A circular garden with a radius of 8 feet is surrounded by a circular path with a width of 3 feet. What is the approximate area of the path alone? Use 3.14 for [tex]$\pi$[/tex].
A. $172.70 ft ^2$
B. $178.98 ft ^2$
C. $200.96 ft ^2$
D. $379.94 ft ^2$
Answer (2)
Calculate the outer radius by adding the path width to the garden radius: R = 8 + 3 = 11 feet.
Compute the area of the outer circle: A o u t er = 3.14 × 1 1 2 = 379.94 square feet.
Compute the area of the inner circle (garden): A inn er = 3.14 × 8 2 = 200.96 square feet.
Subtract the inner area from the outer area to find the path area: $A_{path} = 379.94 - 200.96 =
178.98 f t 2 .
Explanation
Problem Analysis The problem asks for the area of a circular path surrounding a circular garden. To find this area, we need to calculate the area of the larger circle (garden plus path) and subtract the area of the smaller circle (garden).
Calculate Outer Radius The radius of the garden is 8 feet, and the path is 3 feet wide. Therefore, the radius of the outer circle (garden plus path) is the sum of these two lengths: R = 8 + 3 = 11 feet.
Calculate Outer Area Now, calculate the area of the outer circle using the formula A = π R 2 , where R = 11 feet and π ≈ 3.14 . Thus, the area of the outer circle is A o u t er = π ( 1 1 2 ) = 3.14 × 121 = 379.94 square feet.
Calculate Inner Area Next, calculate the area of the inner circle (the garden) using the formula A = π r 2 , where r = 8 feet and π ≈ 3.14 . Thus, the area of the inner circle is A inn er = π ( 8 2 ) = 3.14 × 64 = 200.96 square feet.
Calculate Path Area Finally, subtract the area of the inner circle from the area of the outer circle to find the area of the path: A p a t h = A o u t er − A inn er = 379.94 − 200.96 = 178.98 square feet.
Final Answer The area of the path alone is approximately 178.98 square feet.
Examples
Imagine you're designing a backyard garden with a circular flower bed in the center and a gravel path surrounding it. Knowing how to calculate the area of such a path allows you to determine how much gravel you'll need to purchase, ensuring you have enough material to complete your landscaping project. This calculation is also useful for estimating the cost of materials and planning the layout of your garden effectively.
To calculate the area of the path surrounding a circular garden, we first determine the outer radius, then find the area of the outer circle and the inner circle, and subtract the latter from the former. The area of the path alone is approximately 178.98 square feet, corresponding to option B. Therefore, the correct answer is B. 178.98 ft².
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