Applications of Integration · Challenge problem

An empty bucket weighs $4$ lb. It is filled with $16$ lb of sand and hoisted straight up a $16$-ft mine shaft at a constant rate. As it rises, sand leaks out of the bucket at a constant rate, so that the bucket holds no sand by the time it reaches the top. At what height $h$ (in feet) has the cumulative work done hoisting the bucket reached exactly $128$ ft-lb?