To find the rate of inflow in ft^3/min, we need to determine the volume of water being added per minute.
The volume of a cone can be calculated using the formula: V = (1/3) * π * r^2 * h, where V is the volume, π is pi (approximately 3.14), r is the radius, and h is the height.
Given:
- The depth of the conical vessel is 15 feet.
- The diameter at the top of the vessel is 7.5 feet, which means the radius (r) is half of the diameter, i.e., 7.5/2 = 3.75 feet.
- The water is rising at a rate of 2 feet per minute when the water depth is 4 feet.
We can calculate the initial height of the water in the conical vessel by subtracting the current water depth from the total depth:
Initial height = Total depth - Current depth = 15 ft - 4 ft = 11 ft
Now, we can calculate the initial volume of the water in the conical vessel using the formula mentioned above:
Initial volume = (1/3) * π * r^2 * h
= (1/3) * 3.14 * (3.75 ft)^2 * 11 ft
≈ 58.35 ft^3
Since the water is rising at a rate of 2 feet per minute, the rate of change of the height (dh/dt) is equal to 2 ft/min.
To find the rate of inflow in ft^3/min, we need to calculate the rate of change of volume with respect to time, dv/dt.
We can use the chain rule of differentiation to calculate dv/dt:
dv/dt = (dv/dh) * (dh/dt)
Differentiating the volume formula with respect to height, we have:
dv/dh = (1/3) * 3.14 * 2 * r^2
= (2/3) * 3.14 * (3.75 ft)^2
≈ 27.875 ft^2
Now, we can calculate the rate of inflow, dv/dt:
dv/dt = (dv/dh) * (dh/dt)
= 27.875 ft^2 * 2 ft/min
= 55.75 ft^3/min
Therefore, the rate of inflow of water is approximately 55.75 ft^3/min.