A tank in the shape of an inverted right circular cone has height 7 meters and r

Question

A tank in the shape of an inverted right circular cone has height 7 meters and radius 4 meters. It is filled with 2 meters of hot chocolate. Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. The density of hot chocolate is 1090 kg/m^3. Your answer must include the correct units.

Explanation / Answer

Note that, by similar triangles, the equation that relates the radius, r, to the height, h, is: r/h = 5/5 ==> r = h. The volume of a slab of chocolate at a distance of x from the bottom of the tank is then: dV = LWH = (x)(x)?x = (x^2)?x m^3. Then, the weight of this slab is: dF = mg = p(dV)g = (1500)(x^2)(9.8) ?x N. = 14700x^2 ?x N. At a distance of x from the bottom of the tank, the chocolate needs to be pumped moved through a distance of 5 - x meters, so the work required is: dW = dF * d = 14700x^2(5 - x)?x J. Integrating this from 0 to 4 yields the required work to be: W = 14700 ? x^2(5 - x) dx (from x=0 to 4) = 14700 ? (5x^2 - x^3) dx (from x=0 to 4) = 14700 * [(5/3)x^3 - (1/4)x^4] (evaluated from x=0 to 4) = (14700)(320/3 - 64) = 627200 J.

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