



 
John, The taper you gave cannot be correct for all three other sides  it must be steeper on the two opposite sloping sides and less steep on the one facingthe vertical side. Fortunately, by Cavalieri's principle, we don't need to know the taper, just the cross section at each level. (Imagine a stack of horizontal cards which together had the shape of the hopper. Provided the cards stay the same size we can slide them around as we like and the total volume doesn't change.) So it's the frustum of a pyramid of base 36" square, untruncated height 36", with a 6" square 6" high pyramid removed from the end. For a pyramid V = 1/3 Bh so the hopper volume is 1/3 (36^{2} × 36)  1/3 (6^{2} × 6) = (36^{3}  6^{3})/3 in^{3} = 15480 in^{3}
RD
 


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