How to Calculate Water Level and Boat Speed in Conical Reservoir Problems?

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The discussion focuses on calculating the rate of change of water level and surface radius in a conical reservoir and the speed of a dinghy approaching a dock. For the conical reservoir, the water is draining at 50 m^3/min, and the problem requires determining how fast the water level is falling when it reaches 5m deep and how fast the radius of the water's surface is changing. In the dinghy scenario, the rope is being pulled in at 2 ft/sec, and the task is to find the boat's approach speed when 10ft of rope is out and the rate of change of the angle theta. The participants express difficulty in formulating the necessary equations for both problems. The thread highlights the need for clear application of volume and geometry principles in solving these related rates problems.
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Homework Statement


A draining conical reservoir. Water is flowing at the rate of 50 m^3/min from a shalloe concrete conical reservoir (vertex down) of base radius 45m and height of 6m.

a. How fast (centimeters per minute) is the water level falling when the water is 5m deep?

b. How fast is the radius of the water's surface changing then? Answer in centimeters per minute.

2. Hauling in a dinghy. A dinghy is pulled toward a dock by a rope from the bow through a ring to the dock 6ft above the bow. The rope is hauled in at the rate of 2 ft/sec.

a. How fast is the boat approaching the dock when 10ft of rope are out?

b. At what rate is the angle theta changing then (see the figure)?

Homework Equations


1.

a. Not given, but cone volume = 1/3*pi*r2*h

The Attempt at a Solution


1.

a. dv/dt = 50/(pi)(45)2(5) or dv/dt = 50/(pi)(45y/6)2(6)

b. Nothing.

2.

a. Nothing.

b. Nothing.
 

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