the conical tank is full of water that has a density of 62.4 lbs per cubic foot how much work is done in pumping the water out over the top round to the nearest whole number (height 9 ft radius 7 ft)
Answer (1)
1. Volume of the Cone: The volume V of a cone is calculated using the formula:[tex]$$V = \dfrac{1}{3} \pi r^2 h$$[/tex]For a radius [tex]$$r = 7$$[/tex] ft and height [tex]$$h = 9$$[/tex] ft:[tex]$$V = 147 \pi \, \text{cubic feet}$$[/tex]2. Weight of the Water:The weight W is determined by multiplying the density (62.4 lbs/ft³) by the volume:[tex]$$W = 62.4 \times 147 \pi \approx 28856.83 \, \text{lbs}$$[/tex]3. Work to Pump Water:[tex]Work $$W_{d}$$[/tex] in pumping involves integrating the weight based on different heights:[tex]$$W_d = \int_0^9 (62.4 \pi \left(\dfrac{7}{9}y\right)^2 (9 - y)) dy$$[/tex] 4. Total Work Calculation:After calculating the integral and substituting back:[tex]$$W_d = 4596.84 \pi \approx 14452.25$$[/tex] 5. Final Result:Rounding gives approximately 14452 lbs-ft of work done in pumping the water out.
