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Here's the answer to your question:
To find the rate of change of the radius when the water level is at 3 meters in an inverted conical tank, we can follow these steps:
1. First, determine the volume of the water in the tank when the water level is at 3 meters. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the water surface and h is the water level height.
2. Substitute the given values into the formula: h = 3 meters and the radius can be calculated using similar triangles. Since the tank is 8 meters tall with a diameter of 4 meters, the radius at the top of the tank is 2 meters. Using similar triangles, we can set up a proportion to find the radius when the water level is at 3 meters.
3. With the radius at 3 meters calculated, plug this value into the volume formula along with the height of 3 meters to find the volume of water in the tank at that level.
4. Given that water is leaking at a rate of 5000 cm^3/min, this represents the rate of change of the volume of water in the tank with respect to time.
5. To find the rate of change of the radius when the water level is at 3 meters, we can differentiate the volume formula with respect to time using implicit differentiation and then solve for dr/dt, the rate of change of the radius with respect to time.
6. Once the derivative is obtained, substitute the known values such as the rate of change of the volume and the values of radius and height at h = 3 meters to find the rate of change of the radius when the water level is at 3 meters.
By following these steps and calculations, you can determine the rate of change of the radius when the water level is at 3 meters in the inverted conical tank.
Hope this helps you out!
