Problem 84 - Algebra and Trigonometry

Question

Gas Mileage The gas mileage \(g\) (measured in mi/gal) for a particular vehicle, driven at \(v\) mi/h, is given by the formula \(g=10+0.9 v-0.01 v^{2},\) as long as \(v\) is between 10 mi/h and 75 \(\mathrm{mi} / \mathrm{h}\) . For what range of speeds is the vehicle's mileage 30 \(\mathrm{mi} / \mathrm{gal}\) or better?

Step-by-Step Solution

Answer

The vehicle's mileage is 30 mi/gal or better between 40 and 50 mi/h.

Step 1: Understand the Problem

We need to find the range of speeds for which gas mileage \( g \) is at least 30 mi/gal, using the given function \( g = 10 + 0.9v - 0.01v^2 \). The speed \( v \) must be between 10 mi/h and 75 mi/h.

Step 2: Set Up the Inequality

To find the range of speeds where the gas mileage is at least 30 mi/gal, we need to solve the inequality \( 10 + 0.9v - 0.01v^2 \geq 30 \).

Step 3: Rearrange the Inequality

Subtract 30 from both sides to set the inequality to zero: \( 10 + 0.9v - 0.01v^2 - 30 \geq 0 \), which simplifies to \( -0.01v^2 + 0.9v - 20 \geq 0 \).

Step 4: Solve the Quadratic Inequality

Solve the quadratic equation \( -0.01v^2 + 0.9v - 20 = 0 \) using the quadratic formula. The formula is \( v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = -0.01 \), \( b = 0.9 \), and \( c = -20 \).

Step 5: Calculate the Discriminant

Calculate the discriminant \( \Delta = b^2 - 4ac = (0.9)^2 - 4(-0.01)(-20) = 0.81 - 0.8 = 0.01 \). Since \( \Delta > 0 \), the equation has two distinct solutions.

Step 6: Find the Roots

Substitute into the quadratic formula: \( v = \frac{-0.9 \pm \sqrt{0.01}}{-0.02} \). Calculate the roots: \( v = \frac{-0.9 + 0.1}{-0.02} = 40 \) and \( v = \frac{-0.9 - 0.1}{-0.02} = 50 \).

Step 7: Determine the Range of Speeds

The solutions \( v = 40 \) and \( v = 50 \) split the number line into intervals. Since the quadratic function opens downwards (\( a < 0 \)), \( g \geq 30 \) in the interval \( 40 \leq v \leq 50 \), ensuring the vehicle's mileage is 30 mi/gal or better.

Key Concepts

Gas Mileage

Gas mileage refers to how efficiently a vehicle uses fuel, typically measured in miles per gallon (mi/gal). Understanding gas mileage can help determine the most economical speed range for driving. In this exercise, we are given a mathematical function of gas mileage expressed as:

• \(g = 10 + 0.9v - 0.01v^2\)

The variable \(v\) represents the speed of the vehicle in miles per hour (mi/h). The function tells us that gas mileage depends on speed, and we are tasked with finding speeds that yield good fuel economy.
For gas mileage to reach a certain level, such as 30 mi/gal or more, it is necessary to solve the inequality derived from this relationship. Knowing the mileage efficiency helps drivers optimize their trips for fuel use, saving both money and energy in the process.
Speeds between 40 mi/h and 50 mi/h provide the best efficiency according to the calculations, so driving within this range can maximize gas mileage.

Quadratic Formula

The quadratic formula is a powerful tool used to solve equations of the form \( ax^2 + bx + c = 0 \). Understanding this formula is essential, especially when dealing with quadratic equations like the one in our exercise:

• \(-0.01v^2 + 0.9v - 20 = 0\)

Here, \(a = -0.01\), \(b = 0.9\), and \(c = -20\). The quadratic formula is given as:

• \(v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

By substituting the values for \(a\), \(b\), and \(c\), we calculate the roots or solutions of the equation that tell us at which speeds the gas mileage is exactly 30 mi/gal.
In this exercise, the discriminant \( b^2 - 4ac \) was calculated to find whether real solutions exist. A positive discriminant, \(0.01\) in this case, indicates two distinct real roots. These roots help in determining the specific speeds that keep the vehicle's mileage at or above your given target. Mastering this formula is crucial to solving similar real-world problems effectively.

Range of Speeds

Identifying the range of speeds that optimize fuel efficiency involves understanding both the quadratic expression and its solutions. With the quadratic formula solved, the roots obtained were \( v = 40 \) and \( v = 50 \). These speeds effectively mark the boundaries for the vehicle achieving at least 30 mi/gal.
In simpler terms, these boundaries help to establish intervals. In this case, since the parabola opens downwards (indicated by \(a < 0 \)), the mileage is greatest between the roots. Therefore,

• Speeds from \(40\) mi/h to \(50\) mi/h give the desired gas mileage.

Driving at speeds within this range will ensure you maintain a fuel efficiency of 30 mi/gal or better. This analysis teaches us how changes in speed can affect fuel efficiency, providing car owners with strategies to get the most out of their gasoline. Determining these ranges not only promotes wise fuel consumption but also contributes to more environmentally savvy driving habits.