Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Suppose that a tire on a truck has an outer radius of \(2.5\) feet. How many revolutions per minute does the tire make when the truck is traveling 60 miles per hour?

Step-by-Step Solution

Answer

The tire makes approximately 336 revolutions per minute.

Step 1: Convert Speed to Feet per Minute

First, we need to convert the speed from miles per hour to feet per minute. We know that there are 5280 feet in a mile and 60 minutes in an hour. Thus:\[60 \text{ miles/hour} = 60 \times 5280 \text{ feet/hour} = 316800 \text{ feet/hour}\]To convert to feet per minute:\[\frac{316800 \text{ feet/hour}}{60 \text{ minutes/hour}} = 5280 \text{ feet/minute}\]

Step 2: Calculate the Circumference of the Tire

The circumference of the tire can be calculated using the formula for the circumference of a circle, which is given by \( C = 2\pi r \). Here, \( r = 2.5 \text{ feet} \), so:\[C = 2 \times \pi \times 2.5 = 5\pi \text{ feet}\]

Step 3: Find Revolutions per Minute

Revolutions per minute (RPM) can be found by dividing the truck's speed in feet per minute by the circumference of the tire in feet:\[\text{RPM} = \frac{5280 \text{ feet/minute}}{5\pi \text{ feet/revolution}}\]Simplifying this gives:\[\text{RPM} \approx \frac{5280}{15.70796} \approx 336\]

Step 4: State the Answer

The tire makes approximately 336 revolutions per minute when the truck travels at a speed of 60 miles per hour.

Key Concepts

Circumference Calculation

When dealing with circles, one of the most important measurements is the circumference, which is the distance around the circle. Understanding how to calculate the circumference is crucial in many real-life applications, such as determining how far a tire rolls in one complete turn.
The formula to calculate circumference is: \[ C = 2\pi r \] Where:

• \( C \): Circumference
• \( \pi \): A constant roughly equal to 3.14159
• \( r \): Radius of the circle

By simply plugging the radius of the tire into this formula, you can find the circumference. For our problem, with a radius of 2.5 feet, the circumference is \(5\pi\) feet. This means that each full revolution of the tire covers \(5\pi\) feet.

Unit Conversion

Understanding unit conversion is essential when dealing with different measurement systems. It allows you to compare and compute with numbers in the same units.
In our problem, we need to convert speed from miles per hour to feet per minute. Here's how to do it:

• First, note that 1 mile is equal to 5280 feet.
• Also, recognize that there are 60 minutes in an hour.

Therefore, convert miles per hour (mph) to feet per minute (fpm) by first converting miles to feet, and then hours to minutes. For example, 60 mph is equivalent to 316,800 feet per hour. Dividing by 60 gives us 5,280 feet per minute. This conversion is crucial for calculating revolutions per minute (RPM) where measurements need to be consistent.

Speed Calculation

Speed is a measure of how quickly something moves from one place to another, typically expressed in units such as miles per hour (mph) or feet per minute (fpm).
In the context of this problem, knowing the truck's speed in fpm helps calculate how many times the tire will spin within a given timeframe. The speed we've converted earlier tells us that when a truck moves at 60 mph, it covers 5,280 feet every minute.
For practical purposes, the speed calculation bridges the distance the truck covers with how many complete revolutions the tire makes, giving us the RPM. By dividing the speed in feet per minute by the tire's circumference, you'll discover how many tire revolutions occur per minute.

Radius of a Circle

The radius of a circle is a fundamental measurement, representing the distance from the circle's center to any point on its boundary. It's crucial for many calculations related to circles.
For our exercise, the tire's radius is specified as 2.5 feet, which is essential in determining the circumference.

• Remember, a larger radius results in a larger circle and hence a larger circumference.
• Conversely, a smaller radius yields a smaller circle.

In this exercise, using the radius in combination with the circumference formula \(C = 2\pi r\) helps us understand how far the tire will travel in one complete revolution. Therefore, knowing the radius provides clarity in calculations, informing further steps such as computing revolutions per minute.