Problem 64 - College Algebra with Corequisite Support

Question

For the following exercises, determine the function described and then use it to answer the question. The volume of a right circular cone, \(V\), in terms of its radius, \(r,\) and its height, \(h,\) is given by \(V=\frac{1}{3} \pi r^{2} h .\) Express \(r\) in terms of \(V\) if the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.

Step-by-Step Solution

Answer

The radius of the cone is approximately 2.0 inches.

Step 1: State the volume formula for a cone

The volume of a right circular 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: Substitute the height into the formula

Since we know the height \( h = 12 \) feet, substitute 12 for \( h \) into the volume formula: \[ V = \frac{1}{3} \pi r^2 (12) \].

Step 3: Simplify the formula

Simplify the formula: \( V = 4\pi r^2 \). This is the formula expressing the volume in terms of the radius when the height is 12 feet.

Step 4: Solve for \( r \) in terms of \( V \)

Rearrange the simplified equation to solve for \( r \): \[ r^2 = \frac{V}{4\pi} \]. Then, take the square root of both sides to find \( r \): \[ r = \sqrt{\frac{V}{4\pi}} \].

Step 5: Substitute known value of V into the equation

Given a cone volume \( V = 50 \) cubic inches, substitute this into the equation: \[ r = \sqrt{\frac{50}{4\pi}} \].

Step 6: Calculate the radius

Evaluate the expression \( \sqrt{\frac{50}{4\pi}} \) to find the radius. Use a calculator to obtain \( r \approx 2.0 \) inches.

Key Concepts

Right Circular Cone

A right circular cone is a 3-dimensional geometric shape with a circular base and a pointed top, known as the apex. The axis of the cone passes through the center of the base and the apex, and this axis is perpendicular to the base, forming a right angle. This is why it's called a right circular cone.
In geometry, cones are important because they can be used to explore the relationships between height, radius, and volume. Understanding these relationships helps in calculating physical properties of cone-shaped objects in real-world applications.

• **Base**: A flat, circular shape, lying in a plane, that serves as the foundation.
• **Apex**: The point at the top of the cone which is directly above the center of the base.
• **Height (h)**: The perpendicular distance from the apex to the plane of the base.

Visualizing these elements helps in understanding not only the structure of the cone but also the calculations involved when dealing with its volume.

Volume Formula

The volume of a right circular cone depends on its radius and height. These dimensions define how much space is contained within the cone.
The formula for the volume is given by: \[ V = \frac{1}{3} \pi r^2 h \] where:

• \(V\) is the volume of the cone.
• \(r\) is the radius of the circular base.
• \(h\) is the height of the cone.

Understanding this formula is crucial for anyone trying to determine how to calculate the space inside a cone. This formula reveals that the volume is directly proportional to both the square of the radius and the height, as well as the constant \(\pi\). It indicates that by increasing either the radius or height, the volume of the cone increases.

Radius Calculation

To find the radius of the cone given the volume and height, rearrange the cone's volume formula to express \(r\) in terms of \(V\) and \(h\). This involves algebraic manipulation.
Starting with the volume formula: \[ V = \frac{1}{3} \pi r^2 h \] If the height is given (in this case, 12 feet), substitute it into the formula: \[ V = \frac{1}{3} \pi r^2 (12) \] Simplifying, we obtain: \[ V = 4\pi r^2 \] To solve for \(r\), rearrange the equation: \[ r^2 = \frac{V}{4\pi} \] Taking the square root of both sides: \[ r = \sqrt{\frac{V}{4\pi}} \] This formula allows us to find the radius when we know the volume and the height. For a volume of 50 cubic inches, substitute in and calculate: \[ r = \sqrt{\frac{50}{4\pi}} \] Using a calculator gives \(r \approx 2.0\) inches. This radius tells us the size of the base for a cone of known volume and height.

Algebraic Manipulation

Algebraic manipulation is the process of transforming equations to isolate and solve for a specific variable. In the context of the cone volume problem, algebraic manipulation involves using operations like substitution, simplification, and rearranging terms.
Initially, you have the cone volume equation: \[ V = \frac{1}{3} \pi r^2 h \] After substituting the given height (\(h = 12\) ft), the equation becomes: \[ V = 4\pi r^2 \] From here, to isolate \(r\), divide both sides by \(4\pi\): \[ r^2 = \frac{V}{4\pi} \] Finally, solve for \(r\) by taking the square root of both sides: \[ r = \sqrt{\frac{V}{4\pi}} \] This methodical approach makes it easier to solve problems that involve unknown variables, enabling quick and efficient calculations. It showcases the power of algebra to transform and solve equations in a way that reveals unknowns.