A bed with a uniformly sloped top calls for volume, the space the soil fills. Because the depth changes at a constant rate, use the average of the two end depths, then multiply by the bed’s length and width. The price is per cubic yard, so cube the feet-per-yard conversion before multiplying by the price. Desmos can handle the volume and cost arithmetic.
Hints
- Hint 1
With a uniform slope, the depth increases at a constant rate. Pair a slice near the shallow end with one equally far from the deep end. What is the average depth of each pair?
- Hint 2
The bed’s volume is its rectangular base area times its average depth. Which two measurements give the area of the ground the bed covers?
- Hint 3
The price is per cubic yard, not per cubic foot. A cubic yard measures feet along each of its three edges. How many cubic feet are in one cubic yard?
Step-by-step
Average the depth, then convert the volume
Step 1Find the average soil depth
Because the depth increases uniformly, a slice near the shallow end pairs with an equally distant slice near the deep end. Each pair has the same average depth as the two ends. Let be that depth in feet. Type ; Desmos shows . Adding the end depths without dividing by would double the amount of soil.
- Step 2
Find the volume in cubic feet
The base area is the rectangle the bed covers on the ground: square feet. Multiply it by the average depth for the soil volume. Type under the first line; Desmos shows , meaning cubic feet. Using feet throughout would count soil above the sloped surface.
- Step 3
Convert the cubic units
A cubic yard is feet long, feet wide, and feet tall, so . Convert all three dimensions when you convert a volume. Dividing by would convert only one length.
- Step 4
Apply the price per cubic yard
The price is per cubic yard, so divide the cubic feet by before multiplying by per cubic yard. Type , where is the cost in dollars. Desmos shows , so the topsoil costs $400. Choice D.