Question

PROBLEM SOLVING For a drag race car with a total weight of 3500 pounds, the speed \(s\) (in miles per hour) at the end of a race can be modeled by \(s=14.8 \sqrt[3]{p}\), where \(p\) is the power (in horsepower). Graph the function. a. Determine the power of a 3500 -pound car that reaches a speed of 200 miles per hour. b. What is the average rate of change in speed as the power changes from 1000 horsepower to 1500 horsepower?

Short answer

The power of the car that reaches a speed of 200 miles per hour can be found by cubing \(\frac{200}{14.8}\). The graph of the function is a curve that increases as \(p^{1/3}\). The average rate of change in speed as the power changes from 1000 to 1500 horsepower is given by \(\frac{s(1500) - s(1000)}{500}\), where \(s(p) = 14.8 \sqrt[3]{p}\)

Step-by-step solution

Setting Up the Equation

Given that the speed \(s\) of a 3500 -pound car is modeled by \(s = 14.8 \sqrt[3]{p}\), where \(p\) is the power in horsepower. For the speed of 200 miles per hour, set the equation as \(200 = 14.8 \sqrt[3]{p}\)

Solving for Power

Isolate \(p\) on one side. To do this, divide both sides by 14.8, giving \( \sqrt[3]{p} = \frac{200}{14.8}\), then cube both sides to find \(p = \left(\frac{200}{14.8}\right)^3\)

Graphing the Function

Plot the function \(s = 14.8\sqrt[3]{p}\) using a graphing tool. The horizontal axis represents power \(p\) in horsepower, and the vertical axis represents speed \(s\) in miles per hour.

Step 4:Calculating the Average Rate of Change

The average rate of change in speed as the power changes from 1000 horsepower to 1500 horsepower is given by \(\frac{\Delta s}{\Delta p} = \frac{s(1500) - s(1000)}{1500 - 1000}\), where \(s(p) = 14.8 \sqrt[3]{p}\)

Key concepts

Graphing Functions

Graphing functions is a crucial skill in algebra that allows us to visualize the relationship between two variables. In our exercise, the speed of the car (\(s\)) is a function of the power (\(p\)). To graph this function, we need to plot several points where we calculate speed for various values of power, and then draw a smooth curve through these points. Using graphing software or a graphing calculator can simplify this process. The graph of this function would show us how speed increases with the increase in power, and it will generally have a rising curve, given that speed increases as the cube root of power increases.

Students often struggle with plotting non-linear functions like cube roots. A tip for graphing such functions is to first calculate the function's values at regular intervals—for example, every 100 horsepower—and then use those points to sketch the curve. Understanding how the function behaves (increases or decreases, concavity, etc.) helps in accurately representing it on a graph.

Solving Cubic Roots

To solve cubic roots in algebra, we are often looking for the value that, when cubed, gives us a particular number. In our equation, we aim to isolate the cube root of power (\(p\)) on one side to solve for power. Remember, the cube root of a number is the value that, when raised to the third power, gives the original number. In this case, after isolating the cube root, we'd cube both sides to find the value of power that would give us the speed of 200 mph for the car.

In our exercise, cubing both sides of the equation \( \(\sqrt[3]{p} = \frac{200}{14.8}\) \) gets us to the value of \(p\). This operation reverses the cube root and provides us with the exact power needed. Working with cubic roots can sometimes be tricky, so double-checking calculations using a calculator is often a wise choice. Moreover, understanding this process is vital for solving real-world problems involving volumes and finding roots of algebraic expressions.

Average Rate of Change

The average rate of change is a concept that helps us understand how one variable changes on average with respect to another variable over a specified interval. In algebra, it is similar to finding the slope of the secant line between two points on a curve. In the context of our given problem, we're interested in knowing how much the speed of the car changes on average for the increase in horsepower from 1000 to 1500.

To calculate this, we need to find the difference in speed at these two power levels and divide by the change in power. Using the function \(s(p) = 14.8 \sqrt[3]{p}\), we can substitute the power values to find the speeds and then apply the formula for the average rate of change. Remember, this average rate gives us an overall idea, but the rate can vary at different points along the power spectrum.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a particular value. In our exercise, the expression \(s = 14.8 \sqrt[3]{p}\) is an algebraic expression that models the speed of a car based on its power. When working with algebraic expressions, it’s important to understand how to manipulate them in accordance with algebraic rules to isolate variables and solve equations.

To work effectively with algebraic expressions, one should be proficient in operations such as expansion, factoring, and simplifying expressions. Also, when dealing with equations involving roots, as in this exercise, it is essential to be comfortable with applying inverse operations to eliminate the roots and isolate the desired variable. Strengthening these skills is key to solving algebra problems and is foundational for advanced mathematical studies.