The horsepower H required to overcome wind drag on a certain...

The horsepower $H$ required to overcome wind drag on a certain automobile is approximated by $$H(x)=0.002 x^{2}+0.005 x-0.029, \quad 10 \leq x \leq 100$$ where $x$ is the speed of the car in miles per hour. (a) Use a graphing utility to graph $H$ (b) Rewrite the power function so that $x$ represents the speed in kilometers per hour. [Find $H(x / 1.6) . ]$

The horsepower $H$ required to overcome wind drag on a certain automobile is approximated by
$$H(x)=0.002 x^{2}+0.005 x-0.029, \quad 10 \leq x \leq 100$$
where $x$ is the speed of the car in miles per hour.
(a) Use a graphing utility to graph $H$
(b) Rewrite the power function so that $x$ represents the speed in kilometers per hour. [Find $H(x / 1.6) . ]$

Key Concepts

Function Composition and Transformation

Function composition is the process of applying one function to the results of another. When rewriting a function to account for a change in the input variable—such as converting units—you substitute the new variable expression into the original function. This transformation adjusts the input to reflect the new context (in this case, changing speed units from miles per hour to kilometers per hour).

Graphing Functions

Graphing functions involves plotting the relationship defined by a mathematical expression to visualize how the output values depend on the input. In this context, using a graphing utility allows you to see the behavior of the horsepower function over its defined domain, which aids in understanding trends, turning points, and other important characteristics like intercepts.

Unit Conversion in Function Applications

Unit conversion in function applications is the process of substituting one unit measure with another by using appropriate conversion factors. Here, since the speed was originally in miles per hour and needs to be expressed in kilometers per hour (knowing that 1 mile is equivalent to approximately 1.6 kilometers), the conversion is performed by replacing the input variable with its equivalent expression, ensuring that the function properly reflects the new unit of measurement.

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