Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that \(\cos (x-\pi)=-\cos x\) for all \(x\). \(\approx\) C 35. 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 336.36 revolutions per minute.

Step 1: Understanding the Relationship Between the Distance and Revolutions

When a tire makes one complete revolution, it travels a distance equal to the circumference of the tire. The circumference formula is given by \(C = 2 \pi r\). Here, \(r\) is the radius of the tire.

Step 2: Calculate the Circumference of the Tire

The tire has a radius of 2.5 feet. Using the circumference formula, \(C = 2 \pi \times 2.5 = 5\pi\) feet.

Step 3: Convert the Speed from Miles per Hour to Feet per Minute

First, convert the speed from miles per hour to feet per hour. 1 mile equals 5280 feet, so 60 miles per hour is \(60 \times 5280 = 316800\) feet per hour. Convert this to feet per minute by dividing by 60, which gives \(316800 / 60 = 5280\) feet per minute.

Step 4: Calculate the Number of Revolutions per Minute

To find the number of revolutions per minute, divide the speed in feet per minute by the tire's circumference: \(5280 \div 5\pi \approx 336.36\). So, the tire makes approximately 336.36 revolutions per minute.

Key Concepts

Revolutions Per Minute Calculation

When evaluating the speed at which the tire spins, you are determining how many times it makes a complete loop, or revolution, every minute. The calculation of revolutions per minute (RPM) is essential for understanding the link between the distance traveled by the truck and the drumbeat of the tire's rotation.

Here's a straightforward way to think about it:

• A revolution is one full circle around the circumference of the tire.
• The RPM tells us how many circles the tire completes in a minute.
• To find this figure, we need to calculate how far the truck travels in one minute and then how many tire circumferences fit into that distance.

This forms the basis of the RPM calculation. By dividing the speed of the truck by the tire's circumference, you translate linear movement into rotational terms.

Circumference Formula

The circumference of a circle—the distance around it—can be calculated using the formula \(C = 2\pi r\). In this scenario, the radius \(r\) of the truck's tire is a crucial measurement to determine the path length the tire covers in one rotation.

Let's break it down:

• \(r = 2.5\) feet, so the diameter (which is double the radius) is 5 feet.
• Substituting the radius into the circumference formula gives \(C = 2\pi \times 2.5 = 5\pi\) feet.
• This result, \(5\pi\), is the distance the tire travels in one complete circle.

The circumference is a cornerstone concept because it converts the spinning of the tire into the distance the truck moves. By understanding this, solving any RPM question becomes much more intuitive.

Unit Conversion

Analyzing this problem involves not only mathematics but also effective use of unit conversions. Distinct measures often need to be harmonized—you must switch units to ensure everything aligns.

Consider the steps to convert:

• Speed is given in miles per hour (mph), a unit of large distances, which must be converted to a smaller scale initially using feet and eventually per minute.
• To convert miles to feet, remember 1 mile is 5280 feet.
  Thus, 60 mph becomes 60 \(\times\) 5280 feet per hour, resulting in 316800 feet per hour.
• Next, to find out how far the truck goes each minute, divide by 60.
  This gives you 5280 feet per minute.

Transforming the distance this way simplifies the revolutions calculation, putting everything in the same measurement context. Seamlessly handling unit conversions ensures clarity and correctness in solving real-world mathematical dilemmas.