A conical tank sits point down, at ground level, so that the circular top of the tank is parallel
to the ground. The tank is 5.0 meters tall, and the top of the tank has a diameter of 3.0 meters. The tank is half full of water. What is the water level (depth)?
The volume of similar solids is proportional to the cube of the linear ratios.
For example, a cube of side x has a volume of (1/2)³=1/8 of another cube of side 2x.
Look for (1/?)³=1/2.
To find the water level (depth) in the tank, we need to use the formula for the volume of a cone.
The volume of a cone is given by the formula:
V = (1/3) * π * r^2 * h
Where:
V is the volume of the cone
π is the mathematical constant Pi (approximately 3.14159)
r is the radius of the circular top of the cone
h is the height (or depth) of the cone
In this case, we know that the height of the cone is 5.0 meters and the diameter of the top is 3.0 meters. Since the diameter is twice the radius, the radius would be 1.5 meters.
Now, since the tank is half full of water, we need to find the volume of the entire tank and then divide it by 2.
The volume of the entire tank is given by the formula for a cone:
V_full = (1/3) * π * r^2 * h
Plugging in the values, we have:
V_full = (1/3) * 3.14159 * 1.5^2 * 5.0
= 11.7809725 cubic meters
Now, since the tank is half full, the volume of water in the tank would be:
V_water = V_full / 2
= 11.7809725 / 2
= 5.89048625 cubic meters
To find the water level (depth), we need to rearrange the formula for the volume of a cone to solve for h:
h = V_water / (π * r^2)
Plugging in the values, we have:
h = 5.89048625 / (3.14159 * 1.5^2)
≈ 0.832 meters
Therefore, the water level (depth) in the tank is approximately 0.832 meters.