The driveway of the Santiagos' new home is to be bounded by semicircular arcs as shown. The distance from X to Y is 100 feet. The concrete driveway is 10 feet wide and one-third feet thick.
A diagram.Short description, A diagram.,Long description,
The driveway is a semicircle that curves from the road to the house and back to the road. The outer semicircular arc is from point X to point Y at the edge of the driveway where it touches the road at either end. The inner semicircular arc is unlabeled. The area between the two arcs is the driveway.
Question
Using 3.14 as an approximation for pi, how many cubic feet of concrete are needed to build the driveway? (Round to the nearest cubic foot.)
Answer options with 5 options
A.
471 feet cubed
D.
1308 feet cubed
B.
942 feet cubed
E.
1884 feet cubed
C.
1060 feet cubed
To find the volume of the concrete needed for the driveway, we first need to find the area of the driveway's cross-section.
The outer semicircular arc is the same as a half-circle, so the area of this arc is (1/2)(π)(r^2).
The inner semicircular arc is also a half-circle, but with a smaller radius. Let's call this radius r1. The area of this arc is (1/2)(π)(r1^2).
The area between the two arcs is the difference between the area of the outer arc and the area of the inner arc: (1/2)(π)(r^2) - (1/2)(π)(r1^2).
Since the distance from X to Y is 100 feet, the radius r is half of that, so r = 50 feet.
The width of the driveway is 10 feet, so the radius of the inner arc is 40 feet (50 - 10 = 40).
The thickness of the concrete is one-third of a foot, so we need to multiply the above expression by 1/3 to account for the thickness.
Therefore, the volume of concrete needed is (1/3)[(1/2)(π)(50^2) - (1/2)(π)(40^2)].
Simplifying this expression, we get (1/3)[(1/2)(3.14)(2500) - (1/2)(3.14)(1600)] = (1/3)[(3925 - 2512)] = (1/3)(1413) = 471 cubic feet.
So the correct answer is A. 471 feet cubed.