Q. 27 - Calculus

Question

The work required to pump all of the water out of a cone with top radius 3.5 feet and height 4 feet. 

Step-by-Step Solution

Answer

$Work done =15,073,100 J$

Step1. Given Information

Top radius=3.5feet

height=4feet

Step2. Calculate

$$\begin{gathered} 1). dVol=Area dy \\ =\pi r^{2}dy \\ =\pi (5^2-y^2\left)dy \\ 2\right). dF=\pi (5^2-y^2)*9800 \\ 3). dw=dF*(4-y) \\ W={\int }_{y1}^{y2}\pi (5^2-y^2)dy*9800*(4-y) \\ y1=-5,y2=0 \\ =9800{\int }_{-5}^0(5^2-y^2\left)\right(4-y)dy \\ =9800{\int }_{-5}^0(100-25y-4y^2+y^3)dy \\ =9800{{[100y-\frac{25y^2}{2}-\frac{4y3}{3}+\frac{y4}{4}]}_{-5}}^0 =0-9800[100(-5)-\frac{25{(-5)}^2}{2}-\frac{4{(-5)}^3}{3}+\frac{{(-5)}^4}{4}] \\ =15,073,100 J \end{gathered}$$