Question

An ideal diameter d, in feet, of a ship's propeller is given by the formula d=ch¹/⁵r^⁻³/⁵. Here h is the horsepower of the engine driving the propeller, r is the (maximum) number of revolutions per minute of the propeller, and c is a constant. In both parts, give your answer in terms of a percentage. (Round your answers to two decimal places.)

(a) If the horsepower is increased by 30% while the number of revolutions per minute remains the same, how is the propeller diameter affected?

Answer 1

Increasing the horsepower by 30% while keeping the revolutions per minute the same leads to a 5.58% increase in the propeller's diameter, according to the provided formula.

Step-by-step explanation:

The propeller diameter formula given is d = ch1/5r-3/5. This equation indicates that the diameter d is dependent on the horsepower h and the revolutions per minute r, along with a constant c. When the horsepower is increased by 30%, represented as h → 1.30h, and r remains the same, the new diameter d' can be found by plugging the increased horsepower into the given formula: d' = c(1.30h)1/5r-3/5. This results in d' = c⋅(1.30)1/5h1/5r-3/5 = (1.30)1/5d.

To determine the percentage increase in diameter, we calculate (1.30)1/5 and subtract 1, then multiply by 100 to convert to a percentage. Numerically, this works out to approximately a 5.58% increase in the diameter of the propeller: ((1.30)1/5 - 1) × 100 ≈ 5.58%.

Hence, if the horsepower increases by 30% with the number of revolutions per minute staying constant, the diameter of the ship's propeller would increase by about 5.58%.