Question

Construction The town of West Mountfort recently built a new cylindrical water tower. If the tower is 275 feet tall and has a diameter of 87 feet, how many cubic feet of water can the tank hold? Round the answer to the nearest cubic foot. (Lesson 12-3)

Short answer

The water tower can hold approximately 3,271,023 cubic feet of water.

Step-by-step solution

Identify Given Dimensions

The problem provides us with the height and diameter of the cylindrical water tower. Specifically, the height is 275 feet and the diameter is 87 feet.

Calculate the Radius

The radius of the cylinder can be found by dividing the diameter by 2. Thus, the radius \( r = \frac{87}{2} = 43.5 \) feet.

Use the Volume Formula for a Cylinder

The volume \( V \) of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height of the cylinder.

Substitute Values into the Formula

Substitute the known values into the volume formula: \( V = \pi (43.5)^2 (275) \).

Calculate \( r^2 \)

First, calculate \( (43.5)^2 \). This results in \( 43.5^2 = 1892.25 \).

Calculate the Volume

Substitute \( 1892.25 \) for \( r^2 \) and 275 for \( h \) in the equation: \( V = \pi (1892.25) (275) \).

Multiply Values

Perform the multiplication inside the parentheses first: \( 1892.25 \times 275 = 520368.75 \).

Multiply by \( \pi \)

Multiply the result by \( \pi \): \( V = \pi \times 520368.75 \approx 3,271,023.373 \).

Round the Volume

Round the volume to the nearest cubic foot: approximately \( 3,271,023 \).

Key concepts

Cylindrical Water Tank

A cylindrical water tank is a common structure used to store water for various purposes, including municipal supply and agriculture. Its shape, a perfect cylinder, allows it to hold a significant amount of water while maintaining stability. This cylindrical shape provides uniform stress distribution when the tank is full of water. Understanding the dimensions, such as height and diameter, is crucial to determining its capacity. The overall capacity of such a tank is not only important for its construction but also for utility planning in terms of water supply requirements. For the problem discussed, West Mountfort's cylindrical water tank has a height of 275 feet and a diameter of 87 feet, which are the initial key measurements necessary for calculating how much water it can hold.

Volume Calculation

Volume calculation is an essential concept in geometry, particularly when working with three-dimensional shapes such as cylinders. Calculating the volume helps determine how much space a substance, like water, can occupy within a given shape. To calculate the volume of a cylinder, we need two main pieces of information:

• The radius of the base, which is half of the diameter.
• The height of the cylinder itself.

By using these measurements, the volume can be computed, allowing for practical applications such as designing storage facilities or even water systems like our cylindrical water tank. In the exercise, once the radius and height were identified, the volume was calculated step-by-step to achieve an answer that meets specific rounding requirements.

Geometry Problem-Solving

Geometry problem-solving is a process that involves applying mathematical concepts to solve real-world problems. When approaching geometry tasks, it's crucial to first identify important dimensions and shapes involved. In our example about the water tower, the cylindrical shape drives the problem-solving approach to use cylinder-specific formulas. Good problem-solving also entails:

• Breaking down the problem step by step.
• Accurate calculation practices.
• Use of correct mathematical formulas and operations.

By following these steps, like identifying the radius and applying the correct volume formula, problems become more manageable, leading to precise solutions, such as the tank's maximum water capacity.

Volume Formula for Cylinder

The volume formula for a cylinder is a fundamental part of solving geometry problems related to cylindrical shapes. The formula is expressed as \[ V = \pi r^2 h \] where:

• \( V \) is the volume of the cylinder.
• \( r \) represents the radius of the circular base.
• \( h \) refers to the height of the cylinder.

To use this formula effectively, it's crucial to accurately compute the radius, which is half of the diameter, thus ensuring precise calculations. In the example, the radius was determined as 43.5 feet, leading to an intermediate step of calculating \( r^2 \), which is then used along with the height to find the full volume. This formula underpins many practical problems and is used to determine quantities needed for further logistical planning.