Problem 50 - Algebra Form and Function

Question

The volume of a cone of height 2 and radius \(r\) is \(V=\frac{2}{3} \pi r^{2} .\) What is the radius of such a cone whose volume is \(3 \pi ?\)

Step-by-Step Solution

Answer

Answer: The radius of the cone is \(\frac{3}{\sqrt{2}}\).

Step 1: Write down the formula for the volume of the cone

The given formula for the volume of the cone is \(V=\frac{2}{3} \pi r^{2}\).

Step 2: Substitute the given volume into the formula

We know that the volume of the cone is \(3\pi\). Substitute this value into the volume formula: $$3\pi=\frac{2}{3}\pi r^{2}$$

Step 3: Solve for the radius (r)

Now, we need to solve for r in the equation $$3\pi=\frac{2}{3}\pi r^{2}$$. First, divide both sides by \(\pi\): $$3=\frac{2}{3}r^{2}$$ Next, multiply both sides by \(\frac{3}{2}\) to isolate \(r^2\): $$r^2=\frac{9}{2}$$ Now find the square root of both sides to get the value of r: $$r=\sqrt{\frac{9}{2}}$$

Step 4: Simplify the value of the radius (r)

To further simplify, we can express the square root of a fraction as the square root of the numerator divided by the square root of the denominator: $$r=\frac{\sqrt{9}}{\sqrt{2}}$$We know that the square root of 9 is 3, so our final answer for the radius is: $$r=\frac{3}{\sqrt{2}}$$ Thus, the radius of the cone with a volume of \(3\pi\) is \(\frac{3}{\sqrt{2}}\).

Key Concepts

Radius

Understanding the concept of a radius is key when working with shapes like cones. In the context of a cone, the radius refers to the distance from the center of the circular base to its edge.
It's a straight line that helps define the size of the base. Since the radius is squared in the cone's volume formula, even small changes in the radius can significantly affect the volume.
Knowing how to find or calculate the radius when given other measurements, such as volume or height, can be vital in solving related problems involving cones or other circular shapes.

Volume Formula

The volume formula for a cone gives us a way to calculate how much space is inside the cone. Specifically, the volume of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
This formula utilizes the radius (\(r\)) of the base of the cone, the height (\(h\)) of the cone, and \(\pi\), which is a constant approximately equal to 3.14159. However, in this particular exercise, the formula provided is slightly different since the height (\( h = 2 \)). The formula simplifies to:
\[ V = \frac{2}{3} \pi r^2 \]
The simplified version makes it easy to solve problems once the volume is known, by simply substituting the value of the volume into the equation and solving for the radius or another missing variable.

Solving Equations

Solving equations is a fundamental skill in mathematics. To find the radius of the cone in the given problem, we start with the equation derived from the volume formula:
\[ 3\pi = \frac{2}{3} \pi r^2 \]
To solve for \(r\), we first eliminate \(\pi\) from both sides, which simplifies the equation to \( 3 = \frac{2}{3} r^2 \). Next, multiplying both sides by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \), allows us to isolate \( r^2 \):\[ r^2 = \frac{9}{2} \]
Finally, taking the square root of both sides gives us the radius:\[ r = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}} \]
This step-by-step method provides a clear approach to isolating and solving for a variable, an essential technique in algebra.