Question 90·200 Super-Hard SAT Math Questions·Geometry and Trigonometry
A raised garden bed is 18 feet long and 10 feet wide. The bottom of the bed is flat, but the top surface of the soil slopes so that the soil depth increases uniformly from 1 foot at one end of the bed to 1.5 feet at the other end.
Topsoil costs $48 per cubic yard. If yard feet, which choice gives the total cost, in dollars, of the topsoil needed to fill the bed to the sloped top surface?
When a solid has a dimension that changes uniformly, treat it as a prism with a trapezoidal cross-section (equivalently, use the average of the two end depths). Compute the trapezoid’s area, multiply by the remaining dimension to get volume, and only then do unit conversions—remember that feet-to-yards conversions for volume require cubing (27, not 3 or 9).
Hints
Use a cross-section
Imagine slicing the bed with a vertical plane along its 18-foot length. What 2D shape describes the side view of the soil?
Handle the changing depth efficiently
Because the depth increases uniformly from 1 ft to 1.5 ft, you can treat the lengthwise cross-section as a trapezoid (or use the average of the two depths).
Be careful with cubic units
Convert cubic feet to cubic yards using , not 3 or 9.
Desmos Guide
Compute the trapezoid cross-sectional area
Enter A=((1+1.5)/2)*18 to represent the trapezoid area in square feet.
Find the volume in cubic feet
Enter V=A*10 to multiply by the width (10 ft).
Convert to cubic yards
Enter Y=V/27 to convert cubic feet to cubic yards.
Multiply by the price per cubic yard
Enter C=48*Y. The value of C is the total cost in dollars.
Step-by-step Explanation
Model the bed as a trapezoidal prism
Look at a cross-section along the 18-foot length of the bed. The soil depth changes uniformly from 1 ft to 1.5 ft, so this cross-section is a trapezoid with parallel sides 1 and 1.5 and distance between them 18.
Find the cross-sectional area
Area of a trapezoid:
So,
Convert that area to a volume
Multiply by the bed’s width (10 ft) to get volume:
Convert to cubic yards and compute the cost
Since yard feet, then .
Cost:
So the total cost is 400 dollars.