Problem 118 - A Graphical Approach to Precalculus with Limits

Question

The speedometer of a pickup truck is designed to be accurate with tires of radius 14 inches. (a) Find the number of rotations, to the nearest whole number, of a tire in 1 hour if the truck is driven \(55 \mathrm{mph}\) (b) Suppose that oversize tires of radius 16 inches are placed on the truck. If the truck is now driven for 1 hour with the speedometer reading 55 mph, how far has the truck gone? If the speed limit is \(60 \mathrm{mph}\) did the driver exceed the speed limit?

Step-by-Step Solution

Answer

(a) 39,647 rotations; (b) traveled 62.89 miles, exceeding the speed limit.

Step 1: Calculate the Circumference of the Initial Tire

The circumference of a circle (tire in this case) is given by the formula \(C = 2 \pi r\). With a radius \(r = 14\) inches, calculate the circumference: \[ C = 2 \pi \times 14 = 28\pi \text{ inches} \]

Step 2: Calculate Rotations for 55 MPH with 14-inch Tires

First, convert 55 mph to inches per hour by knowing there are 5280 feet in a mile and 12 inches in a foot: \[ 55 \text{ mph} = 55 \times 5280 \times 12 = 3,484,800 \text{ inches per hour} \] Now, calculate the number of rotations the tire would make using the formula: \[ \text{Number of rotations} = \frac{\text{Distance}}{\text{Circumference}} = \frac{3,484,800}{28\pi} \approx 39,647 \] Round to the nearest whole number: 39,647 rotations.

Step 3: Calculate the Circumference of the Oversized Tire

Calculate the circumference for the new tire radius of 16 inches using the same formula: \[ C = 2 \pi \times 16 = 32\pi \text{ inches} \]

Step 4: Calculate Actual Distance Travelled with 16-inch Tires

Using the 16-inch tires, calculate the distance with the same number of rotations calculated in Step 2: \[ \text{Actual Distance} = 39,647 \times 32\pi \approx 3,991,680 \text{ inches} \] Convert the distance to miles: \[ \frac{3,991,680}{12 \times 5280} \approx 62.89 \text{ miles} \]

Step 5: Compare Speed Against Speed Limit

The truck traveled approximately 62.89 miles in one hour. The speed limit was 60 mph, so the driver did exceed the speed limit by about 2.89 miles.

Key Concepts

Circumference Calculation

Calculating the circumference of a tire is crucial in understanding how far a vehicle moves after one complete tire rotation. To compute the circumference, you use the formula \( C = 2 \pi r \), where \( r \) is the radius of the tire. This formula essentially represents the perimeter of a circle, which is the path that the tire's outer edge travels in one full turn.
In this exercise, the initial tire has a radius of 14 inches. Plugging these values into the formula, we get:

• For the 14-inch tire: \( C = 2 \pi \times 14 = 28\pi \ \text{inches} \).
• For the oversized 16-inch tire: \( C = 2 \pi \times 16 = 32\pi \ \text{inches} \).

By understanding circumference, you can determine how many rotations a tire makes over a set distance, an essential step in calculating tire rotations.

Unit Conversion

Understanding unit conversion is key to determining how measurements change relative to each different unit system. In this problem, we need to convert speed from miles per hour (mph) to inches per hour to match the unit of tire circumference (inches).
Here's the conversion breakdown:

• 1 mile = 5280 feet
• 1 foot = 12 inches
• Therefore, 1 mile = 5280 × 12 = 63,360 inches
• 55 mph = 55 × 63,360 = 3,484,800 inches per hour

This conversion is indispensable because it allows you to calculate the number of tire rotations by comparing the total distance covered (in inches) to the tire’s circumference (also in inches). Properly managing unit conversions ensures accurate calculations regardless of the measurement system.

Speed Limit

In cases involving a speed limit, it's crucial to understand how changes in tire size can affect vehicle speed and distance traveled. The exercise demonstrates how using larger tires impacts the accuracy of a speedometer reading.
With the original 14-inch tires, the truck's speedometer reads accurately at 55 mph. However, with 16-inch tires, the truck actually travels further per rotation due to the larger circumference, resulting in the truck covering more distance than indicated by the speedometer.
The recalculated distance with oversized tires showed the truck traveled approximately 62.89 miles in one hour, exceeding the 60 mph speed limit by roughly 2.89 miles. This situation highlights why it's important for drivers to be aware of potential discrepancies in speedometer readings when changing tire sizes, as this could lead to inadvertently surpassing speed restrictions.