Problem 30 - Precalculus Mathematics for Calculus

Question

A spherical balloon is being inflated. Find the rate of change of the surface area \(\left(S=4 \pi r^{2}\right)\) with respect to the radius \(r\) when \(r=2 \mathrm{ft}\).

Step-by-Step Solution

Answer

The rate of change is \( 16\pi \) square feet per foot when \( r = 2 \) ft.

Step 1: Understand the Given Information

We are given the formula for the surface area of a sphere, which is \( S = 4\pi r^2 \). We need to find the rate of change of the surface area \( S \) with respect to the radius \( r \) when \( r = 2 \) ft. This means we need to calculate the derivative \( \frac{dS}{dr} \) and evaluate it at \( r = 2 \) ft.

Step 2: Differentiate the Surface Area Formula

Differentiate the surface area formula \( S = 4\pi r^2 \) with respect to \( r \). Using the power rule for differentiation, the derivative is \( \frac{dS}{dr} = 8 \pi r \).

Step 3: Evaluate the Derivative at Given Radius

Now, substitute \( r = 2 \) into the derivative \( \frac{dS}{dr} = 8\pi r \). This gives us \( \frac{dS}{dr} = 8\pi \times 2 = 16\pi \).

Step 4: Interpret the Result

The result \( 16\pi \) represents the rate of change of the surface area of the balloon with respect to its radius. When the radius is 2 ft, the surface area increases at a rate of \( 16\pi \) square feet per foot increase in the radius.

Key Concepts

Rate of Change

The "rate of change" refers to how one quantity changes with respect to another. In this case, we examine how the surface area of a sphere changes as the radius changes. The mathematical tool used to determine such changes is calculus, specifically "derivatives." Derivatives show the rate at which one variable is changing in relation to another.

To calculate the rate of change of the sphere's surface area with respect to its radius, we differentiate the surface area formula. The formula for the surface area of a sphere is:

• \( S = 4\pi r^2 \)

Using the power rule, the derivative \( \frac{dS}{dr} = 8\pi r \) describes how fast the surface area \( S \) changes as the radius \( r \) changes. When the radius is 2 feet, as in the exercise, the calculation shows that the surface area increases by \( 16\pi \) square feet for each additional foot added to the radius.

Surface Area

The surface area of a geometric shape is what gives the measure of how much material is needed to cover it. In the exercise, the focus is on a sphere, a perfectly round shape where every point on the surface is equidistant from the center.

For spheres, the formula to calculate surface area is:

• \( S = 4\pi r^2 \)

This specific formula reveals that the surface area is directly dependent on the square of the radius \( r \). As the radius increases, the surface area grows at a rate proportional to the square of the radius. Thus, even small changes in the radius can lead to large adjustments in the surface area. Understanding this relationship is essential in fields such as physics and engineering, where optimizing material use based on geometric size is crucial.

Spherical Geometry

Spherical geometry deals with the properties and equations of circles and spheres. Unlike planar geometry, which is flat, spherical geometry is curved and applies to three-dimensional objects like spheres. Common in astronomy and navigation, it plays a role in a wide range of analytical applications.

A sphere, the main object of interest for spherical geometry, is a perfectly symmetric object in three-dimensional space. Some key properties include:

• All points on the surface are equidistant from the center.
• It has no edges or vertices.
• The surface area and volume formulas (\(4\pi r^2\) and \(\frac{4}{3}\pi r^3\), respectively) depend entirely on the radius \(r\).

Spherical geometry forms a foundation for understanding natural phenomena, such as planetary motion and the shape of droplets. By using spherical geometry, we can delve deeper into the concepts of symmetry and curvature in our universe.