Problem 8
Question
A bicycle wheel is 26 inches in diameter. When the brakes are applied the bike wheel makes \(2.2\) revolutions before coming to a halt. How far has the bike traveled? (Assume the bike does not skid.)
Step-by-Step Solution
Verified Answer
The bike travels about \(179.69\) inches before coming to a halt when the brakes are applied.
1Step 1: Calculate the circumference of the wheel
The circumference (C) of a circle can be determined using the formula \(C = \pi d\), where \(d\) is the diameter of the circle. In this problem, the diameter of the bicycle wheel is 26 inches, so substituting this into the formula gives us \(C = \pi * 26\) inches.
2Step 2: Calculate the total distance travelled
The total distance (D) travelled by the bike can be calculated using the formula \(D = n * C\), where \(n\) is the number of revolutions the wheel makes. In this problem, the wheel makes 2.2 revolutions, so substituting \(n = 2.2\) and the calculated circumference from step 1 into the formula gives us \(D = 2.2 * C\) inches.
3Step 3: Solve for D
Substitute the calculated circumference \(C\) from Step 1 into the equation in Step 2 and solve for \(D\).
Key Concepts
Understanding CircumferenceRevolutions Made ClearCalculating the Distance TraveledUnderstanding Diameter
Understanding Circumference
The circumference of a circle is the distance around it, much like the perimeter of a square or rectangle. It's an important concept when dealing with circular objects like wheels. To calculate the circumference, you can use the formula \(C = \pi d\), where \(\pi\) is a constant (approximately 3.14159) and \(d\) represents the diameter.
Applying this to our problem, with a bicycle wheel diameter of 26 inches, the circumference becomes \(C = \pi \times 26\). This measures the distance the wheel covers in one full revolution. Knowing the circumference helps us determine the total distance traveled when the wheel completes multiple revolutions.
Applying this to our problem, with a bicycle wheel diameter of 26 inches, the circumference becomes \(C = \pi \times 26\). This measures the distance the wheel covers in one full revolution. Knowing the circumference helps us determine the total distance traveled when the wheel completes multiple revolutions.
Revolutions Made Clear
When we refer to revolutions in the context of bicycle wheels or other circular motion, we are talking about how many full spins or turns the wheel makes. Each revolution moves the bicycle forward by a distance equal to the circumference of the wheel.
In the bicycle scenario, understanding how to calculate the number of revolutions is crucial for determining total distance. For example, with a wheel making 2.2 revolutions before stopping, each of these complete turns uses the circumference to cover corresponding ground distance. This measurement helps us understand just how far the bicycle traveled before coming to a stop.
In the bicycle scenario, understanding how to calculate the number of revolutions is crucial for determining total distance. For example, with a wheel making 2.2 revolutions before stopping, each of these complete turns uses the circumference to cover corresponding ground distance. This measurement helps us understand just how far the bicycle traveled before coming to a stop.
Calculating the Distance Traveled
Distance traveled, in problems like these, is a function of the number of revolutions and the wheel's circumference. The formula used is \(D = n \times C\), where \(D\) is the distance, \(n\) is the number of revolutions, and \(C\) is the circumference.
This formula tells us that whenever the wheel turns, it covers a proportionate distance based on its size and the number of spins. For our bicycle example, multiplying the calculated circumference by the revolutions (2.2 in this case) gives us the total distance. This simple multiplication provides an efficient method of converting rotational motion into linear travel distance.
This formula tells us that whenever the wheel turns, it covers a proportionate distance based on its size and the number of spins. For our bicycle example, multiplying the calculated circumference by the revolutions (2.2 in this case) gives us the total distance. This simple multiplication provides an efficient method of converting rotational motion into linear travel distance.
Understanding Diameter
The diameter of a circle is a straight line going from one side of the circle to the other, passing through the center. In the context of a bicycle wheel, it is a critical measurement. The diameter determines not only the wheel's overall size but also its circumference.
In our bicycle wheel example, the diameter is 26 inches. This size is used directly in calculating the circumference, which is central to all other calculations relating to distance and revolutions. A larger diameter implies a longer circumference and means more ground is covered with every revolution. This fundamental dimension underpins how we calculate and understand wheel dynamics and distances traveled.
In our bicycle wheel example, the diameter is 26 inches. This size is used directly in calculating the circumference, which is central to all other calculations relating to distance and revolutions. A larger diameter implies a longer circumference and means more ground is covered with every revolution. This fundamental dimension underpins how we calculate and understand wheel dynamics and distances traveled.
Other exercises in this chapter
Problem 7
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 \rightar
View solution Problem 7
Find all \(x\) such that (a) \(\tan x=1\). (b) \(\tan x=-1\). Try to do this using the unit circle definitions.
View solution Problem 8
Graph the following. (a) \(y=|\sin x|\) (b) \(y=\sin |x|\)
View solution Problem 8
Sketch the graph of \(g(x)=\tan 2 x\) on \([0,2 \pi]\).
View solution