Question

The force required to row a boat at constant velocity is proportional to square of its speed. If a speed v km/h requires $ 4 $ kW, how much power does a speed of $ 2v $ km/h require?
(A) $ 32\;{\text{KW}} $
(B) $ 8\;{\text{KW}} $
(C) $ 24\;{\text{KW}} $
(D) $ 16\;{\text{KW}} $

Answer 1

Hint : Convert the given word statement in the form of the mathematical expression. And then derive the relation for the two different velocities and then place the given values in the equation and then simplify the equation for the required value.

Complete Step By Step Answer:
force required to row a boat at constant velocity is proportional to square of its speed, convert the given statement in the mathematical expression.
 $ F\alpha {v^2} $
Also, power $ P = F \times v $
Using both the equations-
 $ {v^2} \times v = {v^3} $
Therefore,
 $ \dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{{v_1}^3}}{{{v_2}^3}} $
Place the given values –
 $ \dfrac{4}{{{P_2}}} = \dfrac{{{1^3}}}{{{2^3}}} $
Perform cross multiplication, where the numerator of one side is applied to the denominator of the opposite side and vice-versa.
 $ {P_2} = 4 \times 8 $
Simplify the above expression finding the product of the terms.
 $ {P_2} = 32KW $
From the given multiple choices, the option A is the correct answer.

Note :
Read the given equation twice and frame the ratio properly. Since it is the first and basic step for the correct solution. Be good in multiples and division. Since it is the most important fundamental to solve and simplify any mathematical expression. Remember to multiply till twenty numbers for an accurate and efficient solution.