Problem 82 - Elementary Algebra Within Reach

Question

Geometry A circular target is attached to a rectangular board, as shown in the figure. The radius of the circle is \(4 \frac{1}{2}\) inches, and the measurements of the board are 12 inches by 15 inches. What percent of the board is covered by the target? (The area of a circle is \(A=\pi r^{2}\), where \(r\) is the radius of the circle.)

Step-by-Step Solution

Answer

To solve this problem, you must first calculate the area of the circle and the rectangle and then find the percentage of the rectangle's area that the circle occupies.

Step 1: Calculate Area of the Circle

Using the formula for the area of a circle, \(A=\pi r^{2}\), given that the radius \(r\) is \(4 \frac{1}{2}\) inches, we get: \[A_{circle}=\pi (4 \frac{1}{2})^{2}\].

Step 2: Calculate the Area of the Rectangle

The area of a rectangle is given by the formula \(Area = length \times width\). In our case, it becomes \(Area_{rectangle} = 12 \, inches \times 15 \, inches\).

Step 3: Calculate Percentage Area Covered

Now that we have the area of the circle (target) and the area of the rectangle (board), we can find out what percentage of the board is covered by the target using the formula: \[Percentage = \left(\frac{Area_{circle}}{Area_{rectangle}}\right) \times 100\%\]

Key Concepts

Area of a Circle

When dealing with circular shapes, knowing how to calculate the area is essential. The formula to find the area of a circle is given by \( A = \pi r^2 \), where \( A \) represents the area and \( r \) is the radius of the circle. In this exercise, the radius is expressed as a mixed number: \( 4 \frac{1}{2} \ or \ 4.5 \). So, we square the radius to find the area: \( (4.5)^2\).

• First, convert the mixed number to an improper fraction or a decimal for easy handling. In this case, \( 4 \frac{1}{2} \) is \( 4.5 \).
• Then, calculate \( 4.5^2 \ = 20.25 \).
• Finally, multiply this result by \( \pi \) to get the area of the circle: \( \pi \times 20.25 \).

Understanding this process allows us to determine how much space the circle (or target) occupies on a given surface.

Area of a Rectangle

Calculating the area of a rectangle is straightforward. Rectangles have two dimensions: length and width. The formula for the area is \( \text{Area} = \text{length} \times \text{width} \). In this exercise, the rectangle represents the board with dimensions of 12 inches by 15 inches.

• To find the area, simply multiply the length (12 inches) by the width (15 inches).
• This gives us \( 180 \) square inches \( 12 \times 15 = 180 \).

This calculation shows the total surface area of the board, which is essential for further calculations to determine how much of this area is covered by the circle.

Percentage Calculation

Once you have both the areas of the circle and the rectangle, you can calculate what portion of the rectangle is covered by the circle using a percentage. Percentages are a way to express a number as a fraction of 100, and the formula used here is: \( \text{Percentage} = \left( \frac{\text{Area of circle}}{\text{Area of rectangle}} \right) \times 100\% \).

• First, ensure you have calculated the area of the circle using the previously discussed method.
• Next, use the area of the rectangle which is \( 180 \) square inches from the previous step.
• Then, divide the area of the circle by the area of the rectangle.
• Finally, multiply this fraction by 100 to find the percentage.

This calculation tells you what percentage of the board's area is occupied by the target, letting you understand coverage visually and quantitatively.