Problem 49 - University Calculus: Early Transcendentals

Question

Write a differential formula that estimates the given change in volume or surface area. The change in the volume \(V=\pi r^{2} h\) of a right circular cylinder when the radius changes from \(r_{0}\) to \(r_{0}+d r\) and the height does not change

Step-by-Step Solution

Answer

\( dV \approx 2\pi r_0 h \times dr \).

Step 1: Identify the Function

We start with the volume formula of the right circular cylinder, which is given by \[ V = \pi r^2 h \]. This formula shows that the volume \( V \) depends on the radius \( r \) and height \( h \) of the cylinder.

Step 2: Express Volume as Function of Radius

Since the height does not change, here, the volume can be expressed as a function of the radius for a fixed height \( h \): \[ V(r) = \pi r^2 h \].

Step 3: Determine the Derivative of Volume

To find the rate of change in the volume with respect to the radius, we take the derivative of \( V \) with respect to \( r \): \[ \frac{dV}{dr} = \frac{d}{dr}(\pi r^2 h) = 2\pi r h \].

Step 4: Estimate Change in Volume

The estimated change in volume \( dV \) when the radius changes from \( r_0 \) to \( r_0 + dr \) is given by the differential formula: \[ dV \approx \frac{dV}{dr}\bigg|_{r=r_0} \times dr = 2\pi r_0 h \times dr \].

Key Concepts

Right Circular Cylinder

A right circular cylinder is a 3-dimensional geometric shape often used in math and science problems. It has two parallel circular bases and a perpendicular side known as the height. The cylinder stands upright, much like a standard soup can. The critical features of a right circular cylinder include:

• Radius ( ): The distance from the center to the edge of the circular base.
• Height ( h): The straight distance between the circular bases.

Since the bases are perfectly circular, the word "circular" describes the shape of the bases while "right" implies that the sides are vertical to the bases. Therefore, the volume of a right circular cylinder relies heavily on its radius and height.

Rate of Change

The rate of change in this context refers to how quickly the volume of the cylinder changes as you adjust certain dimensions, in this case, the radius. In mathematical terms, it represents how a small change in the radius (dr) affects the overall volume. The rate of change is often compared to a vehicle's speed—how fast or slow a particular aspect (e.g., volume) shifts over time or another variable.

• An increase in the radius will result in an increase in volume.
• If you know the rate of change, you can predict the effect of small adjustments to the cylinder without calculating the volume directly each time.

This kind of analysis is beneficial when one aspect is changing while others stay constant, like the height here.

Derivative

A derivative is a fundamental concept in calculus that measures how a function changes as its input changes. Here, it represents how the volume formula reacts to changes in radius. When you differentiate the volume formula \(V = \pi r^2 h\) with respect to \(r\), you get:\[ \frac{dV}{dr} = 2\pi rh \]This derivative tells us the rate at which the volume is changing with respect to the radius when the height remains constant. It’s like asking, "For a tiny increase in the radius, how much more of the cylinder will be filled?"

• The derivative helps in understanding the immediate or instantaneous rate of volume change.
• It simplifies the prediction of changes in real-world applications without recalculating complex formulas.

Understanding derivatives is crucial for solving many physics and engineering problems.

Volume Formula

The volume formula for a right circular cylinder is \(V = \pi r^2 h\). This formula gives you the capacity, or the amount of space inside the cylinder. Every part of it has specific meaning:

• \(\pi\) is a constant (~3.14), representing the ratio of the circumference to the diameter of a circle.
• \(r^2\) comes from the area of the circle, showcasing how important the radius is to defining the space within the cylinder.
• \(h\) is the height, controlling how tall the cylinder is and thus how much it can hold vertically.

When working through problems involving the cylinder volume, this formula helps predict changes and understand the 3D space you are dealing with. It is especially useful in real-life applications such as packaging, manufacturing, and construction, where the best use of space often saves cost and materials.