Problem 92 - 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 \(\mathrm{mi} / \mathrm{h}\) 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 when traveling between 40 mi/h and 50 mi/h.

Step 1: Understand the Equation

The given formula describes how gas mileage \( g \) depends on the vehicle's speed \( v \). Specifically, \( g = 10 + 0.9v - 0.01v^{2} \). We are tasked with finding the range of speeds at which \( g \) is at least 30 mi/gal.

Step 2: Set Up the Inequality

We want the mileage \( g \) to be 30 or greater. Therefore, we set up the inequality: \[ 10 + 0.9v - 0.01v^{2} \geq 30 \].

Step 3: Simplify the Inequality

First, subtract 30 from both sides of the inequality to rearrange it: \[ 10 + 0.9v - 0.01v^{2} - 30 \geq 0 \]. This simplifies to: \[ -0.01v^{2} + 0.9v - 20 \geq 0 \].

Step 4: Find the Roots of the Quadratic Equation

Rewrite the inequality as a quadratic equation to find its roots: \[ -0.01v^{2} + 0.9v - 20 = 0 \]. Using the quadratic formula \( v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -0.01 \), \( b = 0.9 \), and \( c = -20 \).

Step 5: Apply the Quadratic Formula

Calculate the discriminant \( b^{2} - 4ac \): \[ (0.9)^{2} - 4(-0.01)(-20) = 0.81 - 0.8 = 0.01 \]. Then use the quadratic formula: \[ v = \frac{-0.9 \pm \sqrt{0.01}}{-0.02} \]. This gives: \[ v = \frac{-0.9 \pm 0.1}{-0.02} \].

Step 6: Solve for v

Calculate both solutions:- First solution: \[ v = \frac{-0.9 + 0.1}{-0.02} = \frac{-0.8}{-0.02} = 40 \].- Second solution: \[ v = \frac{-0.9 - 0.1}{-0.02} = \frac{-1.0}{-0.02} = 50 \].

Step 7: Determine the Interval

From the roots calculated, the inequality \( -0.01v^{2} + 0.9v - 20 \geq 0 \) holds in the interval between 40 and 50. This interval represents the speeds \( v \) at which the gas mileage is at least 30 mi/gal.

Step 8: Confirm Range Within Given Limits

The problem specifies that \( v \) is between 10 and 75 mi/h. The found interval \( 40 \leq v \leq 50 \) is within these limits, confirming that no adjustments are needed.

Key Concepts

Gas Mileage Calculation

Gas mileage calculation is an essential concept for understanding how efficiently a vehicle consumes fuel. In many cases, the relationship between speed and mileage can be expressed through mathematical formulas. For this particular exercise, the gas mileage \( g \) is determined by the equation \( g = 10 + 0.9v - 0.01v^{2} \), where \( v \) is the speed of the vehicle in miles per hour. This equation shows that gas mileage depends not only on the speed itself but also on the square of the speed.
This relationship is quadratic, which means that the mileage initially increases with speed, reaches a peak, and then starts to decrease. It's crucial to identify the optimal speed range where mileage stays above a certain value—in this case, 30 mi/gal. By understanding and applying these principles, one can determine the most fuel-efficient way to drive, helping to save on fuel costs and reduce environmental impact.
Maximizing gas mileage is not only economical but also ecological, as better mileage leads to lower emissions over time.

Vehicle Speed Optimization

Optimizing vehicle speed is all about finding that sweet spot where your car operates most efficiently. Every vehicle has a speed range where it reaches optimal fuel efficiency. Driving too fast or too slow can lead to inefficient fuel consumption.

• Look at how vehicle mileage changes with speed using a mathematical model.
• For the optimal range, drivers should aim to keep their speed such that their mileage stays at or above target values, such as 30 mi/gal.
• Consider external factors like traffic conditions, road types, and vehicle load, which can all influence optimal speeds.

In our exercise, we derived that the optimal speed range for the best mileage is between 40 mph to 50 mph. Staying within this range ensures that vehicles run at peak efficiency within the given constraints.
Optimizing vehicle speed can significantly impact a driver's gas mileage, making a notable difference in fuel economy and overall driving expenses.

Quadratic Formula Application

The quadratic formula is a mathematical tool that allows us to solve quadratic equations, which are equations in the form \( ax^2 + bx + c = 0 \). These types of equations appear often in mathematical modeling and scientific calculations, including gas mileage prediction.

• The formula \( v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is applied to find the roots of a quadratic equation.
• Discriminant \( b^2 - 4ac \) helps determine the nature and number of roots.
• In our exercise, the quadratic formula was used to calculate speeds where gas mileage is at least 30 mi/gal.

By solving the equation \( -0.01v^2 + 0.9v - 20 = 0 \) using the quadratic formula, we find the precise speed range where the vehicle's mileage meets the target.
This powerful formula is not limited to fuel efficiency problems; it is also used widely in physics, engineering, economics, and various fields to find points where a quadratic relationship holds. Understanding its application is a valuable skill in analyzing and solving real-world problems.