Problem 110 - College Algebra with Modeling and Visualization

Question

Dimension of a Cone (Refer to Exercise \(109 .\) ) A conical water tank holds 100 cubic feet of water and has a diameter of 6 feet. Estimate its height to the nearest tenth of a foot.

Step-by-Step Solution

Answer

The height of the cone is approximately 10.6 feet.

Step 1: Identify the Formula

The volume of a cone is given by the formula \( V = \frac{1}{3} \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height of the cone.

Step 2: Find the Radius

Given the diameter of the cone is 6 feet, we find the radius by dividing the diameter by 2. Therefore, the radius \( r = \frac{6}{2} = 3 \) feet.

Step 3: Plug in Known Values

Insert the known values into the cone volume formula: \( 100 = \frac{1}{3} \pi (3)^2 h \). Here, the volume \( V \) is 100 cubic feet, and the radius \( r \) is 3 feet.

Step 4: Simplify the Equation

First, calculate \( (3)^2 \) to get \( 9 \). Insert this into the formula: \( 100 = \frac{1}{3} \pi \cdot 9 \cdot h \). Simplify further: \( 100 = 3\pi h \).

Step 5: Solve for Height

Isolate \( h \) by dividing both sides by \( 3\pi \). Thus, \( h = \frac{100}{3\pi} \).

Step 6: Calculate the Height

Use \( \pi \approx 3.1416 \) to calculate: \( h = \frac{100}{3 \times 3.1416} \approx 10.6 \) feet.

Key Concepts

Conical Water Tank

A conical water tank is a type of storage container shaped like a cone. This shape is practical for various uses because it allows efficient storage of liquids, such as water, in a more space-saving form. Many water tanks are designed with a conical base to ensure that all the liquid can be effectively drained out.

Such tanks are frequently utilized in both residential and industrial settings. Their unique geometric shape provides a self-cleaning mechanism for sediments that may settle over time. Understanding the geometry of a conical water tank is essential when it comes to calculations involving their volume or dimensions, such as height and radius.

Volume of a Cone Formula

To calculate the volume of a cone, you use the formula: \[ V = \frac{1}{3} \pi r^2 h \]Here:

• \( V \) represents the volume of the cone.
• \( \pi \) is a mathematical constant roughly equal to 3.1416.
• \( r \) is the radius of the base of the cone.
• \( h \) is the height of the cone.

This formula is derived from the concept that a cone's volume is one-third of the volume of a cylinder with the same base and height. Understanding this formula is key to solving many geometry problems involving conical shapes. It is crucial to memorize this and apply it correctly by plugging in the right values for the radius and height.

Solving for Height

When you have the volume of a cone and need to find its height, you can rearrange the volume formula. In situations where the radius and volume are known, like in our exercise, you isolate \( h \). First, substitute the known values into the equation:\[ 100 = \frac{1}{3} \pi (3)^2 h \]Next, simplify it step-by-step:

• Calculate the square of the radius: \( (3)^2 = 9 \).
• Insert this into the equation to simplify it to \( 100 = 3\pi h \).

Now, solve for \( h \) by dividing both sides by \( 3\pi \):\[ h = \frac{100}{3\pi} \]Finally, calculate this expression using \( \pi \approx 3.1416 \) to find:
\( h \approx 10.6 \) feet. Breaking down this calculation step-by-step ensures you get an accurate answer, which is essential for applications like determining a tank's capacity.