Farmer John stores grain in a large silo located at the edge of his farm. The cylinder-shaped silo has one flat, rectangular face that rests against the side of his barn. The height of the silo is 30 feet and the face resting against the barn is 10 feet wide. If the barn is approximately 5 feet from the center of the silo, determine the capacity of Farmer John’s silo in cubic feet of grain. Hint: Look for connections between this problem and the problem you solved in Part 1 of this week’s Discussion (a square inscribed in a circle).
The volume of any prism is the area of the face multiplied by the height. The height is given, so all you need to find is the area of the base. Start off by drawing a circle. One side has a chord of length 10 feet cut across it. Its given that the chord is 5 feet from the center of the circle. Therefore, you draw a line from the center bisecting the chord, as well as a line from the center to each end of the chord (the radius), and you have two equal isosceles triangles with two sides of length 5. Pythagorean's relation tells us that the third side has a length of 5*sqroot(2) ft. Now that you have the radius, you can find the area of the circle. However, its not a true circle you're concerned with. You want to exclude the area on the outside of the chord. Using trig, you can find that the total angle subtended by the chord is 90°. Therefore, the area that you're concerned with is the circular area for the outside 270° plus the area of the triangles. 270° is (3/4) of the circle, so its area is (3/4) the area of the entire circle. A = (3/4)*π*r? = (3/4)*π*(5*sqroot(2)ft)? = 118ft?. The area of each of the smaller triangles is given by (1/2)*b*h. Since there's two triangles, the total area of the two is 2*(1/2)*b*h = b*h = 5ft*5ft = 25ft?. The total area of the base is 118ft? + 25ft? = 143ft?. Multiplied by the height will give the total volume of the prism, 143ft?*30ft = 4290ft?.
