the tank above is full of water that has a density of 62.4 lbs per cubic foot. how much work is done in…

Explanation:

Step1: Recall the work - formula for pumping liquid

The work (W) done in pumping a liquid is given by (W=\int_{a}^{b}F(y)dy), where (F(y)) is the force required to lift a thin slice of the liquid at height (y) and (dy) is the thickness of the slice. The force (F) to lift a slice of liquid is equal to its weight, and weight (=mg), where (m = \rho V) ((\rho) is density and (V) is volume). For a conical tank of radius (r) and height (h), the volume of a thin slice of water at height (y) from the bottom of the tank with thickness (dy) can be related to the similar - triangles property. For a cone with radius (r = 3) and height (h=9), the radius (x) of a cross - section of the cone at height (y) from the bottom is given by the proportion (\frac{x}{y}=\frac{r}{h}), so (x=\frac{r}{h}y). Here, (r = 3) and (h = 9), so (x=\frac{1}{3}y). The volume of a thin slice of water of thickness (dy) at height (y) from the bottom is (dV=\pi x^{2}dy=\pi(\frac{1}{3}y)^{2}dy=\frac{\pi}{9}y^{2}dy). The mass of the slice of water is (dm=\rho dV), and the force (dF = gdm=\rho g dV). Since (g) is already included in the density unit ((\rho = 62.4) lbs/ft³), the force to lift the slice of water at height (y) is (dF=\rho\cdot dV=62.4\cdot\frac{\pi}{9}y^{2}dy). The distance the slice of water needs to be lifted is (d = 9 - y).

Step2: Set up the work integral

The work done in lifting the slice of water is (dW=dF\cdot d). So (dW=62.4\cdot\frac{\pi}{9}y^{2}(9 - y)dy=62.4\cdot\frac{\pi}{9}(9y^{2}-y^{3})dy). We integrate from (y = 0) to (y = 9) to find the total work done: [ \begin{align*} W&=62.4\cdot\frac{\pi}{9}\int_{0}^{9}(9y^{2}-y^{3})dy\ &=62.4\cdot\frac{\pi}{9}\left[9\cdot\frac{y^{3}}{3}-\frac{y^{4}}{4}\right]{0}^{9}\ &=62.4\cdot\frac{\pi}{9}\left(3y^{3}-\frac{y^{4}}{4}\right)\big|{0}^{9}\ &=62.4\cdot\frac{\pi}{9}\left(3\times9^{3}-\frac{9^{4}}{4}\right)\ &=62.4\cdot\frac{\pi}{9}\left(3\times729-\frac{6561}{4}\right)\ &=62.4\cdot\frac{\pi}{9}\left(2187-\frac{6561}{4}\right)\ &=62.4\cdot\frac{\pi}{9}\left(\frac{8748 - 6561}{4}\right)\ &=62.4\cdot\frac{\pi}{9}\cdot\frac{2187}{4}\ &=62.4\cdot\frac{\pi\times2187}{36}\ &=62.4\cdot\frac{2187\pi}{36}\ &=62.4\times\frac{243\pi}{4}\ &=62.4\times60.75\pi\ &\approx62.4\times60.75\times3.14\ &=62.4\times190.755\ & = 11803.012\approx11803 \end{align*} ]

Answer:

11803