Problem 48 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The change in lateral surface area is \(dS = \pi \left(\sqrt{r_0^2 + h^2} + \frac{r_0^2}{\sqrt{r_0^2 + h^2}}\right) dr\).

Step 1: Identify the Given Information

We are given the lateral surface area formula of a right circular cone: \(S = \pi r \sqrt{r^2 + h^2}\). The change in the radius is from \(r_0\) to \(r_0 + dr\), and the height remains constant.

Step 2: Determine What is Needed

We need to find the differential formula \(dS\) to estimate the change in surface area when the radius changes by a small amount \(dr\).

Step 3: Differential of the Lateral Surface Area

First, find the derivative of the lateral surface area \(S\) with respect to the radius \(r\). Then, use the chain rule to determine the differential \(dS\).

Step 4: Differentiate the Function

Differentiating \(S = \pi r \sqrt{r^2 + h^2}\) with respect to \(r\) gives us \(\frac{dS}{dr} = \pi \left(\sqrt{r^2 + h^2} + \frac{r^2}{\sqrt{r^2 + h^2}}\right)\).

Step 5: Apply the Chain Rule

The differential \(dS\) is given by \(dS = \frac{dS}{dr} \cdot dr\). Substitute the derivative found in the previous step: \(dS = \pi \left(\sqrt{r^2 + h^2} + \frac{r^2}{\sqrt{r^2 + h^2}}\right) dr\).

Step 6: Substitute the Initial Values

Substitute \(r = r_0\) into the expression for \(dS\): \[dS = \pi \left(\sqrt{r_0^2 + h^2} + \frac{r_0^2}{\sqrt{r_0^2 + h^2}}\right) dr.\]

Step 7: Conclusion

The change in the lateral surface area of the cone, when the radius is altered from \(r_0\) to \(r_0 + dr\), is estimated by: \[dS = \pi \left(\sqrt{r_0^2 + h^2} + \frac{r_0^2}{\sqrt{r_0^2 + h^2}}\right) dr.\].

Key Concepts

Lateral Surface Area

The lateral surface area of a right circular cone is a significant concept in geometry. It refers to the area that wraps around the sides of the cone, excluding the base. Understanding this helps in visualizing and calculating the space involved in practical applications like wrapping or painting cylindrical objects.

• The formula for the lateral surface area of a right circular cone is given by \( S = \pi r \sqrt{r^2 + h^2} \).
• Here, \( r \) is the radius of the base, and \( h \) is the height of the cone.

The square root component \( \sqrt{r^2 + h^2} \) represents the slant height of the cone. So, the formula combines the circumference of the circular base with the slant height to compute the lateral area. By focusing on both dimensions (radius and slant height), this formula gives a comprehensive measure of the cone's curved surface.

Right Circular Cone

A right circular cone is a 3D geometric shape that features prominently in various branches of mathematics and engineering. It's characterized by its circular base and a point called the apex or vertex that is perpendicular to the base.

• A distinguishing property of a right circular cone is that the line from the apex to the center of the base - the height, \( h \) - forms a right angle with the base.
• The slant height is crucial in determining the lateral surface area, which is calculated along with the radius.

Studying right circular cones provides insight into spatial relationships and volume calculations in real-world contexts, such as designing cones for funnels and similar structures.

Derivative

The derivative is a central concept in differential calculus. It represents the rate of change or how a function changes as its input changes. In the context of the lateral surface area of a right circular cone, the derivative tells us how the surface area changes with respect to a small change in the radius.

• The derivative of the lateral surface area \( S = \pi r \sqrt{r^2 + h^2} \) with respect to \( r \) is \( \frac{dS}{dr} = \pi \left(\sqrt{r^2 + h^2} + \frac{r^2}{\sqrt{r^2 + h^2}}\right) \).

By calculating the derivative, we understand how sensitive the surface area is to changes in radius, providing a foundational understanding of the cone's geometry and helping predict changes due to physical alterations.

Chain Rule

The chain rule is a fundamental tool in calculus used for differentiating compositions of functions. It's especially important when dealing with problems that require multiple variables or steps. In our context, it allows us to find the differential of the cone's lateral surface area when the radius changes.

• The chain rule states that if a function \( y \) is composed of another function, say \( u \), then \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \).

In our exercise, after finding the derivative \( \frac{dS}{dr} \), we apply the chain rule to relate the small change in radius \( dr \) to the change in surface area \( dS \). The final expression \( dS = \pi \left(\sqrt{r_0^2 + h^2} + \frac{r_0^2}{\sqrt{r_0^2 + h^2}}\right) dr \) captures this relationship elegantly, allowing us to approximate changes in the lateral surface area effectively.