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Question:

A metro train from Mehrauli to Gurgoan has capacity to board 900 people. The fare charged (in RS.) is defined by the function

f = ${\left(54-\frac{x}{32}\right)}^2$

where ‘x’ is the number of the people per trip. How many people per trip will make the marginal revenue equal to zero?

Explanation:

Fare per person f = ${\left(54-\frac{x}{32}\right)}^2$

∴ Total fare f₁ = x${\left(54-\frac{x}{32}\right)}^2$

∴ Marginal Revenue is the change in total fare due to change in number of people by one unit. It is given by

$\frac{d{f}_1}{dx}$ = x$\left[2\left(54-\frac{x}{32}\right)\left(-\frac{1}{32}\right)\right]$ + ${\left(54-\frac{x}{32}\right)}^2$

= $\frac{-x}{16}$$\left(54-\frac{x}{32}\right)$ + ${\left(54-\frac{x}{32}\right)}^2$

= $\left(54-\frac{x}{32}\right)$ $\left(\frac{-x}{16}+54-\frac{x}{32}\right)$

= $\left(54-\frac{x}{32}\right)$$\left(54-\frac{3x}{2}\right)$

Now  $\frac{d{f}_1}{dx}$ = 0

∴ $\left(54-\frac{x}{32}\right)$ $\left(54-\frac{3x}{2}\right)$ = 0

∴ x = 1728 or x = 576

But x ≤ 900

∴ x = 576

Hence, option (b).