Problem 23 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. The radius of a cylinder is twice as long as the radius of a cone, and the height of the cylinder is half as long as the height of the cone. What is the ratio of the volume of the cylinder to that of the cone?

Step-by-Step Solution

Answer

The ratio of the volume of the cylinder to the cone is 6:1.

Step 1: Identify Variables

Let the radius of the cone be \( r \) and the height of the cone be \( h \). Then the radius of the cylinder is \( 2r \) and the height of the cylinder is \( \frac{h}{2} \).

Step 2: Formula for Volume of Cone

The formula for the volume of a cone is \( V_{cone} = \frac{1}{3} \pi r^2 h \).

Step 3: Formula for Volume of Cylinder

The formula for the volume of a cylinder is \( V_{cylinder} = \pi (\text{radius})^2 (\text{height}) \). Here, use \( 2r \) and \( \frac{h}{2} \) for the cylinder's radius and height, respectively.

Step 4: Calculate Volume of Cylinder

Substitute the values into the formula: \( V_{cylinder} = \pi (2r)^2 \left( \frac{h}{2} \right) = \pi \cdot 4r^2 \cdot \frac{h}{2} = 2\pi r^2 h \).

Step 5: Expression for Volume of Cone

Using the cone's formula, we have: \( V_{cone} = \frac{1}{3} \pi r^2 h \).

Step 6: Compute Ratio of Volumes

Take the ratio of the cylinder's volume to the cone's volume: \( \frac{V_{cylinder}}{V_{cone}} = \frac{2\pi r^2 h}{\frac{1}{3} \pi r^2 h} \).

Step 7: Simplify the Ratio

Simplify the expression: \( \frac{2\pi r^2 h}{\frac{1}{3} \pi r^2 h} = \frac{2}{\frac{1}{3}} = 6 \).

Key Concepts

Cylinder Volume Exploration

Understanding the volume of a cylinder is crucial when dealing with geometric problems. A cylinder is a 3-dimensional shape with two parallel circular bases and a straight side called the lateral surface. To find the volume of a cylinder, you use the formula:

• Volume of a Cylinder, \( V_{cylinder} = \pi \, (\text{radius})^2 \, (\text{height}) \)

In this problem, you start with the given dimensions where the radius of the cylinder is twice as long as that of the cone, and its height is half of the cone's height. Plug these new values into the cylinder's formula. The radius becomes \(2r\) and the height \(\frac{h}{2}\).
So the calculation for the cylinder's volume is: \[ V_{cylinder} = \pi (2r)^2 \left(\frac{h}{2}\right) = 2\pi r^2 h \] This modification allows the volume to factor in how a change in dimensional ratios uniquely alters the cylinder's volume compared to its original formulation.

Cone Volume Examination

The volume of a cone can sometimes seem tricky because the formula includes the factor \(\frac{1}{3}\). However, it's quite intuitive. The cone can be thought of as a pyramid-like shape with a circular base. Its volume is one-third the volume of a cylinder with the same base and height. The formula is:

• Volume of a Cone, \( V_{cone} = \frac{1}{3} \pi r^2 h \)

Here, \(r\) and \(h\) represent the radius and height of the cone, respectively. Using these values, we find:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \] This calculation expresses how the volume of a cone is precisely a third of what a cylinder's would be, if they both had the identical spatial configuration.

Geometric Problem Solving Approach

Geometric problem-solving requires applying formulas strategically to understand spatial relationships. In exercises like this, it's essential to:

• Identify known variables and relationships.
• Use corresponding formulas to set up expressions.
• Compare these expressions to find quantitative relationships, as seen in volume ratio computations.

For instance, in solving the given problem, knowing that the dimensions of the cylinder and cone are connected through their relational increase or decrease, helps in deducing the volume ratio.
Analyzing their relationship gives an easy path to calculate the final result by simplifying:
\[ \frac{V_{cylinder}}{V_{cone}} = \frac{2\pi r^2 h}{\frac{1}{3} \pi r^2 h} = 6 \]This means the cylinder's volume is six times that of the cone, highlighting how small changes in dimensions impact volume considerably.