Question

The radius of each wheel of a car is 15 inches. If the wheels are turning at the rate of 3 revolutions per second, how fast is the car moving? Express your answer in inches per second and in miles per hour.

Short answer

The car is moving at \90\pi \text{ inches per second} \(approx. 282.7 \text{ inches/second} \ and \approx 16.07 \text{ miles/hour}\.

Step-by-step solution

Find the Circumference of the Wheel

The circumference of a circle is given by the formula \(C = 2\pi r\), where \(r\) is the radius. Here, the radius \(r = 15 \text{ inches}\). Therefore, the circumference \(C\) can be calculated as \C = 2\pi \times 15 = 30\pi \text{ inches}\.

Calculate the Distance the Wheel Travels per Second

If the wheel makes 3 revolutions per second, the distance it covers in one second is \(3 \times C\). Using the circumference found in Step 1, the distance is \3 \times 30\pi = 90\pi \text{ inches per second}\.

Convert Inches per Second to Miles per Hour

First, convert inches per second to inches per hour by multiplying by 3600 (the number of seconds in an hour). So, \(90\pi \text{ inches/second} \times 3600 \text{ seconds/hour} = 324000\pi \text{ inches/hour}\). Next, convert inches to miles by noting that there are 63,360 inches in a mile: \[\frac{324000 \pi}{63360} \text{ miles/hour} \approx 16.07 \text{ miles/hour}\]

Key concepts

Circumference Calculation

To find the distance around a circular wheel, we calculate its circumference. The formula for the circumference of a circle is: \( C = 2\pi r \).
The symbol \( C \) represents the circumference, \( \pi \) (pi) is approximately 3.14159, and \( r \) is the radius of the circle. For this exercise, the radius \( r \) is given as 15 inches.
So, we substitute 15 for \( r \) in the formula: \[ C = 2\pi \times 15 = 30\pi \text{ inches} \].
This means the circumference of one wheel is \( 30\pi \approx 94.25 \text{ inches} \). This value tells us the distance one wheel covers in one complete revolution. It's an essential step for calculating how far the car moves for each wheel turn.

Unit Conversion

Sometimes, we need to change units to make comparisons easier or to provide solutions in more familiar terms. Here, we need to convert inches per second to miles per hour.
First, convert inches per second to inches per hour. We do this because there are 3600 seconds in an hour. For instance, if the wheel covers \( 90\pi \text{ inches/second} \), we multiply by 3600:\[ 90\pi \text{ inches/second} \times 3600 \text{ seconds/hour} = 324000\pi \text{ inches/hour} \].
Now, we convert inches to miles, knowing that there are 63,360 inches in a mile. This step helps in converting large values into more understandable units like miles per hour. Divide the total inches by 63,360:\[ \frac{324000\pi}{63360} \approx 16.07 \text{ miles/hour} \].
This means the car is moving at approximately 16.07 miles per hour either way you want to represent it.

Speed Calculation

Speed calculation helps determine how fast an object is moving. In this case, we first find out how far the car travels per second because the wheels are turning at 3 revolutions per second.
The circumference of one wheel is \( 30\pi \text{ inches} \). With 3 revolutions per second, the car moves: \[ 3 \times 30\pi = 90\pi \text{ inches/second} \approx 282.74\text{ inches/second} \].
We then convert this speed to miles per hour. There are 3600 seconds in an hour and 63,360 inches in a mile. So, multiply 90\pi by 3600 and divide by 63,360:\[ 90\pi \times 3600 / 63,360 \text{ miles/hour} \approx 16.07 \text{ miles/hour} \].
Understanding speed involves multiple steps, but breaking it down with the help of these basic conversions makes it straightforward and graspable.