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The horsepower, H produced by a truck engine is proportional to the cube of the truck's speed, S.
Calculate the percentage increase in horsepower that is needed to double the speed.

The horsepower, $H$ produced by a truck engine is proportional to the cube of the truck's speed, $S$ . $\newline$ Calculate the percentage increase in horsepower that is needed to double the speed.

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Find derivatives of logarithmic functions

Full solution

Q. The horsepower,

H

produced by a truck engine is proportional to the cube of the truck's speed,

S

.

\newline

Calculate the percentage increase in horsepower that is needed to double the speed.

Initial Equation: Let's denote the initial speed of the truck as $S$ and the initial horsepower as $H$ . According to the problem, the horsepower is proportional to the cube of the truck's speed, which can be expressed as: $\newline$ $H = k \cdot S^3$ $\newline$ where $k$ is the constant of proportionality.
New Speed and Horsepower: Now, we want to find the horsepower when the speed is doubled. Let's denote the new speed as $S'$ and the new horsepower as $H'$ . Since we are doubling the speed, we have: $\newline$ $S' = 2S$
Substitute New Speed: Substituting the new speed into the horsepower equation, we get: $\newline$ $H' = k \times (S')^3$ $\newline$ $H' = k \times (2S)^3$ $\newline$ $H' = k \times 8S^3$
Substitute Initial Horsepower: Since $H = k \cdot S^3$ , we can substitute $H$ into the equation for $H'$ to find the new horsepower in terms of the initial horsepower: $\newline$ $H' = 8 \cdot H$
Calculate Percentage Increase: To find the percentage increase in horsepower, we need to calculate the ratio of the increase in horsepower to the initial horsepower and then multiply by $100$ to get the percentage: $\newline$ Percentage Increase = $\left(\frac{H' - H}{H}\right) \times 100$
Substitute $H'$ into Formula: Substitute $H' = 8H$ into the percentage increase formula: $\newline$ Percentage Increase = $\left(\frac{8H - H}{H}\right) \times 100$ $\newline$ Percentage Increase = $\left(\frac{7H}{H}\right) \times 100$ $\newline$ Percentage Increase = $7 \times 100$ $\newline$ Percentage Increase = $700\%$

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