Question

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 \(H(x / 1.6) .\)

Short answer

The graph of function \(H\) shows that the horsepower required exponentially increases as the speed in miles per hour increases. When you convert speed from mph to kph in the function, you get \(H(x / 1.6)=0.002 (x / 1.6)^{2}+0.005 (x / 1.6)-0.029\).

Step-by-step solution

Graphing the function

We're given the function \(H(x)=0.002 x^{2}+0.005 x-0.029\). This function is a quadratic function. Get a graphing utility and enter the given function to plot the graph.

Understanding the unit conversion

We're given that the speed \(x\) in the function \(H(x)\) is in miles per hour. But we are asked to convert \(x\) to represent the speed in kilometers per hour. We know the conversion factor, which is 1 mile = 1.6 kilometers.

Converting the units in the function

Replacing \(x\) with \(x/1.6\) in the original function converts the speed units from miles/hour to kilometers/hour. So the new function becomes \(H(x / 1.6)=0.002 (x / 1.6)^{2}+0.005 (x / 1.6)-0.029\).

Key concepts

Graphing Utility

Understanding quadratic functions such as horsepower drag calculations on a vehicle can be greatly enhanced by visual aids. This is where a graphing utility comes into play. A graphing utility, such as a graphing calculator or software like Desmos, allows students to input the quadratic equation and instantly see the parabolic curve that represents the relationship between the car's speed and the required horsepower to overcome wind drag.

For example, in the given exercise, entering the function \(H(x)=0.002 x^2 + 0.005 x - 0.029\) into a graphing utility will show how the horsepower needs increase as the speed of the car increases. By graphing, students can visually interpret the rate of change and the impact of speed on horsepower. It's important to adjust the scale to include the interval \(10 \leq x \leq 100\) for an accurate representation, ensuring that students can analyze the relevant speed range.

Unit Conversion

The process of unit conversion is essential in many scientific and mathematical problems, particularly when they involve comparisons or calculations with different measurement systems.

In the given exercise, converting miles per hour to kilometers per hour is necessary to rewrite the function in a different unit. The conversion factor between miles and kilometers is a fundamental value; 1 mile is approximately equal to 1.6 kilometers. To convert a speed from miles per hour to kilometers per hour, one multiplies the speed by 1.6. Conversely, to adjust the function, we divide the speed variable by 1.6 to represent it in kilometers per hour instead.

Unit conversion exercises like these equip students with practical skills to tackle real-world problems where different units of measurement are often used interchangeably. Understanding and applying the correct conversion factor is crucial for accuracy in science, engineering, and daily life.

Power Function

In algebra, a power function is of the form \(f(x) = ax^n\), where \(a\) is a coefficient and \(n\) is a positive or negative integer or rational number. The function \(H(x)=0.002 x^2 + 0.005 x - 0.029\) from the exercise is a type of power function, specifically a quadratic function, because the highest power of \(x\) is 2.

Understanding the characteristics of power functions, such as their symmetry, intercepts, and end behavior, is important for grasping the relationship they describe. In quadratic functions, the graph opens upward if the coefficient of \(x^2\) is positive, as in the exercise problem, or downward if it's negative. These concepts are not only vital in pure mathematics but also have applications in physics, engineering, and economics, where they model various phenomena, including projectile motion and cost-profit analysis.