Final answer:
To calculate the area of the sidewalk, we find the areas of both the larger and smaller circles and then subtract the latter from the former. The area of the sidewalk is found to be 22 m².
Step-by-step explanation:
To find the area of the sidewalk around a circular flower bed that is 6 m in diameter with a circular sidewalk that is 1 meter wide, we first need to calculate the area of the larger circle (including the sidewalk) and then subtract the area of the smaller circle (the flower bed alone).
Step-by-Step Solution
- Calculate the radius of the flower bed, which is half of the diameter. So, the radius (r1) of the flower bed is 6 m / 2 = 3 m.
- Calculate the area (A1) of the flower bed using the formula for the area of a circle, A = πr². A1 = π(3 m)² = π(9 m²) = 28.2743339 m² (rounded to the proper number of significant figures it is 28.3 m²).
- Next, calculate the radius of the larger circle, which is the radius of the flower bed plus the width of the sidewalk. The radius (r2) of the larger area including the sidewalk is 3 m + 1 m = 4 m.
- Calculate the area (A2) of the larger circle including the sidewalk using the same area formula. A2 = π(4 m)² = π(16 m²) = 50.2654825 m² (rounded to the proper number of significant figures it is 50.3 m²).
- The area of the sidewalk alone is the area of the larger circle minus the area of the smaller circle. Area of sidewalk = A2 - A1 = 50.3 m² - 28.3 m² = 22 m².