Problem 83 - Calculus Volume 1

Question

\(d V\) if a circular cylinder of height 3 changes from \(r=2\) to \(r=1.9 \mathrm{cm} .\)

Step-by-Step Solution

Answer

The change in volume is approximately \(1.17\pi\) cm³.

Step 1: Understand the Volume Formula

The volume of a cylinder is given by the formula: \( V = \pi r^2 h \). Here, \( r \) is the radius, \( h \) is the height, and \( \pi \) is a constant (approximately 3.14159). For this problem, the height \( h = 3 \) cm.

Step 2: Calculate the Initial Volume

Calculate the initial volume with radius \( r = 2 \) cm using the formula: \( V_1 = \pi (2)^2 \times 3 = 12\pi \) cm³.

Step 3: Calculate the Final Volume

Calculate the final volume with radius \( r = 1.9 \) cm using the formula: \( V_2 = \pi (1.9)^2 \times 3 = 10.83\pi \) cm³.

Step 4: Find the Change in Volume

Subtract the final volume from the initial volume to find the change in volume: \( \Delta V = V_1 - V_2 = 12\pi - 10.83\pi = 1.17\pi \) cm³.

Key Concepts

Volume Formula

To understand the volume of a cylinder, it's important to grasp the volume formula: \[ V = \pi r^2 h \]This formula calculates the space within a cylinder. Here, \( V \) denotes the volume, \( r \) is the radius of the circular base, \( h \) is the height of the cylinder, and \( \pi \) is a mathematical constant (approximately 3.14159). It is used since the base of the cylinder is a circle.

• The term \( \pi r^2 \) represents the area of the circular base.
• Multiplying by the height \( h \) gives the 3-dimensional space volume.

The formula is simple yet powerful, providing the volume calculation needed for any cylinder with a known radius and height.
Using this formula helps us determine the volume before and after any changes in the cylinder’s dimensions.

Radius Change

A change in the radius of a cylinder significantly impacts its volume. In our problem, the radius changes slightly from \( r = 2 \) cm to \( r = 1.9 \) cm. This change might seem minor, but it affects the entire volume of the cylinder.
Remember that the radius is squared in the volume formula:

• This means even a small change in the radius leads to a larger difference when squared.
• The smaller radius square will produce a smaller base area for the volume calculation.

Understanding how radius changes affect volume is crucial, as it highlights why precise measurements are important in mathematical problems involving physical properties.

Cylinder Height

The height of the cylinder, marked as \( h \) in the volume formula, is straightforward in this exercise. It is given as 3 cm, which remains constant between the calculations for both radii.
Here’s why height is important:

• The height extends the circular base into the third dimension, making it a cylinder.
• Any changes in height would proportionally influence the overall volume.

Even though the height remains unchanged in this exercise, it is an integral part of understanding volume calculation. This ensures that learners focus on both dimensional changes and constants when computing volumes.

Volume Calculation

The calculation of a cylinder's volume involves using the volume formula twice—once for each radius—to determine the initial and final volumes.
Here’s the step-by-step breakdown:

• Calculate the initial volume using the radius of 2 cm: \[ V_1 = \pi \times 2^2 \times 3 = 12\pi \text{ cm}^3 \]
• Calculate the final volume with the adjusted radius 1.9 cm: \[ V_2 = \pi \times 1.9^2 \times 3 = 10.83\pi \text{ cm}^3 \]

After calculating both volumes, find the change in volume by subtraction: \[ \Delta V = V_1 - V_2 = 12\pi - 10.83\pi = 1.17\pi \text{ cm}^3 \]
This simple operation helps demonstrate the effect of changes in the radius on the cylinder's volume.