Problem 243 - Calculus Volume 1

Question

Although some engines are more efficient at given horsepower than others, on average, fuel efficiency decreases with horsepower at a rate of 1\(/ 25\) mpg/ horsepower. If a typical 50 -horsepower engine has an average fuel efficiency of 32 \(\mathrm{mpg}\) , what is the average fuel efficiency of an engine with the following horsepower: 150 , \(300,450 ?\)

Step-by-Step Solution

Answer

For 150 hp: 28 mpg; for 300 hp: 22 mpg; for 450 hp: 16 mpg.

Step 1: Understanding the Problem

We need to calculate the average fuel efficiencies for engines with 150, 300, and 450 horsepower. The given relationship is that fuel efficiency decreases at a rate of \( \frac{1}{25} \) mpg per horsepower.

Step 2: Creating the Formula

If the baseline is a 50-horsepower engine, which has an average fuel efficiency of 32 mpg, then for any horsepower \( h \), the average fuel efficiency \( f(h) \) can be calculated using the formula: \[ f(h) = 32 - \left( h - 50 \right) \times \frac{1}{25} \] Here, \( h - 50 \) is the change in horsepower from 50, and \( \frac{1}{25} \) is the rate of decrease in mpg per horsepower.

Step 3: Calculating for 150 horsepower

Substitute \( h = 150 \) into the formula: \[ f(150) = 32 - \left( 150 - 50 \right) \times \frac{1}{25} \] Calculate the change in efficiency:\[ \Delta f = 100 \times \frac{1}{25} = 4 \] Then, subtract from the baseline efficiency:\[ f(150) = 32 - 4 = 28 \] Thus, the average fuel efficiency is 28 mpg.

Step 4: Calculating for 300 horsepower

Substitute \( h = 300 \) into the formula: \[ f(300) = 32 - \left( 300 - 50 \right) \times \frac{1}{25} \] Calculate the change in efficiency:\[ \Delta f = 250 \times \frac{1}{25} = 10 \] Then, subtract from the baseline efficiency:\[ f(300) = 32 - 10 = 22 \] Thus, the average fuel efficiency is 22 mpg.

Step 5: Calculating for 450 horsepower

Substitute \( h = 450 \) into the formula: \[ f(450) = 32 - \left( 450 - 50 \right) \times \frac{1}{25} \] Calculate the change in efficiency:\[ \Delta f = 400 \times \frac{1}{25} = 16 \] Then, subtract from the baseline efficiency:\[ f(450) = 32 - 16 = 16 \] Thus, the average fuel efficiency is 16 mpg.

Key Concepts

Fuel Efficiency

Fuel efficiency reflects how well an engine converts fuel into renewable motion, measured in miles per gallon (mpg). In general, more fuel-efficient engines use less fuel to travel the same distance compared to less efficient ones. In our context, we're examining how horsepower influences this efficiency.

Fuel efficiency can be impacted by various factors, such as engine design, weight of the vehicle, and the power output, namely horsepower. As seen in the exercise, when horsepower increases, the fuel efficiency tends to decrease. This is because higher horsepower usually allows a vehicle to generate more speed and power, but it comes at the cost of burning more fuel.

To quantify this relationship, we use the formula given, which helps predict the mpg for different horsepower values. The formula takes into account the decrease per horsepower, demonstrating how calculus helps model real-world situations.

Rate of Change

The rate of change is an essential concept in calculus that helps us understand how one quantity affects another. In this exercise, the rate of change is the decrease in fuel efficiency as horsepower increases. Specifically, for each increase of one horsepower, the fuel efficiency decreases by a set amount, 1/25 mpg.

This consistent decrease rate can be thought of as a slope in a linear relationship between horsepower and fuel efficiency. The formula we use is derived by calculating how much the efficiency changes based on horsepower, minus the baseline efficiency of a 50-horsepower engine.

Calculating the rate of change allows us to predict future values and understand underlying trends. It’s like creating a roadmap that outlines how modifications in a car's power output affect fuel consumption, a valuable insight when designing efficient vehicles.

Horsepower

Horsepower is a unit of measure that quantifies the power output of an engine. It helps determine how fast a vehicle can accelerate and climb hills. In this exercise, horsepower serves as an independent variable affecting fuel efficiency.

A 50-horsepower engine demonstrates decent fuel efficiency at 32 mpg. However, as we increase horsepower to values such as 150, 300, or 450, the efficiency drops significantly. This decrease correlates directly to the additional energy required to maintain or increase performance at higher horsepower levels, leading to increased fuel consumption.

By understanding horsepower, manufacturers and engineers can balance the power output with fuel efficiency, seeking a harmonious blend between speed or performance and economical fuel use. Thus, not every engine is about power; sometimes, efficiency holds the key.