Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

The number of miles \(M\) that a certain compact car can travel on 1 gallon of gasoline is related to its speed \(v\) (in mi/hr) by $$ M=-\frac{1}{30} v^{2}+\frac{5}{2} v \text { for } 0

Step-by-Step Solution

Answer

The speeds are between 30 mi/hr and 45 mi/hr.

Step 1: Set Up the Inequality

We want to find the speeds where the miles per gallon, \( M \), is at least 45. Therefore, we set up the inequality: \(-\frac{1}{30}v^2 + \frac{5}{2}v \geq 45\).

Step 2: Clear the Fraction

Multiply the entire inequality by 30 to eliminate the fraction: \(-v^2 + 75v \geq 1350\).

Step 3: Rearrange the Inequality

Reorganize the inequality to have all terms on one side:\(-v^2 + 75v - 1350 \geq 0\).

Step 4: Solve the Quadratic Inequality

Shift terms to make it easier to solve:\(v^2 - 75v + 1350 \leq 0\).Solve the equation \(v^2 - 75v + 1350 = 0\) using the quadratic formula: \( v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \),where \( a = 1, b = -75, c = 1350 \).

Step 5: Apply the Quadratic Formula

Plug into the formula:\( v = \frac{-(-75) \pm \sqrt{(-75)^2 - 4(1)(1350)}}{2(1)} \).Calculate the discriminant: \ \( (-75)^2 = 5625 \) and \ \( 4 \times 1350 = 5400 \). Thus, \ \( \sqrt{5625 - 5400} = \sqrt{225} = 15 \).

Step 6: Compute the Roots

Find the roots:\( v = \frac{75 \pm 15}{2} \).The solutions are \( v = \frac{75 + 15}{2} = 45 \) and \( v = \frac{75 - 15}{2} = 30 \).

Step 7: Determine the Valid Range

The quadratic inequality \(v^2 - 75v + 1350 \leq 0\) signifies that the values of \(v\) between the roots satisfy the inequality. Therefore, \(30 \leq v \leq 45\).

Key Concepts

Miles per Gallon

"Miles per gallon" (MPG) is a measure of how far a vehicle can travel on a single gallon of gasoline. It is an indicator of fuel efficiency, where higher values imply better fuel economy. In the context of the given quadratic inequality, MPG is represented by the variable \( M \). Here, \( M \) changes with speed \( v \), which is expressed in miles per hour (mph).

To understand the relationship between speed and MPG, we are given the equation:

• \( M = -\frac{1}{30} v^2 + \frac{5}{2} v \)

This shows a parabolic relationship, typical of quadratic functions, where the MPG is impacted by the square of the speed. It's important to see that the efficiency not only increases with speed initially but will eventually decrease as speed continues to increase—giving that upside-down parabola shape.

Car Speed Optimization

Finding the optimal speed for a vehicle enables drivers to maximize MPG, striking the best balance between fuel consumption and desired speed. When we speak about optimizing car speed in this context, we are seeking a range of speeds where the MPG is at least 45.

The original exercise involved setting up an inequality to analyze speeds where the vehicle achieves this level of efficiency. Evaluating this will provide drivers with the optimal speed range to achieve the desired fuel efficiency goals. By solving the inequality:

• \( -\frac{1}{30}v^2 + \frac{5}{2}v \geq 45 \),

we identified the critical points, or roots, by converting the problem to a manageable quadratic equation, leading to discovery of the optimal speed range.

Quadratic Formula

The quadratic formula is a tool used to solve quadratic equations set in the standard form \( ax^2 + bx + c = 0 \). It provides the solutions (or roots) using the formula:

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

In our exercise, the equation \( v^2 - 75v + 1350 = 0 \) was utilized to find the speeds where the MPG is 45 or more. Here,

• \( a = 1 \)
• \( b = -75 \)
• \( c = 1350 \)

By calculating the discriminant \( b^2 - 4ac \), the formula helped identify the exact points \( v = 30 \) and \( v = 45 \), which are the speeds that define the valid operational range for optimal driving efficiency.

Efficiency Analysis

Efficiency analysis involves evaluating the relationship between speed and fuel consumption to optimize performance. For the given scenario, after solving the quadratic inequality, the analysis revealed that the most efficient range for high MPG is between 30 and 45 mph.

Unlike linear relationships, quadratic inequalities, like the one involving car efficiency, portray a more complex behavior where efficiency increases and later decreases as speed increases. This requires a careful inspection of speed ranges rather than focusing solely on a single optimal point.

The analysis showed that any speed within this range maximizes the car's MPG to be at least 45, ensuring drivers maintain an efficient ride while minimizing fuel usage.