Problem 45 - Algebra 2

Question

Transportation Suppose the radius of a bicycle wheel is 13 in. (measured to the outside of the tire). Find the number of radians through which a point on the tire turns when the bicycle has moved forward a distance of 12 \(\mathrm{ft.}\)

Step-by-Step Solution

Answer

The point on the tire turns approximately 11.0769 radians when the bicycle moves forward a distance of 12 ft.

Step 1: Convert Distance from Feet to Inches

As we are given that the radius of the bicycle wheel is in inches, we need the distance in the same units. Multiply 12 feet by 12 to convert it to inches. Doing so gives us a total of 144 inches.

Step 2: Apply Formula to Get Radians

The formula to calculate the number of radians through which a point on the bike tire turns is given by radian = distance / radius. Here, the distance is 144 inches and the radius is 13 inches. Dividing the distance by the radius gives us approximately 11.0769 radians.

Key Concepts

Bicycle Wheel

A bicycle wheel is a fundamental part of the bicycle that allows it to move smoothly and efficiently. It consists of several components:

• Rim: The outer part that holds the tire in place.
• Spokes: The parts connecting the rim to the hub, helping with structural integrity.
• Hub: The central part containing the axle that turns as the bike moves.

The tire rolls along the ground as the wheel rotates, allowing the bike to move forward. Understanding the wheel's geometry is crucial when solving problems involving movement, distance, and rotation, as these are all interlinked with the wheel's size and motion.

Radius

The radius in a circle is the distance from the center of the circle to any point on its circumference. This measurement is crucial for circular objects like bicycle wheels, as it directly affects their functionality.

For a bicycle wheel, the radius helps determine:

• The wheel's size: Larger wheels can cover more distance with each rotation compared to smaller wheels.
• The calculation of radians: When a wheel turns, the number of radians is calculated using the radius, showing how far a point on the wheel travels along the circumference.

In mathematical terms, the circumference of a wheel (C) can be determined by the formula:\[ C = 2\pi r \]where \( r \) is the radius. This information is crucial for computations involving circular motion.

Distance Conversion

When dealing with measurements, consistency in units is vital. For this exercise, we have measurements in feet and inches. As the radius is given in inches, we must convert all distances to inches.

Conversion from feet to inches is straightforward:

• 1 foot = 12 inches
• To convert feet to inches, multiply the number of feet by 12.

In the problem, a distance of 12 feet becomes 144 inches upon conversion. By ensuring that all measurements are in the same unit (inches, in this case), we can directly apply formulas without confusion, leading to precise and accurate calculations.

Rotation Measurement

Rotation measurement is assessed in radians, providing insight into how much an object, like a bicycle wheel, has turned.

Radians define the angle through which a point on a circle rotates. This concept is essential when measuring rotational motion without relying on degrees.

• Formula: The number of radians a wheel turns is calculated by dividing the distance a point travels by the radius of the wheel:

\[ \text{radians} = \frac{\text{distance}}{\text{radius}} \]
For a bicycle wheel moving 12 feet (converted to 144 inches), with a radius of 13 inches, the calculation is:
\[ \text{radians} = \frac{144}{13} \approx 11.0769 \]
This result tells us how far along the circumference a point on the wheel has rotated, providing insights into the bike's movement.