Question

Suppose that you work a summer job as a landscaper. One of your clients wants you to

dig a ditch in their yard to relieve the flooding that happens after heavy rainfall. Your
boss tells you to make a ditch that is 18 inches deep and 8 feet long. The ditch should
be 12 inclys wide at the bottom and 24 inches wide at the top.
(a) Your boss tells you that you need to line the ditch with plastic sheeting. Estimate
the minimum amount of plastic you will need to line the ditch. Explain how you
determined your answer.
(b) Estimate the maximum amount of water that the ditch can hold without overflowing.
Explain how you determined your answer.
(c) If you increase the length of the ditch by 1 foot, by how much does the volume of the
ditch change?
1
(d) The client decides that they want to line the ditch with a - inch layer of rocks to make it look nicer. Estimate how much rock you would need, in cubic inches.

Answer 1

(a) To estimate the minimum amount of plastic sheeting needed to line the ditch, we need to calculate the surface area of the ditch's inner lining. To do this, we can break the ditch down into a trapezoid and a rectangular prism.

The trapezoid at the top of the ditch has a width of 24 inches, a height of 3 inches (half of the 6-inch difference between the top and bottom widths), and a length of 8 feet. Its area is:

1/2 x (24 + 12) x 3 x 96 = 432 square inches

The rectangular prism that makes up the rest of the ditch has a width of 12 inches, a height of 18 inches, and a length of 8 feet. Its area is:

12 x 18 x 96 = 20,736 square inches

To find the total surface area, we add these two areas together:

432 + 20,736 = 21,168 square inches

Therefore, we need at least 21,168 square inches of plastic sheeting to line the ditch.

(b) To estimate the maximum amount of water that the ditch can hold without overflowing, we need to calculate its volume. We can break the ditch down into the same trapezoid and rectangular prism as before.

The trapezoid has a volume of:

1/2 x (24 + 12) x 3 x 96 = 3,456 cubic inches

The rectangular prism has a volume of:

12 x 18 x 96 = 20,736 cubic inches

To find the total volume, we add these two volumes together:

3,456 + 20,736 = 24,192 cubic inches

Therefore, the ditch can hold a maximum of 24,192 cubic inches of water without overflowing.

(c) Increasing the length of the ditch by 1 foot means that we need to add an extra foot to the length of the rectangular prism. The new length would be 9 feet instead of 8 feet. The width and depth of the ditch remain the same. Therefore, the volume of the ditch would increase by:

12 x 18 x 12 = 2,592 cubic inches

(d) To estimate the amount of rock needed to line the ditch, we need to calculate the volume of the layer of rocks. The layer has the same shape as the ditch, with a width of 12 inches, a height of 0.5 inches (1/2 inch), and a length of 8 feet. Its volume is:

12 x 0.5 x 96 = 5,760 cubic inches

Therefore, we would need 5,760 cubic inches of rock to line the ditch with a 1/2 inch layer.