Problem 133 - Calculus Volume 3

Question

Express the volume of a right circular cylinder as a function of two variables: a. its radius \(r\) and its height \(h\). b. Show that the rate of change of the volume of the cylinder with respect to its radius is the product of its circumference multiplied by its height. c. Show that the rate of change of the volume of the cylinder with respect to its height is equal to the area of the circular base.

Step-by-Step Solution

Answer

Volume: \( V = \pi r^2 h \). Rate with respect to \( r \): \( 2\pi rh \). Rate with respect to \( h \): \( \pi r^2 \).

Step 1: Write the Formula for Volume of Cylinder

The volume of a right circular cylinder can be expressed in terms of its radius \(r\) and height \(h\) using the formula for the volume of a cylinder: \[ V = \pi r^2 h \] This means the volume \(V\) is a function of the two variables \(r\) and \(h\).

Step 2: Differentiate with Respect to Radius

To find the rate of change of volume with respect to the radius \(r\), we take the partial derivative of the volume \(V\) with respect to \(r\):\[ \frac{\partial V}{\partial r} = \frac{\partial}{\partial r}(\pi r^2 h) = 2\pi rh \]This expression, \(2\pi rh\), is the product of the cylinder's circumference, \(2\pi r\), and its height \(h\).

Step 3: Differentiate with Respect to Height

Similarly, to find the rate of change of volume with respect to the height \(h\), we take the partial derivative of the volume \(V\) with respect to \(h\):\[ \frac{\partial V}{\partial h} = \frac{\partial}{\partial h}(\pi r^2 h) = \pi r^2 \]This result, \(\pi r^2\), represents the area of the circular base.

Key Concepts

Partial Derivatives

In calculus, partial derivatives are crucial when dealing with functions of multiple variables. They allow us to understand how a function changes as one of its variables changes, while keeping the other variables constant.

In the case of the volume of a right circular cylinder, which depends on both radius (\( r \) ) and height (\( h \) ), partial derivatives help us determine how the volume changes when either the radius or height changes.

The partial derivative with respect to the radius, noted as \( \frac{\partial V}{\partial r} \), indicates the rate at which the volume increases or decreases as the radius changes, while the height remains constant. Similarly, \( \frac{\partial V}{\partial h} \) tells us the change rate with respect to height changes only.

Rate of Change

The rate of change is a vital concept in calculus that measures how one quantity changes concerning another. It connects to physical concepts like speed and growth rates.

When we analyze how the volume of a cylinder changes with its radius or height, we're discussing rates of change.

• For the radius, the rate of volume change is given by the partial derivative \( \frac{\partial V}{\partial r} = 2\pi rh \), which reveals that the change in volume is directly linked to the cylinder's circumference multiplied by its height.


• For the height, the rate of change is \( \frac{\partial V}{\partial h} = \pi r^2 \), equivalent to the area of the base.

Understanding these rates provides valuable insights into how different dimensions of the cylinder influence its overall volume.

Cylinder Volume Formula

The cylinder volume formula is foundational for determining the space inside a right circular cylinder.

For a cylinder with radius \( r \) and height \( h \), the formula for calculating its volume is \( V = \pi r^2 h \). This formula combines:

• The base area, which is a circle with area \( \pi r^2 \).
• The height \( h \)

By multiplying these two, you find the total volume of space within the cylinder. This formula stands out because it ties geometric dimensions to measurable space, making it a practical tool in various engineering and physics applications.

Right Circular Cylinder

A right circular cylinder is a common geometric shape characterized by a few defining attributes:

• It has two parallel circular bases, both of the same size.
• The axis connecting these bases is perpendicular to the planes in which the bases lie, hence the term "right".

This simple structure makes it a frequently used shape in real-world applications, from designing storage tanks to formulating mathematical problems.

In these contexts, understanding how changes in its dimensions affect properties like volume is essential, which involves studying the impact of radius and height adjustments on the overall structure.

Differentiation with Respect to Variables

Differentiation in calculus is about finding the rate at which a function changes as its input variables change. When we focus on multiple variables, it becomes particularly powerful.

For our cylinder problem, we differentiate with respect to \( r \) and \( h \) separately through the method of partial differentiation.

This process helps extract specific insights about how the cylinder's volume reacts to alterations in each dimension. By calculating \( \frac{\partial V}{\partial r} \) and \( \frac{\partial V}{\partial h} \), we can break down the influence of each variable. This makes differentiation a key tool in understanding and predicting changes in multidimensional scenarios.