Problem 40 - Precalculus: Mathematics for Calculus

Question

Inflating a Balloon 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 \( 16 \pi \text{ square feet per foot} \) when \( r = 2 \text{ ft} \).

Step 1: Understanding the Problem

We are given the formula for the surface area of a spherical balloon: \( S = 4 \pi r^2 \). We need to find the rate of change of this surface area \( S \) with respect to the radius \( r \), which means we need to find \( \frac{dS}{dr} \). Then, evaluate this derivative at \( r = 2 \text{ ft} \).

Step 2: Find the Derivative

Differentiate the surface area formula \( S = 4 \pi r^2 \) with respect to \( r \). Using the power rule, the derivative \( \frac{d}{dr}(r^2) = 2r \), so we have \( \frac{dS}{dr} = \frac{d}{dr}(4 \pi r^2) = 4 \pi \cdot 2r = 8 \pi r \).

Step 3: Evaluate the Derivative at the Given Radius

Now that we have the rate of change formula \( \frac{dS}{dr} = 8 \pi r \), substitute \( r = 2 \text{ ft} \) into the derivative. So, \( \frac{dS}{dr} = 8 \pi \times 2 = 16 \pi \).

Step 4: Interpret the Result

The rate of change of the surface area with respect to the radius when \( r = 2 \text{ ft} \) is \( 16 \pi \text{ square feet per foot} \). This means that for every additional foot the radius increases, the surface area increases by \( 16 \pi \) square feet.

Key Concepts

Surface Area

The surface area of a sphere is a critical concept in understanding spherical geometry. For any sphere, this surface area is calculated using the formula \( S = 4 \pi r^2 \), where \( S \) represents the surface area and \( r \) is the radius of the sphere.
This relationship shows how the surface area increases with the square of the radius. It's important to note that as the radius increases, the surface area grows very quickly. For example, doubling the radius results in a fourfold increase in the surface area. When dealing with physical problems like inflating a balloon, understanding how surface area changes with the radius is crucial. This helps in predicting how much material is required, or how stress might change as a balloon inflates.

Derivative

A derivative represents the rate at which a function is changing at any point. In this case, we are interested in finding how the surface area of the sphere changes with respect to its radius.
To find the derivative here, we take the formula for surface area \( S = 4 \pi r^2 \) and differentiate it with respect to \( r \). Using the power rule, the derivative of \( r^2 \) is \( 2r \), which gives us \( \frac{dS}{dr} = 8 \pi r \).

• This derivative, \( 8 \pi r \), tells us the rate of change of the surface area for any small change in the radius.
• It effectively tells us how much the surface area increases per unit increase in the radius.

Evaluating this derivative at a particular radius, such as \( r = 2 \) feet, helps in understanding the specific change at that instant. For \( r = 2 \), \( \frac{dS}{dr} = 16 \pi \), indicating the surface area increases by \( 16\pi \) square feet per foot increase in radius.

Spherical Geometry

Spherical geometry deals with the properties and relationships of spheres. It differs significantly from planar geometry as it considers surfaces that are curved. Understanding spherical geometry is essential in various scientific fields, from astronomy to engineering.When dealing with practical problems, such as inflating a spherical balloon, spherical geometry allows us to calculate various factors like volume and surface area based on the radius. The surface area of a sphere, governed by \( S = 4 \pi r^2 \), showcases the importance of this geometry.
A unique aspect of spherical geometry is how dimensions such as radius and diameter directly affect calculations. For example, the radius directly determines both volume and surface area, making it a fundamental measure. As spheres scale up, spherical geometry clarifies exactly how different measurements change.

• Spherical geometry is used in many real-world applications, from designing spherical tanks to understanding planetary bodies.
• It helps predict how physical changes like inflation in a balloon affect other dimensions.