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Chapter 6: Problem 70

Wheel Revolutions How many revolutions will a car wheel of diameter 30 in. make as the car travels a distance of one mile?

Short Answer

Expert verified
The wheel makes about 672 revolutions when the car travels a mile.

Step by step solution

01

Understand the Problem

We want to find out how many times a wheel with a diameter of 30 inches will turn when a car travels a distance of one mile.
02

Calculate Wheel Circumference

The circumference of a wheel is the distance it covers in one full revolution. The formula for the circumference of a circle is \( C = \pi \times d \), where \( d \) is the diameter. Here, \( d = 30 \) inches, so \( C = \pi \times 30 = 30\pi \) inches.
03

Convert Mile to Inches

To compare the wheel's circumference to the distance traveled, we need both in the same units. One mile is equal to 5280 feet, and one foot is 12 inches, so one mile is \( 5280 \times 12 = 63360 \) inches.
04

Calculate Number of Revolutions

Divide the total distance traveled by the circumference of the wheel to find the number of revolutions. This is calculated as \( \text{Number of revolutions} = \frac{\text{Distance in inches}}{\text{Circumference in inches}} = \frac{63360}{30\pi} \approx 672.0 \).
05

Finalize Solution

The wheel will make approximately 672 revolutions when the car travels one mile.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wheel Revolution Calculations
To tackle any problem involving wheel revolutions, it's crucial to understand what revolution means in this context. A wheel revolution refers to one complete turn of the wheel, during which the wheel covers a distance equal to its full circumference. When calculating how many revolutions a wheel makes over a certain distance, you first need to determine the wheel's circumference. The circumference is the outer boundary length of the circle formed by the wheel, directly related to its diameter. Once you know how far the wheel travels in one revolution, you can calculate the total number of revolutions by dividing the total distance traveled by the wheel's circumference. This approach helps you quantify wheel rotations for any given travel distance.
Unit Conversion
In problems involving distances or sizes, it's often necessary to convert units to ensure uniformity. Units must match to make meaningful calculations. For a wheel distance problem, if the travel distance is in miles but the wheel size is given in inches, both must be converted to the same unit before proceeding. Typically, we convert miles to inches because wheel dimensions are given in inches. This conversion may involve multiple steps, such as converting miles to feet and then feet to inches. Through consistent unit conversion, we ensure accuracy and coherence in the results.
Mile to Inches Conversion
Understanding how to convert miles to inches is a fundamental skill in many geometric and physical problems. Knowing that one mile equals 5280 feet and each foot is 12 inches helps us find out how many inches make up a mile. By multiplying 5280 feet by 12 inches per foot, we get the total number of inches in a mile: 63360 inches. Breaking it down in parts can help visualize the conversion:
  • 1 mile = 5280 feet
  • 1 foot = 12 inches
  • 1 mile = 5280 x 12 = 63360 inches
Converting miles to inches is key when comparing or relating different measurements or figuring out calculations dependent on distance.
Number of Revolutions Calculation
After determining the wheel's circumference and converting the travel distance into consistent units, calculating the number of revolutions becomes straightforward. To find the number of revolutions a wheel makes over a specified distance, you divide the total travel distance by the wheel's circumference. For example, if a wheel with a circumference of 30蟺 inches covers a mile (63360 inches), the number of revolutions is found by dividing the total inches by the circumference: \[ \text{Number of revolutions} = \frac{63360}{30\pi}\]This expression simplifies to approximately 672 revolutions. This kind of calculation is important not only in theoretical exercises but also in real-world engineering and design applications concerning circular motion and travel distance.

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Most popular questions from this chapter

Bicycle Wheel The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in. the wheel sprocket a radius of 2 in. and the wheel a radius of 13 in. The cyclist pedals at 40 \(\mathrm{rpm}\) . (a) Find the angular speed of the wheel sprocket. (b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.)

The area of a sector of a circle with a central angle of 2 rad is \(16 \mathrm{m}^{2} .\) Find the radius of the circle.

The Leaning Tower of Pisa The bell tower of the cathedral in Pisa, Italy, leans \(5.6^{\circ}\) from the vertical. A tourist stands 105 \(\mathrm{m}\) from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of the tower to be \(29.2^{\circ} .\) Find the length of the tower to the nearest meter.

Airport B is 300 mi from airport A at a bearing \(\mathrm{N} 50^{\circ} \mathrm{E}\) (see the figure). A pilot wishing to fly from \(\mathrm{A}\) to \(\mathrm{B}\) mistakenly flies due east at 200 \(\mathrm{mi} / \mathrm{h}\) for 30 minutes, when he notices his error. (a) How far is the pilot from his destination at the time he notices the error? (b) What bearing should he head his plane in order to arrive at airport B?

Rain Gutter A rain gutter is to be constructed from a metal sheet of width 30 \(\mathrm{cm}\) by bending up one-third of the sheet on each side through an angle \(\theta .\) (a) Show that the cross-sectional area of the gutter is modeled by the function $$ A(\theta)=100 \sin \theta+100 \sin \theta \cos \theta $$ (b) Graph the function \(A\) for \(0 \leq \theta \leq \pi / 2\) (c) For what angle \(\theta\) is the largest cross-sectional area achieved?
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