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Chapter 19: Problem 5

A bicycle gear with radius 4 inches is rotating with a frequency of 50 revolutions per minute. In 2 minutes what distance has been covered by a point on the corresponding chain?

Short Answer

Expert verified
The point on the chain covered a total distance of \(800 \pi\) inches in a span of 2 minutes.

Step by step solution

01

Calculate Circumference of the Gear

The circumference of the gear is given as \(Circumference = 2 \pi r\) where \(r\) is the radius of the gear. Given that \(r = 4 inches\), the circumference of the gear can be found as \(Circumference = 2* \pi * 4\).
02

Calculate Total Rotations in 2 minutes

It's given that the gear is rotating at a rate of 50 revolutions per minute. Therefore, in 2 minutes, the total rotations would be \(50 rotations/minute * 2 minutes = 100 rotations\).
03

Calculate Total Distance Covered

The total distance covered by a point on the chain is the circumference of the gear times the total number of rotations over the 2 minute period. Thus, the total distance covered is \(Circumference * Total Rotations = (2* \pi * 4)*100\).

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Key Concepts

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

Circumference Calculation
The circumference of a circle is an important concept in bicycle gear mathematics.To determine the distance a bicycle gear covers in one complete rotation, you need to calculate its circumference.Circumference is defined as the total distance around the circle, and for a circular gear, it is calculated using the formula:

\[Circumference = 2 \pi r\]

Here, \(r\) represents the radius of the gear, which is the distance from the center of the circle to the outer edge.For example, if the radius of the gear is 4 inches, the circumference would be:
\[Circumference = 2 \times \pi \times 4 = 8\pi \, \text{inches}\]

Knowing the circumference helps to understand how far the gear travels each time it completes a full rotation.
Rotational Motion
Rotational motion plays a crucial role in understanding how a bicycle moves. It describes the movement of the gear as it spins around its axis. Each complete rotation of the gear equates to the gear traversing one full circumference in distance.

Revolving at a frequency of 50 revolutions per minute means the bicycle gear spins 50 times in one minute. So, if we examine a time span of 2 minutes, we find the total number of rotations by multiplying:
  • 50 rotations per minute
  • 2 minutes
This results in a total of 100 rotations.

Understanding rotational motion allows us to infer how many times the gear has circled and thus aids in calculating further distances.
Distance Covered by Rotation
Once we know both the circumference of the gear and the total number of rotations, calculating the distance covered is straightforward.The distance covered by the rotation of a bicycle gear is simply the product of the number of rotations and the circumference of the gear.

For instance, if a gear has a circumference of \(8\pi\) inches, and it completes 100 rotations in a given time:
  • Circumference per rotation = \(8\pi \) inches
  • Total rotations = 100
The total distance covered would be:\[100 \times 8\pi = 800\pi \, \text{inches}\]
By mastering these calculations, we can accurately quantify how far the bicycle has traveled based on gear size and speed.

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Most popular questions from this chapter

(a) Using what you know about the properties of polynomial functions, explain how the graph of \(f(x)=\sin x\) tells you that it is not a polynomial. (Think about the number of roots and the long-term behavior.) (b) Using what you know about the properties of rational functions, explain how the graph of \(f(x)=\tan x\) tells you that it is not a rational function. (Think about the number of roots and vertical asymptotes.) (c) What are characteristics of trigonometric functions that distinguish them from other functions we've studied?

Evaluate the following limits. (a) \(\lim _{x \rightarrow \infty} \cos x\) (b) \(\lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)\) (c) \(\lim _{x \rightarrow 0^{+}} \frac{1}{\sin x}\) (d) \(\lim _{x \rightarrow \infty} \sin \left(\frac{x^{2}}{x+1}\right)\) (e) \(\lim _{x \rightarrow \infty} \cos \left(\frac{\pi x^{3}-99}{x^{3}-x^{2}+7}\right)\)

Let \(f(x)=\frac{\sin x}{x}\). This function will be quite important when we are interested in the derivative of sine and cosine. (a) What is the domain of \(f(x)\) ? (b) Use a graphing calculator or computer to help you sketch the graph of \(f(x)\). (c) Although \(f(x)\) is undefined at \(x=0, \lim _{x \rightarrow 0} f(x)\) exists. What do you think this limit might be? Check out your conjecture numerically. Observe that if \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=L\), then for \(x\) very close to zero, \(\frac{\sin x}{x} \approx L\), or, equivalently, \(\sin x \approx L x\) for \(x\) close to zero.

Suppose we have the equation of a sine curve, \(y=\sin \left(\frac{\pi x}{2}\right)\), with period 4, amplitude 1 . (a) We wish to shift the graph over 1 unit to the left. Which of the following will accomplish this? i. \(y=\sin \left(\frac{\pi}{2} x-1\right)\) ii. \(y=\sin \left(\frac{\pi}{2}(x-1)\right)=\sin \left(\frac{\pi}{2} x-\frac{\pi}{2}\right)\) iii. \(y=\sin \left(\frac{\pi}{2} x+1\right)\) iv. \(y=\sin \left(\frac{\pi}{2}(x+1)\right)=\sin \left(\frac{\pi}{2} x+\frac{\pi}{2}\right)\) Be sure that you get this right. Draw a picture of what you are aiming for, and then try a point or two to verify. For example, you might run a test on \(x=0\). (b) For each of the remaining three choices in part (a), describe in words what happens to the original graph. (c) In order to obtain the desired result in part (a), it is probably simplest to use a cosine function. What cosine function will give the desired result? (d) Suppose \(A, B\), and \(C\) are positive constants and \(y=A \sin (B x+C) .\) What are the period and amplitude of the sine graph? Describe the horizontal shift.

Determine whether or not each function is periodic. If a function is periodic, determine its period. (a) \(f(x)=\cos |x|\) (b) \(g(x)=\sin |x|\) (c) \(h(x)=|\cos x|\) (d) \(j(x)=|\sin x|\) (c) \(k(x)=\sin \left(x^{2}\right)\) (f) \(l(x)=\sin ^{2} x\)
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