For a steamer the consumption of petrol (per hour) varies as...

For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in $\mathrm{km}$ ). If the speed of the water current is steady at $\mathrm{C} \mathrm{km} / \mathrm{h}$, then the most economical speed of the steamer going against the current will be (a) $1.25 \mathrm{C}$ (b) $1.5 \mathrm{C}$ (c) $1.75 \mathrm{C}$ (d) $2 C$

For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in $\mathrm{km}$ ). If the speed of the water current is steady at $\mathrm{C} \mathrm{km} / \mathrm{h}$, then the most economical speed of the steamer going against the current will be (a) $1.25 \mathrm{C}$
(b) $1.5 \mathrm{C}$
(c) $1.75 \mathrm{C}$
(d) $2 C$

Key Concepts

Calculus in Optimization

Calculus, particularly differentiation, is a tool for finding extreme values (minimums or maximums) of functions. By modeling fuel consumption as a function of speed and then differentiating with respect to speed, one can set the derivative equal to zero to find the optimal speed that minimizes consumption. This approach provides a systematic way to solve optimization problems in various physical and economic contexts.

Direct Proportionality to a Power

This concept deals with relationships where one quantity varies as a power of another. In the context of fuel consumption, the idea that consumption varies as the cube of the steamer's speed illustrates that even small increases in speed can result in significantly larger increases in fuel usage. This type of relationship is key in understanding how changes in one variable exponentially affect another.

Relative Motion

Relative motion concerns the speed of an object with respect to another moving medium. In problems involving currents or winds, the effective speed of a vessel is determined by the difference between its own speed and the speed of the current. Understanding relative motion is critical when assessing the true performance or efficiency of vehicles moving through a fluid medium.

Optimization

Optimization involves finding the most efficient or economical value of a variable under given constraints. In this context, it means determining the speed that minimizes fuel consumption per unit distance traveled against a current. This requires a balance between the increase in consumption due to higher speed and the benefits of overcoming opposing motion more quickly.

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