Question

A bucket begins weighing 20 pounds, including the sand it holds. The bucket is to be lifted to the top of a 25 foot tall building by a rope of negligible weight. However, the bucket has a hole in it, and leaks 0.1 pounds of sand each foot it is lifted. Find the work done lifting the bucket to the top of the building

Answer 1

ANSWER

$\begin{equation*} 468.75\text{ foot-pound} \end{equation*}$

EXPLANATION

Let the bucket be lifted x feet.

The weight of the bucket at x feet is given by:

$20-0.1x$

The work done in lifting the bucket by dx feet is:

$dW=(20-0.1x)dx$

The total work done is the integral of the work done in lifting the bucket x feet, that is:

$W=\int(20-0.1x)dx$

Hence, the work done in lifting the bucket 25feet is:

$\begin{gathered} W=\int_0^{25}(20-0.1x)dx \\ \\ W=(20x-\frac{0.1x^2}{2})_0^{25} \\ \\ W=(20*25-\frac{0.1*25^2}{2})-(20*0-\frac{0.1*0^2}{2}) \\ \\ W=500-31.25 \\ \\ W=468.75\text{ foot-pound} \end{gathered}$

That is the answer.