Problem 39 - Calculus Early Transcendentals

Question

Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone with fixed height \(h=6 \mathrm{m}\) when its radius decreases from \(r=10 \mathrm{m}\) to \(r=9.9 \mathrm{m}\) \((S=\pi r \sqrt{r^{2}+h^{2}})\).

Step-by-Step Solution

Answer

Answer: The approximate change in the lateral surface area is about -3.142 square meters.

Step 1: Calculate the lateral surface area with \(r=10\)m

Use the formula for lateral surface area \(S=\pi r \sqrt{r^{2}+h^{2}}\) and plug in \(r=10\) and \(h=6\): \(S_1=\pi \cdot 10\sqrt{10^2+6^2}=100\pi\sqrt{136}\).

Step 2: Calculate the lateral surface area with \(r=9.9\)m

Use the formula for lateral surface area \(S=\pi r \sqrt{r^2+h^2}\) and plug in \(r=9.9\) and \(h=6\): \(S_2=\pi \cdot 9.9\sqrt{9.9^2+6^2}=99\pi\sqrt{137.61}\).

Step 3: Calculate the difference between the two lateral surface areas

Find the difference between the two lateral surface areas, \(S_1\) and \(S_2\): \(\Delta S = S_2 - S_1 = 99\pi\sqrt{137.61}-100\pi\sqrt{136}\).

Step 4: Approximate the difference

Approximate the difference in lateral surface area: \(\Delta S \approx 99\pi\sqrt{137.61}-100\pi\sqrt{136} \approx -3.142 \mathrm{m}^2\). The approximate change in the lateral surface area of the cone when the radius decreases from \(r=10\mathrm{m}\) to \(r=9.9\mathrm{m}\) is about \(-3.142 \mathrm{m}^2\).

Key Concepts

Lateral Surface Area

In geometry, the lateral surface area refers to the area of the sides of a three-dimensional shape, excluding its base. When dealing with a right circular cone, the lateral surface area is a specific measurement important for understanding the cone's surface.
The formula for calculating the lateral surface area of a right circular cone is given by \(S = \pi r \sqrt{r^2 + h^2}\), where \(r\) is the radius of the base, and \(h\) is the height of the cone. This formula essentially derives from the properties of the slant height of the cone, which provides the distance from the base to the apex along the cone's surface.
Understanding lateral surface area is crucial in various applications, such as manufacturing and construction, where knowing how much material covers the sides of an object is needed.

• It helps to calculate the amount of material needed for covering the cone excluding its base.
• The lateral surface area changes as the radius or height of the cone alters.

Right Circular Cone

A right circular cone is a geometric figure featuring a circular base and a vertex not on the base plane, with the vertex directly above the center of the circular base. This makes the cone 'right,' as opposed to an oblique cone where the apex might not line up with the center of the base.
Several key components make up a right circular cone:

• Radius ( oldsymbol{ }): The distance from the center to any point on the circle of the base.
• Height ( oldsymbol{ }): The distance from the base's center to the vertex along the cone's axis.
• Slant Height: The length along the cone's surface from base to tip, forming the hypotenuse of the right triangle formed by the height and radius.

The formula used for the lateral surface area relies on these components to accurately determine how much of the cone's sides extend from the base to the apex. This foundational understanding makes it easier to apply the formula and comprehend changes in the cone's properties as dimensions change.

Approximation

Approximation in calculus and mathematics often helps estimate values that are not easily computed exactly. In the context of this exercise, approximation is used to determine the change in the lateral surface area of a right circular cone when the radius decreases slightly.
The concept revolves around estimating small changes effectively, especially when the difference is not straightforward. By comparing the computed values for two different radii, we can see how a minor decrease in the radius affects the surface area. Calculating these changes precisely would involve complex mathematical operations, but approximation simplifies the process.

• It allows for quick and efficient estimations when exact values are hard to establish without a lot of computational work.
• This method uses nearby values and the concept of linearity for small changes, such as \( oldsymbol{ ext{Change in Radius}} = 0.1 ext{m}\) to make predictions about what the surface might look like.

In practice, this approach is widely applied in fields where precision needs to be balanced with computation and practicality.