Problem 30 - Algebra for College Students

Question

The volume of a cylinder varies jointly as its altitude and the square of the radius of its base. If the volume of a cylinder is 1386 cubic centimeters when the radius of the base is 7 centimeters, and its altitude is 9 centimeters, find the volume of a cylinder that has a base of radius 14 centimeters if the altitude of the cylinder is 5 centimeters.

Step-by-Step Solution

Answer

The new volume is 3077.2 cubic centimeters.

Step 1: Understand the Joint Variation Formula

The volume of a cylinder varies jointly as the altitude (height) and the square of the radius of its base. This can be expressed mathematically as \( V = k \cdot r^2 \cdot h \), where \( V \) is the volume, \( r \) is the radius, \( h \) is the altitude, and \( k \) is a constant of variation.

Step 2: Determine the Constant of Variation

Given that the volume \( V = 1386 \) cubic centimeters when \( r = 7 \) cm and \( h = 9 \) cm, substitute these values into the equation to find \( k \): \[1386 = k \cdot 7^2 \cdot 9\]This simplifies to: \[1386 = k \cdot 441\]Thus, \[k = \frac{1386}{441} = 3.14\]

Step 3: Calculate the Desired Volume with New Dimensions

Now use the constant \( k = 3.14 \) to find the volume of the cylinder with \( r = 14 \) cm and \( h = 5 \) cm:\[V = 3.14 \cdot 14^2 \cdot 5\]Calculate \( 14^2 = 196 \) and then:\[V = 3.14 \cdot 196 \cdot 5\]Simplify to find:\[V = 3.14 \cdot 980 = 3077.2\]Thus, the volume is 3077.2 cubic centimeters.

Key Concepts

Volume of a Cylinder

The volume of a cylinder is the amount of space contained within its circular base and straight sides. To find the volume, we use the formula: \[ V = \pi r^2 h \]In this formula, \( V \) represents the volume, \( r \) the radius of the circular base, and \( h \) the height or altitude of the cylinder. This formula calculates how much cube-shaped space can fit inside the cylinder.
A cylinder with a larger base or a greater height will naturally have a greater volume. When you increase the radius or the height, the volume expands, reflecting the joint variation relationship.
Remember: Volume is always measured in cubic units, such as cubic centimeters or cubic meters, because it encompasses three-dimensional space.

Altitude

Altitude in the context of a cylinder is another name for its height. It represents the distance from the base of the cylinder to the top when standing upright.
Altitude is crucial because it is directly proportional to the volume of a cylinder.

• A taller cylinder (with more altitude) will hold more volume.
• A shorter cylinder will contain less volume, assuming the base's radius remains constant.

Understanding altitude helps visualize how the volume can be affected when the height of the cylinder is changed. This concept underscores the idea of joint variation, where changes in height will directly influence the total volume.

Radius

The radius of a cylinder is the distance from the center of the base circle to its edge. In mathematical terms, it is essential because it is used to calculate the area of the base with \( \pi r^2 \).
The radius is significant in the volume of a cylinder because it is squared in the volume formula: \( V = \pi r^2 h \).

• A larger radius leads to a bigger base, meaning the cylinder can hold more space.
• If the radius doubles, the area of the base and subsequently the volume increase by a factor of four, due to the squaring in the formula.

This relationship between the radius and volume efficiency captures the effect of joint variation, highlighting how changes in the radius have a more noticeable impact on volume than changes in other dimensions.

Constant of Variation

In problems involving joint variation, the constant of variation \( k \) plays a crucial role. For a cylinder, where volume \( V \) depends on the radius squared \( r^2 \) and the altitude \( h \), it is expressed in the equation: \[ V = k \cdot r^2 \cdot h \]This constant \( k \) accounts for the condition under which the other variables interact to determine volume.
To find \( k \), we use given conditions (specific values of \( r, h, \) and \( V \)) to solve for it. Once known, this constant helps in calculating the volume for other sets of \( r \) and \( h \), making it invaluable for predicting and comparing different scenarios.

• Knowing \( k \), you can reliably calculate the volume for any cylinder with different dimensions.
• It stays constant across these calculations unless the characteristics or medium in which these cylinders exist change.

The constant is a fundamental concept, especially in physics and engineering, as it embodies the consistent and predictable nature of certain relationships.