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)
The area of the circular path surrounding the garden is approximately 178.98 ft². This value is found by calculating the area of the outer circle and subtracting the area of the inner circle. Thus, the correct answer is option B: 178.98 ft².
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Calculate the radius of the outer circle by adding the garden's radius and the path's width: R = 8 + 3 = 11 feet.
Determine the area of the outer circle using π = 3.14 : A o u t er = 3.14 × 1 1 2 = 379.94 ft 2 .
Compute the area of the inner circle (garden): A inn er = 3.14 × 8 2 = 200.96 ft 2 .
Subtract the inner circle's area from the outer circle's area to find the path's area: A p a t h = 379.94 − 200.96 = 178.98 ft 2 .
Explanation
Problem Analysis Let's break down this problem. We have a circular garden surrounded by a circular path. We need to find the area of the path only. To do this, we will find the area of the larger circle (garden + path) and subtract the area of the smaller circle (garden).
Find the radius of the outer circle First, we need to find the radius of the outer circle, which includes the garden and the path. The garden has a radius of 8 feet, and the path is 3 feet wide. So, the radius of the outer circle is: R = 8 + 3 = 11 feet
Calculate the area of the outer circle Next, we calculate the area of the outer circle using the formula A = π r 2 , where r is the radius. Using π = 3.14 , we have: A o u t er = π R 2 = 3.14 × ( 11 ) 2 = 3.14 × 121 = 379.94 ft 2
Calculate the area of the inner circle Now, we calculate the area of the inner circle (the garden) using the same formula. The radius of the garden is 8 feet, so: A inn er = π r 2 = 3.14 × ( 8 ) 2 = 3.14 × 64 = 200.96 ft 2
Calculate the area of the path To find the area of the path, we subtract the area of the inner circle from the area of the outer circle: A p a t h = A o u t er − A inn er = 379.94 − 200.96 = 178.98 ft 2
Final Answer Therefore, the area of the path alone is approximately 178.98 ft 2 .
Examples
Imagine you're designing a park with a circular fountain. You want to surround the fountain with a circular walkway. Knowing the radius of the fountain and the desired width of the walkway, you can calculate the area of the walkway. This helps you determine the amount of paving material needed, giving you a practical application of calculating the area between two circles.
