Answer:
a) area of sidewalk: 346.5 ft²
b) 278 bags
Step-by-step explanation:
The formula for the area of a circle is ...
A = π·r² . . . . . where A is the area and r is the radius of the circle. (The radius is half the diameter.)
There are a couple of ways to find the area of a "washer" (a circle with a hole in the middle). One is to subtract the area of the hole from the area of the larger circle. Another way is to find the circumference of the circle whose radius is the average of the inner and outer radii, and multiply that by the width of the washer (the difference of the outer and inner radii).
For the latter purpose, the formula for the circumference of a circle is ...
C = π·d . . . . . or, since the diameter is twice the radius, ...
C = 2π·r
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a) Here, the diameter of the circle that is the center of the walkway is ...
28 ft + 3.5 ft = 31.5 ft
So, the circumference of that circle is ...
C = π·d = (22/7)·(31.5 ft) = 99 ft
Then the area of the walkway is ...
(99 ft)·(3.5 ft) = 346.5 ft²
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b) 0.8 bags of concrete are required for each square foot, so we can find the number of bags by multiplying 0.8 times the number of square feet:
bags = 0.8 × 346.5 = 277.2
If only whole bags are available, then 278 bags of concrete will be the minimum number needed.
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<em>Comment on the two methods of doing this calculation</em>
If r1 and r2 are the inner and outer radii of the circles, then the area of the washer is π(r2² -r1²).
The centerline diameter will be (2r2 +2r1)/2 = r1 +r2, and the width of the washer will be (r2 -r1). Then the washer area will be π·(r1 +r2)·(r2 -r1). This latter expression can be "simplified" to π(r2² -r1²), a formula for the washer area that is identical to the one above.
