A leaky 10-kg bucket is lifted from the ground to a height of 13 m at a constant speed with a rope that weighs 0.5 kg/m. Initially the bucket contains 39 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 13-m level. Find the work done. (Use 9.8 m/s for g.) Show how to approximate the required work by a Riemann sum. (Let x be the height in meters above the ground. Enter x,* as x,.) lim Дх Express the work as an integral. dx Evaluate the integral. (Round your answer to the nearest integer.)

A cement mixer is pouring concrete in a conical pile. At the time when the height and base radius of the concrete cone are, respectively, 3 and 12 feet, the rate at which the height is increasing is 1 feet per minute, and the rate at which the volume of cement in the pile is increasing is 96π cubic feet per minute. At that moment, how fast is the radius of the cone changing? (Volume of a cone with radius r and height h is given by V=13πr2h.) (Hint: Use the multi-variate chain rule)

Need only a handwritten solution only (not a typed one).

A leaky 10-kg bucket is lifted from the ground to a height of 10 m at a constant speed with a rope that weighs 0.7 kg/m. Initially the bucket contains 30 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 10-m level. How much work is done? (Use 9.8 m/s² for g.) Show how to approximate the required work (in J) by a Riemann sum. (Let x be the height in meters above the ground. Enter x.* as x₁.) Σ( j=1 Express the work (in J) as an integral in terms of x (in m). lim n-∞ Ax dx Evaluate the integral (in J). (Round your answer to the nearest integer.)