Aptitude Discussion

Q. |
A Metro train from Mumbai to Gurgaon has capacity to board 900 people. The fare charged (in Rs) is defined by the function: \(f=\left(54-\dfrac{x}{32}\right)^2\) where $x$ is the number of the people per trip. How many people per trip will make the marginal revenue equal to zero? |

✖ A. |
1728 |

✔ B. |
576 |

✖ C. |
484 |

✖ D. |
364 |

**Solution:**

Option(**B**) is correct

The number of people per trip is $x$.

The fare for each person

\(=\left(54-\dfrac{x}{32}\right)^2\)

The revenue from $x$ people

\(=x\left(54-\dfrac{x}{32}\right)^2\)

Let

\(g(x)=x\left(54-\dfrac{x}{32}\right)^2\)

When the revenue is the maximum, the marginal revenue equal to zero.

\(g'(x)=\left(54-\dfrac{x}{32}\right)^2-\dfrac{2x}{32}\left(54-\dfrac{x}{32}\right)\)

For the maximum value of $g(x)$:

\(g'(x)=0\)

\(\Rightarrow \left(54-\dfrac{x}{32}\right)=0\) and

\( \left(54-\dfrac{3x}{32}\right)=0\)

$\Rightarrow x=1728$, $x=576$

Given the capacity of the train is 900, the number of people per trip = **576**

**Anurag**

*()
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How to Calculate marginal revenue ?