Question

Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h.
\includegraphics[width=\linewidth]{latex.png} The relation between fuel consumption \( F \) (liters per 100 km) and speed \( V \) (km/h) under some constraints is given as: \[ F = \frac{V^2}{500} - \frac{V}{4} + 14. \] On the basis of the above information, answer the following questions: [(i)] Find \( F \), when \( V = 40 \, \text{km/h} \). [(ii)] Find \( \frac{dF}{dV} \). [(iii)] (a) Find the speed \( V \) for which fuel consumption \( F \) is minimum.
\hspace*{0.8cm} OR
(b) Find the quantity of fuel required to travel \( 600 \, \text{km} \) at the speed \( V \) at which \( \frac{dF}{dV} = -0.01 \).

Hint:

For optimization problems, always substitute the given value into the equation and perform the necessary arithmetic steps carefully to find the desired result.

Answer 1

(i) Find \( F \) when \( V = 40 \, \text{km/h} \):
Substitute \( V = 40 \) into the given equation: \[ F = \frac{V^2}{500} - \frac{V}{4} + 14. \] \[ F = \frac{40^2}{500} - \frac{40}{4} + 14. \] \[ F = \frac{1600}{500} - 10 + 14 = 3.2 - 10 + 14 = 7.2. \] Solution:
\[ \boxed{F = 7.2 \, \text{liters per 100 km.}} \]