Problem 32 - Elementary Algebra

Question

The International Basketball Federation rules (2006) state, "For all women's competitions in all categories, the circumference of the ball shall be no less than \(724 \mathrm{~mm}\) and no more than \(734 \mathrm{~mm}\) (size 6)." Find the minimum radius of the ball in millimeters. Round to the nearest whole number. (Source: www.fiba.com)

Step-by-Step Solution

Answer

115 mm

Step 1: Understand the Given Condition

The minimum circumference of the women's basketball is given as 724 mm. Use this information to find out the minimum radius.

Step 2: Formula for Circumference of a Sphere

Recall the formula for the circumference of a sphere: \[ C = 2 \pi r \] where \(C\) is the circumference and \(r\) is the radius.

Step 3: Rearrange the Formula to Solve for Radius

Rearrange the circumference formula to solve for \(r\): \[ r = \frac{C}{2 \pi} \]

Step 4: Substitute the Given Circumference

Substitute the minimum circumference value into the rearranged formula: \[ r = \frac{724}{2 \pi} \]

Step 5: Calculate the Radius

Perform the calculation: \[ r = \frac{724}{2 \pi} \approx \frac{724}{6.2832} \approx 115.27 \]

Step 6: Round the Radius to the Nearest Whole Number

Round \(115.27\) to the nearest whole number: \[ r \approx 115 \text{ mm} \]

Key Concepts

circumference of a sphere

In this problem, we first need to understand the concept of the circumference of a sphere. The circumference of a sphere is the distance around the sphere and is given by the formula: \[ C = 2 \pi \times r \] Here, \(C\) stands for the circumference, and \(r\) represents the radius. This formula is crucial because it connects the circumference with the radius, allowing us to find one if we know the other. In this exercise, we are given the circumference and need to find the radius. If you recall the shape of a basketball, the circumference is like measuring around the middle, just like measuring your waist with a tape measure. This visualization can help you understand the importance of the given measurement.

rearranging formulas

Rearranging formulas is a critical skill in algebra and geometry. When given a formula, sometimes we need to solve for a variable that isn't isolated. In the case of the circumference formula \(C = 2\pi r\), we need to solve for \(r\) (the radius). To isolate \(r\), we can rearrange the formula like this: \[ r = \frac{C}{2\pi} \] By doing this, we have isolated \(r\) on one side of the equation, making it easier to substitute in our known values and solve for the radius. This rearranging step transforms the formula from a focus on \(C\) to one on \(r\), making our calculations straightforward.

mathematical rounding

Mathematical rounding helps simplify our answers, making them easier to use in practical situations. After calculating the radius, we got a decimal number, \(115.27\, \text{mm}\). Typically, we round to the nearest whole number to make it simpler. In this case, 0.27 is less than 0.5, so we round down to 115. This is done based on common rounding rules: \ul\ \li\If the digit is less than 5, round down\li\ If the digit is 5 or more, round up \ul\. Rounding helps avoid overly complex numbers and makes the final answer more user-friendly.

measurement units

Using the correct measurement units is essential for accuracy and understanding. In this exercise, all measurements are given in millimeters (\(\text{mm}\)). This unit is part of the metric system and is often used for small measurements. It's important to be consistent with units throughout the calculation to avoid errors. When converting or calculating, always check that your units match. Here, we didn't need to convert units, but if we did, being careful with units would be crucial. Keeping an eye on units ensures clarity and accuracy in mathematical operations and final answers.