Problem 23 - Applied Physics

Question

You are asked to design a cylindrical water tank that holds 500,000 gal with radius \(18.0 \mathrm{ft}\). Find its height (1 \(\mathrm{ft}^{3}=7.50\) gal ).

Step-by-Step Solution

Answer

The height of the tank is approximately 65.5 feet.

Step 1: Convert Gallons to Cubic Feet

First, we need to convert the volume of the water tank from gallons to cubic feet. Given that 1 cubic foot is equivalent to 7.5 gallons, we use the conversion factor:\[ \text{Volume in cubic feet} = \frac{\text{Volume in gallons}}{7.5} = \frac{500,000}{7.5} \approx 66666.67 \text{ cubic feet} \]

Step 2: Recall the Formula for the Volume of a Cylinder

The volume of a cylinder is calculated using the formula:\[ V = \pi r^2 h \]where \(V\) is the volume, \(r\) is the radius, and \(h\) is the height of the cylinder. We know that \(V = 66666.67\) cubic feet and \(r = 18.0\) feet.

Step 3: Solve for the Height

Rearrange the volume formula to solve for the height \(h\):\[ h = \frac{V}{\pi r^2} \]Substitute the values into the equation:\[ h = \frac{66666.67}{\pi \times (18.0)^2} \approx \frac{66666.67}{1017.88} \approx 65.5 \text{ feet} \]

Key Concepts

Volume Conversion

Before diving into calculating the dimensions of a cylindrical tank, it's important to be able to convert between different volume units, particularly when designing or working with storage containers. In this exercise, we are provided with a volume in gallons and need to convert it into cubic feet, which is a more suitable unit for measuring the volume of a physical structure such as a tank. One of the core concepts here is understanding how to convert gallons to cubic feet. The conversion factor given is that 1 cubic foot equals 7.5 gallons. To convert gallons to cubic feet, you simply divide the total number of gallons by 7.5.
For instance, if you have 500,000 gallons, the conversion would be:

• Divide 500,000 by 7.5 to find the cubic feet.
• This calculation results in approximately 66,666.67 cubic feet.

This conversion is crucial as it allows you to use the standard formula for calculating the volume of different geometrical shapes accurately.

Cylinder Volume Formula

The formula used to calculate the volume of a cylinder is an essential mathematical concept when dealing with cylindrical structures. This formula is expressed as:\[V = \pi r^2 h\]where:- \(V\) represents the volume of the cylinder,- \(r\) is the radius, - \(h\) is the height of the cylinder, and- \(\pi\) is a constant approximately equal to 3.14159.Understanding this formula is key when you need to determine any one of the cylinder’s dimensions if the other quantities are known.
In the exercise, you were given the volume in cubic feet and the radius, which allows you to rearrange this formula to solve for the height. Knowing the cylinder volume formula not only helps in this exercise but provides a foundation for understanding other problems involving cylindrical shapes.

Height Calculation

Once you have converted the necessary units and have an understanding of the cylinder volume formula, calculating the height of the cylinder becomes straightforward. From the formula for the volume of a cylinder, \(V = \pi r^2 h\), we can solve for \(h\), the height, by rearranging the equation:\[h = \frac{V}{\pi r^2}\]This rearrangement repositions the unknown variable \(h\) on one side of the equation.
For the given problem, we already know:

• The converted volume is approximately 66,666.67 cubic feet.
• The radius is given as 18.0 feet.

Substitute these values into the rearranged formula:

• Calculate \(\pi r^2\) first, which gives about 1,017.88 square feet.
• Next, divide the volume by \(\pi r^2\) to solve for the height \(h\).
• \(h = \frac{66,666.67}{1,017.88} \approx 65.5\) feet.

Understanding how to calculate height allows you to determine the vertical dimension of the tank, ensuring that it will meet the required volume capacity.