Question:
A closed cylindrical tank contains 36\(\pi\) cubic feet of water and is filled to half its capacity. When the tank is placed upright on its circular base on level ground, the height of the water in the tank is 4 feet. When the tank is placed on its side on level ground, what is the height, in feet, of the surface of the water above the ground?
A closed cylindrical tank contains 36\(\pi\) cubic feet of water and is filled to half its capacity. When the tank is placed upright on its circular base on level ground, the height of the water in the tank is 4 feet. When the tank is placed on its side on level ground, what is the height, in feet, of the surface of the water above the ground?
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In geometry problems on standardized tests, if your calculation results in an irrational number (like \(\sqrt{9/\pi}\)) but the options are all clean integers, double-check the problem statement. It's highly likely that a \(\pi\) was intended to be part of a given value to allow for cancellation.
Updated On: Oct 3, 2025
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The Correct Option is B
Solution and Explanation
*(Note: The OCR of the question likely omitted a \(\pi\) symbol from the volume, which is a common issue. The solution proceeds assuming the volume of water is \(36\pi\) cubic feet, as this leads to one of the integer answers provided.)*
Step 1: Understanding the Concept:
The problem involves a cylinder and its properties (volume, radius, height). We need to use the information from when the tank is upright to find its radius. Then, we use this radius to determine the water level when the tank is on its side.
Step 2: Key Formula or Approach:
The volume of a cylinder is given by the formula: \[ V = \pi r^2 h \] where \(r\) is the radius of the circular base and \(h\) is the height.
Step 3: Detailed Explanation:
Part 1: Find the radius of the tank.
We are given that when the tank is upright, the volume of the water is \(36\pi\) cubic feet and the height of the water is 4 feet.
Using the volume formula for the water in the tank: \[ V_{\text{water}} = \pi r^2 h_{\text{water}} \] Substitute the known values: \[ 36\pi = \pi r^2 (4) \] To solve for \(r\), we can first divide both sides by \(4\pi\): \[ \frac{36\pi}{4\pi} = r^2 \] \[ 9 = r^2 \] \[ r = 3 \text{ feet} \] So, the radius of the cylindrical tank is 3 feet.
Part 2: Determine the water height when the tank is on its side.
The problem states that the tank is filled to half its capacity.
When a cylinder is placed on its side, its total "height" from the ground is its diameter (\(2r\)).
If the cylinder is exactly half-full, the water level will be a flat surface passing through the central axis of the cylinder.
Therefore, the height of the water surface above the ground is equal to the radius of the cylinder.
\[ \text{Height of water surface} = r = 3 \text{ feet} \] Step 4: Final Answer:
When the tank is placed on its side, the height of the surface of the water above the ground is 3 feet.
Step 1: Understanding the Concept:
The problem involves a cylinder and its properties (volume, radius, height). We need to use the information from when the tank is upright to find its radius. Then, we use this radius to determine the water level when the tank is on its side.
Step 2: Key Formula or Approach:
The volume of a cylinder is given by the formula: \[ V = \pi r^2 h \] where \(r\) is the radius of the circular base and \(h\) is the height.
Step 3: Detailed Explanation:
Part 1: Find the radius of the tank.
We are given that when the tank is upright, the volume of the water is \(36\pi\) cubic feet and the height of the water is 4 feet.
Using the volume formula for the water in the tank: \[ V_{\text{water}} = \pi r^2 h_{\text{water}} \] Substitute the known values: \[ 36\pi = \pi r^2 (4) \] To solve for \(r\), we can first divide both sides by \(4\pi\): \[ \frac{36\pi}{4\pi} = r^2 \] \[ 9 = r^2 \] \[ r = 3 \text{ feet} \] So, the radius of the cylindrical tank is 3 feet.
Part 2: Determine the water height when the tank is on its side.
The problem states that the tank is filled to half its capacity.
When a cylinder is placed on its side, its total "height" from the ground is its diameter (\(2r\)).
If the cylinder is exactly half-full, the water level will be a flat surface passing through the central axis of the cylinder.
Therefore, the height of the water surface above the ground is equal to the radius of the cylinder.
\[ \text{Height of water surface} = r = 3 \text{ feet} \] Step 4: Final Answer:
When the tank is placed on its side, the height of the surface of the water above the ground is 3 feet.
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