Problem 58 - Algebra 2

Question

Open-Ended. The height \(h\) of a cylinder varies directly with its volume \(V\) and inversely with the square of its radius \(r .\) Find at least four ways to change the volume and radius of a cylinder so that its height is quadrupled.

Step-by-Step Solution

Answer

Four ways to four times the height of the cylinder are: 1. Increasing the volume four times while keeping the radius constant. 2. Halving the radius with a constant volume. 3. Increasing both the volume eight times and the radius two times. 4. Halving the volume and decreasing the radius by \(1/\sqrt{2}\).

Step 1: Increase the volume, keeping the radius constant

If the radius is constant, a fourfold increase in the volume will lead to a fourfold increase in the height. So one solution could be to quadruple the volume while keeping the radius the same.

Step 2: Halve the radius, keeping the volume constant

If the volume is constant, then reducing the radius by half would quadruple the height (since the height varies inversely with the square of the radius). So halving the radius while keeping the volume the same is another solution.

Step 3: Increase the volume and radius proportionately

One could also increase the volume and the radius. For example, to keep the height quadrupled, one might increase the volume eightfold and double the radius. This increases the volume by eight times but because the height varies inversely with the square of the radius, the overall effect is that the height is still quadrupled.

Step 4: Decrease the volume and increase the radius proportionately

One could decrease the volume and increase the radius. For example, if one halves the volume and decreases the radius by a factor of \(1/\sqrt{2}\), the height would be quadrupled. This decreases the volume by half but because the height varies inversely with the square of the radius, the overall effect is that the height is still quadrupled.

Key Concepts

Cylinder Volume

The volume of a cylinder is one of the key aspects when analyzing changes in a cylinder's structure. To understand it better, let's explore the formula for the volume of a cylinder: \[ V = \pi r^2 h \] where \( V \) represents the volume, \( r \) is the radius of the circular base, and \( h \) is the height of the cylinder.
The volume is directly proportional to the height and the square of the radius. This means if you increase the volume, while keeping one of these variables constant, the other variable must change to maintain balance.

• Increasing the volume while keeping the radius constant leads to a larger height.
• Decreasing the volume while keeping the height constant necessitates a smaller radius.

Understanding these relationships is crucial for solving problems involving the cylinder's dimensions and volume dynamics.

Radius Squared

The square of a cylinder's radius plays a significant role in determining the volume and height. In the formula \( V = \pi r^2 h \), the term \( r^2 \) is squared, highlighting its importance.
When the radius is squared, small changes in the radius can have large effects on the volume and height. Hence, understanding this relationship is vital to predict how changes can affect the structure:

• If the radius doubles, the term \( r^2 \) quadruples, affecting the volume significantly.
• If the radius is halved, the term \( r^2 \) becomes one-fourth, which can dramatically alter the cylinder's height when the volume is kept constant.

Every adjustment in the radius leads to a squared impact on the volume, affecting the cylinder's height as well.

Proportional Change

Proportional change refers to maintaining a balance in variables that influence the cylinder's dimensions. This is central to making specific changes in volume and radius to achieve desired alterations in height. To quadruple a cylinder's height through proportional change, different strategies can be employed:

• Quadrupling the volume, maintaining the radius, effects a proportional change in height.
• When the volume remains constant, halving the radius results in a proportional, inverse change that quadruples the height.

The key here is to perform adjustments that balance the effects of radius and volume changes, allowing the height to reach the desired state.

Height Variation

Height variation in a cylinder, based on changes in its volume and radius, includes understanding direct and inverse proportionality. The height of a cylinder directly relates to its volume and inversely to the square of its radius.
Direct Variation:

• When the volume increases, if the radius stays the same, the height also increases.
• An increase in height is directly proportional to an increase in volume, assuming constant radius.

Inverse Variation:

• If the radius decreases, the height must increase if the volume stays constant, due to the relationship being inversely proportional to the square of the radius.

Understanding these concepts allows for strategic manipulation of cylinder dimensions to achieve specific height variations, such as quadrupling the height by adjusting volume and radius effectively.