Question

A water utility manager is determining what shaped storage tank should be used to store water for a small mobile home park. The mobile home park needs 110,000 gallons of storage. There is room for a 25 -foot diameter and 30 foot tall cylinder shaped tank or a 30 foot diameter sphere shaped tank. Which tank will provide the adequate storage?

Short answer

The cylinder shaped tank should be used as it provides 14,726.2 cubic feet of storage.

Step-by-step solution

- Identify Formulas

The formula for the volume of a cylinder is: \[ V_{cylinder} = \pi r^2 h \] The formula for the volume of a sphere is: \[ V_{sphere} = \frac{4}{3} \pi r^3 \]

- Convert Units

1 cubic foot of water is approximately 7.48 gallons. The park needs 110,000 gallons of storage, which is: \[ 110,000 \div 7.48 \approx 14,705 \text{ cubic feet} \]

- Calculate Cylinder Volume

The diameter of the cylinder is 25 feet, so the radius is \[ r = \frac{25}{2} = 12.5 \text{ feet} \] The height (h) is 30 feet. Substitute in the volume formula: \[ V_{cylinder} = \pi (12.5)^2 \times 30 \approx 14,726.2 \text{ cubic feet} \]

- Calculate Sphere Volume

The diameter of the sphere is 30 feet, so the radius is \[ r = 15 \text{ feet} \] Substitute in the volume formula: \[ V_{sphere} = \frac{4}{3} \pi (15)^3 \approx 14,137.2 \text{ cubic feet} \]

- Compare Volumes

The required volume of storage is 14,705 cubic feet. Cylinder volume ≈ 14,726.2 cubic feet Sphere volume ≈ 14,137.2 cubic feet

- Conclusion

The cylinder has a volume of 14,726.2 cubic feet, which is more than the required 14,705 cubic feet. The sphere has a volume of 14,137.2 cubic feet, which is less than the required 14,705 cubic feet. Therefore, the cylinder shaped tank should be used.

Key concepts

Volume Calculation

Volume calculation is essential in determining how much space a shape can hold. For different shapes, we use different formulas. It helps in contexts like deciding water storage for a park.

The volume of a cylinder is calculated using: \ \( V_{cylinder} = \pi r^2 h \) . Here, \( r \) is the radius and \( h \) is the height of the cylinder.

The volume of a sphere, on the other hand, uses the formula: \ \( V_{sphere} = \frac{4}{3} \pi r^3 \) , where \( r \) is the radius.

In this exercise, the main goal was to compare these volumes to figure out the best tank for water storage. Don't forget that converting units may be necessary before volume comparison. Let's dive deeper into individual volume calculations.

Cylinder Volume

For a cylindrical tank, start with its radius and height. Here’s how to proceed:

The diameter of the cylinder is given as 25 feet. To find the radius, divide diameter by 2: \ \( r = \frac{25}{2} = 12.5 \) feet.
The height \( h \) is 30 feet.

Substituting these into the volume formula for a cylinder: \ \( V_{cylinder} = \pi (12.5)^2 \times 30 \).
This simplifies to approximately 14,726.2 cubic feet.

This result indicates that the cylindrical tank can hold over 14,700 cubic feet of water, just about what we need for the storage requirements of the mobile home park.

Sphere Volume

Next, let’s figure out the volume of a sphere-shaped tank.

The diameter of the sphere is given as 30 feet. To find the radius: \ \( r = \frac{30}{2} = 15 \) feet.

Using the volume formula for a sphere: \ \( V_{sphere} = \frac{4}{3} \pi (15)^3 \).
Evaluating this gives approximately 14,137.2 cubic feet.

This means the spherical tank can hold a bit over 14,100 cubic feet of water, which is slightly less than required. Comparing this with our cylinder volume, we see why the sphere isn’t the best choice here.

Unit Conversion

Converting units is crucial when working with volume. In our example, the water utility manager needed to convert gallons to cubic feet.

1 cubic foot is approximately 7.48 gallons. Given 110,000 gallons of water:

Conversion calculation: \ \( 110,000 \div 7.48 \approx 14,705 \) cubic feet.

This conversion ensures that we are comparing volumes accurately. Accurate unit conversion bridges different measurement systems and provides a common framework for comparison.