Problem 64 - College Algebra

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 \(h\) 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 inches.

Step 1: Substitute known height into the volume formula

We know the height of the cone is 12 feet, hence we substitute \(h = 12\) into the cone volume formula. This gives us \(V = \frac{1}{3} \pi r^{2} \times 12\). Simplifying, the formula for volume becomes \(V = 4 \pi r^{2}\).

Step 2: Solve for radius in terms of volume

We need to express the radius \(r\) in terms of the volume \(V\). From the equation \(V = 4 \pi r^{2}\), we solve for \(r^{2}\): \(r^{2} = \frac{V}{4 \pi}\). Then, take the square root to find \(r\): \(r = \sqrt{\frac{V}{4 \pi}}\).

Step 3: Use the new formula to find the radius for a specific volume

Given a cone with a volume of 50 cubic inches, we substitute \(V = 50\) into the formula derived in the previous step: \(r = \sqrt{\frac{50}{4 \pi}}\). Calculate this to find \(r\).

Step 4: Simplify and calculate

Simplify \(r = \sqrt{\frac{50}{4 \pi}}\) to \(r = \sqrt{\frac{50}{4 \times 3.14159}}\). This gives \(r \approx \sqrt{3.97887}\). Calculate \(r\) to find \(r \approx 1.995\).

Key Concepts

Right Circular Cone

A right circular cone is a three-dimensional geometric shape. It features a circular base connected by a curved surface to a single point, known as the apex. This shape is called "right circular" because the axis (from the apex to the center of the base) is perpendicular to the base.

Understanding components of a right circular cone is crucial when dealing with mathematical problems involving volume or surface area. The key components are:

• Base: A flat circular surface of the cone.
• Height (h): The perpendicular distance from the base to the apex.
• Radius (r): The distance from the center of the base to its edge.

While solving problems involving cones, visualization of these components can significantly aid in conceptual understanding.

Radius in Terms of Height

When solving problems related to cones, you may encounter scenarios where you need to express one dimension in terms of another. This exercise asked to express the radius in terms of height and provided an example, where the height is directly given, simplifying the problem.

To find the radius using the volume formula and a specified height, we first substitute given values into the volume formula for a cone: \[ V = \frac{1}{3} \pi r^2 h \]With a known height, like 12 feet, the volume formula becomes: \[ V = 4 \pi r^2 \]This results from substituting the given height into the formula and simplifying the equation. This simplified form helps in directly relating radius and volume when height is constant, making it easier to solve for the radius.

Solving Equations

Solving equations is a fundamental skill in many mathematical problems. In the context of cones, you often need to rearrange the cone's volume formula to solve for different parameters.

• With the simplified volume equation, \( V = 4 \pi r^2 \), you can rearrange it to solve for \( r^2 \), giving us \( r^2 = \frac{V}{4 \pi} \).
• Taking the square root of both sides then allows you to solve explicitly for the radius, \( r = \sqrt{\frac{V}{4 \pi}} \).

These steps showcase important algebraic manipulation, including division and square rooting, allowing for the determination of one variable in terms of known quantities. Practice on these steps enhances algebraic fluency.

Substitution Method

The substitution method is a useful algebraic technique to replace variables with known values or other expressions, simplifying the original equation. In our exercise, we used substitution to solve for the radius given a specific volume of the cone.

Here's how substitution works in this context:

• First, insert the known value for the height, as done with \( h = 12 \), turning the volume formula to \( V = 4 \pi r^2 \).
• Next, substitute a specific volume, like \( V = 50 \), into the rearranged formula: \( r = \sqrt{\frac{50}{4 \pi}} \).
• This substitution enables you to directly solve for \( r \), resulting in a numerical value.

Through substitution, complex equations can become manageable, allowing easier calculation of variables you're looking to find.