Aleaky 10-kg bucket is lifted from the ground to a height of 16 m at a constant speed with a rope that weighs 0.7 kg/m. initially the bucket contains 48 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 16-m level. find the work done. (use 9.8 m/s2 for g.) show how to approximate the required work by a riemann sum. (let x be the height in meters above the ground. enter xi* as xi.)

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The mass comes in three parts:

Both the mass of the rope and the mass of the water in the bucket varies with the height of the bucket. Express the two masses as a function of :

The water in the bucket behaves like yet another rope of density . The mass of the water left in the bucket at height will be

The mass of the bucket is . Combining the three:

Weight of the bucket at height :

The bucket moves upward at a constant speed. As a result,

Express the work required as a definite integral:

Rewrite the definite integral as a Riemann Sum:

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