the tank is half full of oil with a density of 50 pounds per cubic foot. how much work is done in pumping…

Explanation:

Step1: Define variables and setup

Cylinder radius ( r = 4 , \text{ft} ), height ( h = 3 , \text{ft} ), oil height ( 1.5 , \text{ft} ). Density ( \rho = 50 , \text{lb/ft}^3 ). Use horizontal slices at height ( y ) (0 at bottom), thickness ( \Delta y ).

Step2: Calculate volume and force of a slice

Volume of slice: ( \Delta V = \pi r^2 \Delta y = 16\pi \Delta y ).
Force (weight): ( \Delta F = \rho \Delta V = 50 \cdot 16\pi \Delta y = 800\pi \Delta y ).

Step3: Determine lifting distance for each slice

Distance to lift slice to top: ( d = 3 - y ).

Step4: Set up work integral

Work for slice: ( \Delta W = \Delta F \cdot d = 800\pi (3 - y) \Delta y ).
Total work: ( W = \int_0^{1.5} 800\pi (3 - y) , dy ).

Step5: Evaluate the integral

[ W = 800\pi \int_0^{1.5} (3 - y) , dy = 800\pi \left[ 3y - \frac{1}{2}y^2 \right]_0^{1.5} ]
Substitute limits:
[ = 800\pi \left( 3(1.5) - \frac{1}{2}(1.5)^2 \right) = 800\pi \left( 4.5 - 1.125 \right) = 800\pi (3.375) ]
Calculate value:
[ = 2700\pi \approx 2700 \cdot 3.1416 \approx 8482.3 ]

Answer:

8482