I'm really confused? help?

Question

Find the depth of a hopper to hold 15 bu. of grain, if it is built in the shape of the frutum of a square pyramid with the upper and lower bases measuring 7 and 4 ft. on a side, respectively. (A bushel = 1 1/4 cu. ft.)

Answer 1

The volume formula for the frustum of a pyramid is this:

V = h[B + B' + √(B∙B')]/3,

where h is the height of the frustum, and B and B' are the areas of the top and bottom of the frustum.

Since we are given the volume of the frustum, and the measure of a side of each base, we can calculate their areas and the square root of the product of the areas.  Then all we need to do is plug those values into the equation and crank out h:

V = h[B + B' + √(B∙B')]/3 ----> h = 3V/[B + B' + √(B∙B')].

First we convert bushels to cubic feet and plug that into the equation on the right above.  Then we can solve for h:

15 bu = 15∙(5/4 ft³) = 75/4 ft³ = 18.75 ft³

h = 3∙(18.75 ft³)/{(7)² + (4)² + √[(7)²∙(4)²]}

h = 56.25 ft³/[49 + 16 + √(49∙16)] ft²

h = 56.25 ft³/(49 + 16 + √784) ft²

h = 56.25 ft³/(49 + 16 + 28) ft²

h = 56.25 ft³/93 ft²

h ≈ 0.605 ft

h ≈ 7.26 inches

So the hopper is approximately 0.605 ft or 7.26 inches deep.

Answer 2

the volume of a square pyramid = 0.2357*a^3 where a is its base dimension. the volume of the 7-foot sq pyr = 0.2357*7^3 = 80.845 cubic feet and that of the 4-foot sq pyr = 0.2357*4^3 = 15.08 cubic feet. the difference is 80.845 - 15.08 = 65.76 cubic feet = 65.76/1.25 = 52.6 bushels