}}$

$\frac{du}{dv}$ = 0 ̃ v = 0, $\frac{3c}{2}$ is found to given a minimum for u.