Show how to approximate the required work by a Riemann sum. Then expre

veudeje 2021-11-13 Answered
Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5lb/ft^3.)
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Expert Answer

Jick1984
Answered 2021-11-14 Author has 12 answers
Circular pool has diameter 24 ft, so radius r=12 ft.
The sides are 5 ft high, water depth is 4 ft.
A horizotal "slice" of the weater with thickness x" has volume
V=πr2x=π(12)2x=144πx ft3
The force is obtained by multiplying by the weight per unit volume
F=144πx ft362.5lbft3=9000πx
If we let x = the heigh from the bottom of the pool, then each slice of water has to move a distance of
d=5x
Because the side of the pool is 5 ft high.
If xi represents this height for the i th then we can write the Riemann sum
W=limni=0n9000πxix
To sum the total work, we integrate multiplying the force times distance. The slices become infinitely thin. The water is 4ft, so the limits are from 0 to 4.
W=04(9000π)(5x)dx
=9000π[5x12x2]04
=900π[208(0)]
=108000π339292 ft-lb
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