A water heater in the shape of a circular cylinder is standing vertically, resting on one of the round ends. The tank is 60 inches tall and has a volume of 30 gallons. If water is flowing into the tank at the rate of 5 gallons per minute, at what rate is the level of the water in the tank increasing?
the volume of a 1" layer of water is 1/2 gal. So, when the height of the water is h inches, the volume of water is
v = 1/2 h
dv/dt = 1/2 dh/dt
5 = 1/2 dh/dt
so,
dh/dt = 10 in/min
You can go through all that pi r^2 stuff, but it's not needed here.
To find the rate at which the level of the water in the tank is increasing, we need to use the formula for the volume of a cylinder.
The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the volume of the water heater is 30 gallons and the tank is in the shape of a circular cylinder, we convert the volume from gallons to cubic inches.
1 gallon = 231 cubic inches
So, the volume of the tank in cubic inches is 30 * 231 = 6930 cubic inches.
We are given that water is flowing into the tank at a rate of 5 gallons per minute. To find the rate at which the level of the water is increasing, we need to find the rate at which the volume is increasing.
Let's call the rate at which the level of the water is increasing, dh/dt (in inches per minute).
We are also given that the tank is 60 inches tall. So, the height of the tank, h, at any given time, is 60 inches.
Since we want to find the rate of change of the volume with respect to time, we can differentiate the volume formula with respect to height:
dV/dt = πr^2(dh/dt)
We want to find dh/dt (the rate at which the height is increasing), so we rearrange the equation:
dh/dt = (dV/dt) / (πr^2)
Now, we substitute the values we know into the equation. We know that dV/dt (the rate of change of volume with respect to time) is 5 gallons per minute. Again, we need to convert this to cubic inches per minute:
1 gallon = 231 cubic inches
So, dV/dt = 5 gallons/minute * 231 cubic inches/gallon = 1155 cubic inches/minute.
We are not given the radius, but we can calculate it using the formula for the volume of a cylinder:
V = πr^2h
6930 = πr^2 * 60
r^2 = 6930 / (π * 60)
r^2 ≈ 11.62
r ≈ √(11.62) ≈ 3.41 inches
Now we can substitute the values we know into the equation to find dh/dt:
dh/dt = (1155 cubic inches/minute) / (π * (3.41 inches)^2)
dh/dt ≈ 32.54 cubic inches/minute
Therefore, the rate at which the level of the water in the tank is increasing is approximately 32.54 cubic inches per minute.
To find the rate at which the level of water in the tank is increasing, we need to use related rates.
Let's start by finding the rate at which the volume of the water in the tank is increasing. We know that water is flowing into the tank at a rate of 5 gallons per minute.
The volume of a circular cylinder can be calculated using the formula: V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the water heater has a volume of 30 gallons and 1 gallon is approximately 231 cubic inches, we can convert the volume to cubic inches:
V = 30 gallons * 231 cubic inches/gallon ≈ 6930 cubic inches.
Now, we can differentiate both sides of the volume formula with respect to time t:
dV/dt = d(πr^2h)/dt.
Since the radius r is constant (as the shape of the tank is not changing), we can differentiate only the height h with respect to time t:
dV/dt = πr^2 * dh/dt.
We know that dV/dt = 5 gallons per minute, and we want to find dh/dt. Therefore, we have the equation:
5 = πr^2 * dh/dt.
To find dh/dt, we need to know the radius r. However, it is not provided in the question. Without the radius, we cannot determine the exact rate at which the water level is increasing. The radius is essential to find the value of dh/dt because the volume formula involves both the radius and the height of the cylinder.
Hence, without the radius, we cannot find the rate at which the water level in the tank is increasing.