1)An excavation is 12 ft. deep and has trapezoidal sides (faces). The upper base is horizontal rectangle 400 ft. by 180 ft., and the lower base is a horizontal rectangle 350 ft. by 150 ft. How many cubic yards of earth were removed in digging the excavation?
2)A chimney in the shape of a frustum of a regular pyramid is 186.3 ft. high. Its upper base is a square 10 ft. on a side, and its lower base is a square 16 ft. on a side. The flue is of uniform square cross section. 7 ¼ by 7 ¼ ft. find the weight of the chimney if the material weighs 112.8 lb. per cu. ft.
Number (1)
Upper base area
$A_1 = 400 \times 180 = 72\,000 ~ \text{ft}^2$
Lower base area
$A_2 = 350 \times 150 = 25\,500 ~ \text{ft}^2$
Mid section
$a = \frac{1}{2}(180 + 150) = 165 ~ \text{ft}$
$b = \frac{1}{2}(400 + 350) = 375 ~ \text{ft}$
$A_m = ab = 165 \times 375 = 61\,875 ~ \text{ft}^2$
Distance between A1 and A2
$L = 12 ~ \text{ft}$
Volume of excavation (Prismatoid)
$V = \frac{1}{6}L(A_1 + 4A_m + A_2)$
$V = \frac{1}{6}(12) [ \, 72\,000 + 4(61\,875) + 25\,500 \, ]$
$V = 690\,000 ~ \text{ft}^3$
$V = 690\,000 ~ \text{ft}^3 \left( \dfrac{1 ~ \text{yd}}{3 ~ \text{ft}} \right)^3$
$V = 25\,555.56 ~ \text{yd}^3$
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