To find the number of gallons of oil in a horizontally placed cylindrical tank, we need to calculate the volume of oil in the tank first and then convert that volume from cubic feet to gallons.

The volume of oil in a horizontally oriented cylindrical tank can be determined using the formula for the segment of a circle. Since the oil only partially fills the tank, the volume of the liquid is calculated as a segment of a circular cross-section. The formula for the volume $V$ of the liquid in the cylindrical tank is:

$V = L \times A_{\text{segment}}$

Where:

• $L$ is the length of the tank,
• $A_{\text{segment}}$ is the area of the circular segment that corresponds to the liquid level.

Step 1: Area of the circular segment

The area of a circular segment can be calculated using the following formula:

$A_{\text{segment}} = R^2 \cos^{-1}\left(\frac{R-h}{R}\right) - (R-h) \sqrt{2Rh - h^2}$

Where:

• $R = \frac{d}{2}$ is the radius of the tank (half of the diameter),
• $h$ is the depth of the oil,
• $d$ is the diameter of the tank.

Given:

• Diameter $d = 6$ feet, so $R = \frac{6}{2} = 3$ feet,
• Depth of oil $h = 2$ feet,
• Length of tank $L = 12$ feet.

Step 2: Convert volume to gallons

Once we calculate the volume in cubic feet, we can convert it to gallons using the conversion factor:

$1 \, \text{cubic foot} = 7.48 \, \text{gallons}$

Now let's compute the result.The volume of oil in the tank is approximately 740.54 gallons.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. How would the formula change if the tank were placed vertically?
2. What would the volume be if the oil depth increased to 3 feet?
3. How can you calculate the volume if the tank has an elliptical cross-section?
4. What if the tank were a sphere instead of a cylinder—how would you compute the volume?
5. How do you handle units when converting between metric and imperial measurements for volume?

Tip: When calculating the volume of irregular shapes, breaking them down into simpler geometric components can help you apply known formulas more easily.