Conic Tank and Cylindrical Tank: Rate of Change of Water Level
What is the rate of change of the water level in the cylindrical tank?
Given the data, what can we find about the rate of change of the water level in the cylindrical tank as water is being filled from the conic tank?
Rate of Change of Water Level in the Cylindrical Tank
The rate of change of the water level in the cylindrical tank is 0. The water level in the cylindrical tank does not change as water is being filled from the conic tank.
We are filling water from a conic tank (with vertex down) to a cylindrical tank. The conic tank has a height of 18 m and the radius of the base is 6 m. The diameter of the cylindrical tank's base is 10 m.
The water level in the conic tank is decreasing at 1m/s when the water level is 12 m. To find the rate of change of the water level in the cylindrical tank, we can use the concept of similar triangles and the relationship between the volumes of a cone and a cylinder.
Calculating the Rate of Change of Water Levels
1. Calculate the rate of change of the volume of water in the conic tank:
- The volume of a cone is given by V = (1/3)π*r^2*h, where r is the radius of the base and h is the height.
- Differentiating the volume equation with respect to time, we get dV/dt = (1/3)π(2rh)(dh/dt), where dV/dt represents the rate of change of the volume of the cone, and dh/dt represents the rate of change of the height of the water in the cone.
- Substituting the given values, we have dV/dt = 48π m^3/s.
2. Calculate the rate of change of the volume of water in the cylindrical tank:
- The volume of a cylinder is given by V = π*r^2*h, where r is the radius of the base and h is the height.
- Using similar triangles, we find the relationship between the height of the water in the cone and the height of the water in the cylinder.
- By setting up the proportion, we determine that the height of the water in the cylindrical tank is 20/3 m.
- Differentiating the volume equation with respect to time, we get dV/dt = 0, as the radius of the cylinder is constant and there is no change in the volume of the cylinder.
In conclusion, the rate of change of the water level in the cylindrical tank is 0. The water level in the cylindrical tank remains constant as water is being filled from the conic tank.