A reservoir has the shape of a right prism whose parallel bases are isosceles tr

Question

A reservoir has the shape of a right prism whose parallel bases are isosceles triangles, as shown in the figure above. It measures 7 m high, 6 m wide and 6 m long, and it is filled with water. We want to calculate the amount of work required to pump all of its water through the hose standing 2 m above the reservoir. Let x be the height in meters measured from the base of the reservoir. The height of a thin water layer between height x and x + Delta x is approximately P(x) Delta x. What is P(x)? Express the answer using a formula. Recall that the density of water is rho = 000 Kg/m^3 and the acceleration due to gravity on the earth's surface is g 9.8 m/s^2. The work in Joules required to pump the thin water layer to 2 m above the reservoir is approximately w (x) = The work in Joules required to jump the thin water layer to 2m above the reservoir is approximately w(x) Delta x. What is w(x)? Express the answer as a formula. Using previous results, determine the amount of work required to pump all of the water to 2 m above the reservoir. Give the numerical value with 3 significant digits. .

Explanation / Answer

i) P(x) dx = rho V g

P(x) = 1000*6*6x/7*9.8

= 50400 x

ii) w(x) = P(x) *h = 50400x*(9-x)

= 453600x - 50400x^2

iii) W = integral 453600x - 50400x^2 dx

= 453600x^2/3 - 50400x^3/4

= 453600*7^2/3 - 50400*7^3/4

= 3087000 J

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