Problem 49 - Basic Technical Mathematics with Calculus

Question

The rim on a basketball hoop has an inside diameter of 18.0 in. The largest cross section of a basketball has a diameter of 12.0 in. What is the ratio of the cross-sectional area of the basketball to the area of the hoop?

Step-by-Step Solution

Answer

The ratio of the basketball's cross-sectional area to the hoop's area is \( \frac{4}{9} \).

Step 1: Calculate Area of the Basketball Cross-Section

The cross-section of the basketball is a circle with a diameter of 12.0 inches. To find the area, use the formula for the area of a circle: \[ A = \pi r^2 \] where \( r \) is the radius of the circle. First, calculate the radius: \( r = \frac{12.0}{2} = 6.0 \) inches.Now, find the area:\[ A_{basketball} = \pi (6.0)^2 = 36\pi \text{ square inches} \]

Step 2: Calculate Area of the Hoop Cross-Section

The cross-section of the hoop is also a circle, and it has a diameter of 18.0 inches. Again, use the circle area formula:\[ A = \pi r^2 \]Calculate the radius: \( r = \frac{18.0}{2} = 9.0 \) inches.Find the area:\[ A_{hoop} = \pi (9.0)^2 = 81\pi \text{ square inches} \]

Step 3: Calculate the Ratio of Areas

To find the ratio of the cross-sectional area of the basketball to the area of the hoop, divide the area of the basketball by the area of the hoop:\[ \text{Ratio} = \frac{A_{basketball}}{A_{hoop}} = \frac{36\pi}{81\pi} \]The \( \pi \) terms cancel out, simplifying the expression:\[ \text{Ratio} = \frac{36}{81} \]Now simplify the fraction by dividing the numerator and the denominator by the common factor of 9:\[ \text{Ratio} = \frac{4}{9} \]

Key Concepts

Circle Area Calculation

Calculating the area of a circle is a crucial skill, especially in geometry. Remember, a circle's area can be found using the formula: \[ A = \pi r^2 \] where \( A \) represents the area and \( r \) is the radius.
To find the radius when given the diameter, simply divide the diameter by 2. For example, if we have a circle with a diameter of 12.0 inches, the radius would be \( \frac{12.0}{2} = 6.0 \) inches.
Now, use the radius to calculate the area:

• Plug \( 6.0 \) into the area formula: \( A = \pi (6.0)^2 \)
• This results in an area of \( 36\pi \) square inches.

This basic understanding allows you not only to determine the area of a single circle but also equips you for more complex multi-step problems.

Ratio of Areas

The ratio of areas is essentially a way to compare different sizes of shapes, often showcasing how much larger or smaller one area is relative to another.
When comparing two circles – say, a basketball's cross-section to a hoop's cross-section – calculating the ratio begins with finding each circle's area, as we've shown above.
Next, you simply divide the area of one shape by the area of the other. For instance, if the basketball's area is \( 36\pi \) and the hoop's area is \( 81\pi \), the formula we use is:

• \( \text{Ratio} = \frac{36\pi}{81\pi} \)
• The \( \pi \) terms cancel each other out, leaving the simpler \( \frac{36}{81} \)

The last step is to simplify the fraction. Both terms can be divided by a common factor, 9 in this case, reducing it to \( \frac{4}{9} \).
This ratio tells us the basketball's cross-sectional area is 4 parts out of a total 9 parts when compared to the hoop.

Cross-Sectional Area

A cross-sectional area refers to the shape that is revealed when you slice straight through an object. For round objects, like basketballs, this slice is a circular shape.
Understanding cross-sectional area is essential for those dealing with structures or objects where the cross-section size can affect performance or fitting, such as in pipelines or fitting a basketball through a hoop.
For the basketball, with a diameter of 12 inches, the cross-sectional area is a circle with a radius of 6 inches (half the diameter). You calculate the area to understand how it fits into spaces or how it compares to other circles, like the hoop.
The hoop's diameter, at 18 inches, translates to a radius of 9 inches, creating a larger cross-sectional area.

• Basketball cross-section: \( 36\pi \) square inches
• Hoop cross-section: \( 81\pi \) square inches

Grasping the concept of cross-sectional areas expands your ability to solve real-world problems by visualizing how objects fit and relate to each other in three-dimensional space.