Calculating Height Increase for Constant Cylinder Volume

The problem asks us to find the percentage increase in the height of a cylinder required to keep its volume constant, given that its radius is decreased by 16 percent.

The volume of a cylinder is given by the formula:

\(V = \pi r^2 h\)

Where:

• \(V\) is the volume
• \(r\) is the radius
• \(h\) is the height

Let's denote the original radius and height as \(r_1\) and \(h_1\), respectively. The original volume is \(V_1 = \pi r_1^2 h_1\).

The radius is decreased by 16 percent. So, the new radius, \(r_2\), is:

\(r_2 = r_1 - 0.16 r_1\)

\(r_2 = (1 - 0.16) r_1\)

\(r_2 = 0.84 r_1\)

Let the new height be \(h_2\). The new volume, \(V_2\), is \(V_2 = \pi r_2^2 h_2\).

According to the problem, the volume of the cylinder remains the same, which means \(V_1 = V_2\).

So, we have:

\(\pi r_1^2 h_1 = \pi r_2^2 h_2\)

We can cancel \(\pi\) from both sides:

\(r_1^2 h_1 = r_2^2 h_2\)

Now, substitute \(r_2 = 0.84 r_1\) into the equation:

\(r_1^2 h_1 = (0.84 r_1)^2 h_2\)

\(r_1^2 h_1 = (0.84)^2 r_1^2 h_2\)

Since \(r_1\) is the original radius, it is not zero, so we can divide both sides by \(r_1^2\):

\(h_1 = (0.84)^2 h_2\)

Now, we need to find \(h_2\) in terms of \(h_1\):

\(h_2 = \frac{h_1}{(0.84)^2}\)

Let's calculate \((0.84)^2\):

\((0.84)^2 = 0.7056\)

So,

\(h_2 = \frac{h_1}{0.7056}\)

To find the percentage increase in height, we use the formula:

\(\text{Percentage increase} = \left(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}\right) \times 100\%\)

\(\text{Percentage increase in height} = \left(\frac{h_2 - h_1}{h_1}\right) \times 100\%\)

We can rewrite this as:

\(\text{Percentage increase in height} = \left(\frac{h_2}{h_1} - 1\right) \times 100\%\)

From our equation \(h_2 = \frac{h_1}{0.7056}\), we can get the ratio \(\frac{h_2}{h_1}\):

\(\frac{h_2}{h_1} = \frac{1}{0.7056}\)

Now, substitute this ratio into the percentage increase formula:

\(\text{Percentage increase in height} = \left(\frac{1}{0.7056} - 1\right) \times 100\%\)

Calculate the value:

\(\frac{1}{0.7056} \approx 1.417163\)

\(\text{Percentage increase in height} \approx (1.417163 - 1) \times 100\%\)

\(\text{Percentage increase in height} \approx 0.417163 \times 100\%\)

\(\text{Percentage increase in height} \approx 41.7163\%\)

Rounding to two decimal places, the height must be increased by approximately 41.72 percent.

Let's summarise the key steps:

• Identify the volume formula for a cylinder: \(V = \pi r^2 h\).
• Set up the initial and final volumes with corresponding dimensions.
• Use the given percentage decrease in radius to find the new radius.
• Equate the initial and final volumes.
• Solve for the ratio of the new height to the original height.
• Calculate the percentage increase using this ratio.

Revision Table: Cylinder Volume Calculations

Additional Information on Volume and Percentage Changes

This problem illustrates an important concept in geometry and quantitative aptitude: how percentage changes in dimensions affect the volume of a 3D shape. For a cylinder, the volume depends on the square of the radius and the height. When the volume is kept constant, there is an inverse relationship between the square of the radius and the height. If the radius decreases, the height must increase to compensate, and vice versa.

Specifically, since \(V = \pi r^2 h\) is constant, \(r^2 h = \text{constant}\). If the radius changes from \(r_1\) to \(r_2\) and the height changes from \(h_1\) to \(h_2\), we have \(r_1^2 h_1 = r_2^2 h_2\). This implies \(\frac{h_2}{h_1} = \left(\frac{r_1}{r_2}\right)^2\). The percentage increase in height is derived directly from this ratio.

Understanding how different dimensions contribute to the volume (linear, square, cubic) is crucial for solving problems involving percentage changes in geometric figures like cylinders, cones, spheres, cubes, and cuboids.