EXAMPLE 8 Find the volume of a pyramid whose base is a square with side B and whose height is h. SOLUTION We place the origin O at the vertex of the pyramid and the x-axis along its central axis as in the figure. Any plane P_x that passes through x and is perpendicular to the x-axis intersects the pyramid in a square with side of length s. We can express s in terms of x by using similar triangles. x/h = (s/2)/(B/2) = s/B and so s = [Another method is to observe that the line OP has slope B/(2h) and so its equation is y = Bx/(2h).] Thus the cross-sectional area is A(x) = s^2 = The pyramid lies between x = 0 and x = , so its volume is V = int_0^h A(x) dx = int_0^h (B^2/h^2)x^2 dx = B^2/h^2 [ ]_0^h =
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EXAMPLE 8 Find the volume of a pyramid whose base is a square with side L and whose height is h. SOLUTION We place the origin O at the vertex of the pyramid and the x-axis along its central axis as in the top figure. Any plane Px that passes through x and is perpendicular to the x-axis intersects the pyramid in a square with side of length s, say. We can express s in terms of x by observing from the similar triangles in the bottom figure that x/h = (s/2)/(L/2) = s/L and so s = [ ]. [Another method is to observe that the line OP has slope L/(2h) and so its equation is y = Lx/(2h).] Thus the cross-sectional area is A(x) = s^2 = [ ]. The pyramid lies between x = 0 and x = [ ], so its volume is V = ∫[0 to h] A(x) dx = ∫[0 to h] (L^2/h^2) x^2 dx = [ ]|[0 to h] = [ ].
Adi S.
The volume of a triangular pyramid is given by the formula $V=\frac{1}{3} B h,$ where $B$ represents the area of the triangular base and $h$ is the height of the pyramid. Find the volume of a triangular pyramid whose height is given and whose base has the coordinates shown. Assume units are in m. $$h=7.5 \mathrm{m} ; \text { vertices }(-2,3),(-3,-4), \text { and }(-6,1)$$
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Consider a tank in the shape of pyramid with height 12 m and square base of length 2 m. The base is sitting on the ground. It is partially filled with water the water is 6 m deep. (A). Sketch the pyramid, choose a coordinate system, and label it on the sketch. Slice the pyramid into horizontal slices each with height Δx (or Δy) cm and sketch one of the slices. (B). Find the approximate volume of the slice that is a vertical distance xi (or yi) cm from the tip (or base) of the pyramid. The only variables in your answer should be (xi and Δx) or (yi and Δy). (C). Set up, but do not evaluate, a definite integral that calculates the volume of water inside the tank. (Recall that the water is 6 m deep.)
Frank D.
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