Problem 112 - College Algebra with Modeling and Visualization

Question

Surface Area of a Balloon The surface area \(A\) of a balloon with radius \(r\) is given by \(A(r)=4 \pi r^{2} .\) Suppose that the radius of the balloon increases from \(r\) to \(r+h\) where \(h\) is a small positive number. (a) Find \(A(r+h)-A(r) .\) Interpret your answer. (b) Evaluate your expression in part (a) when \(r=3\) and \(h=0.1 .\) Then evaluate it for \(r=6\) and \(h=0.1\) (c) If the radius of the balloon increases by \(0.1,\) does the surface area always increase by a fixed amount or does the amount depend on the value of \(r ?\)

Step-by-Step Solution

Answer

The surface area increase depends on the radius, as shown by differing results for different radii.

Step 1: Find the Expression for A(r+h)

To find \(A(r+h)\), substitute \(r+h\) in place of \(r\) in the original surface area formula. \[A(r+h) = 4\pi(r+h)^2\] Next, expand the square in the equation: \((r+h)^2 = r^2 + 2rh + h^2\). Substitute this into the expression: \[A(r+h) = 4\pi(r^2 + 2rh + h^2) = 4\pi r^2 + 8\pi rh + 4\pi h^2\].

Step 2: Calculate A(r+h) - A(r)

Subtract \(A(r)\) from \(A(r+h)\): \[A(r+h) - A(r) = (4\pi r^2 + 8\pi rh + 4\pi h^2) - 4\pi r^2\]. This simplifies to \[A(r+h) - A(r) = 8\pi rh + 4\pi h^2\]. This result represents the change in surface area when the radius increases by \(h\).

Step 3: Evaluate for r = 3 and h = 0.1

Substitute \(r = 3\) and \(h = 0.1\) into the expression \(8\pi rh + 4\pi h^2\): \[A(3+0.1) - A(3) = 8\pi(3)(0.1) + 4\pi(0.1)^2\]. Calculate each term: \(8\pi(3)(0.1) = 2.4\pi\) and \(4\pi(0.1)^2 = 0.04\pi\). Add these together for the final change: \[A(3.1) - A(3) = 2.44\pi\].

Step 4: Evaluate for r = 6 and h = 0.1

Substitute \(r = 6\) and \(h = 0.1\) into the expression \(8\pi rh + 4\pi h^2\): \[A(6+0.1) - A(6) = 8\pi(6)(0.1) + 4\pi(0.1)^2\]. Calculate each term: \(8\pi(6)(0.1) = 4.8\pi\) and \(4\pi(0.1)^2 = 0.04\pi\). Add these together for the final change: \[A(6.1) - A(6) = 4.84\pi\].

Step 5: Interpret the Results

In parts (b) and (c), the results show that the increase in surface area, \(A(r+h) - A(r)\), depends on the initial radius \(r\). As seen in the calculations, the increase differs when \(r = 3\) and \(r = 6\), even though \(h = 0.1\) is constant. This indicates that the change in surface area is not constant but increases with a larger radius \(r\).

Key Concepts

Understanding Radius

The term "radius" refers to the distance from the center of a sphere or circle to any point on its surface. In the context of a balloon, the radius is a crucial measurement that helps to determine the size and volume of the balloon.

• The radius is typically denoted by the letter \(r\).
• In a two-dimensional circle, the radius plays a key role in any calculations involving the circle's circumference and area.
• For a three-dimensional sphere, like a balloon, the radius helps determine the sphere's surface area and volume.

As the radius increases or decreases, other properties of the sphere, such as its surface area, change accordingly. Understanding the concept of radius is foundational when working with geometrical figures.

Surface Area Formula

The formula to calculate the surface area of a sphere is pivotal for this exercise. It's given by: \[ A(r) = 4\pi r^2 \] This equation tells you how much area is covered by the surface of a sphere with radius \(r\). The "\(4\pi r^2\)" part of the equation means:

• "\(4\pi\)" is a constant that represents the relationship between the radius and the entire surface area of a sphere.
• The term "\(r^2\)" shows that the surface area depends quadratically on the radius. Doubling the radius increases the surface area by four times its original size \((2^2 = 4)\).

In practical terms, this formula is used to find out how much material would be needed to completely cover the outside of a spherical object like a balloon. Understanding this formula is crucial when determining how changes in radius affect the surface area.

Change in Surface Area

When the radius of a balloon increases slightly, say from \(r\) to \(r+h\), the surface area will change. The change can be calculated using the expression \(A(r+h) - A(r)\). Breaking down this process:

• Start with substituting \(r+h\) into the surface area formula: \[ A(r+h) = 4\pi (r+h)^2 \]
• Next, expand \((r+h)^2\) to get \(r^2 + 2rh + h^2\).
• Substitute back into the surface area formula: \[ A(r+h) = 4\pi(r^2 + 2rh + h^2) = 4\pi r^2 + 8\pi rh + 4\pi h^2 \]
• Subtract the original surface area \(A(r) = 4\pi r^2\) from this expanded expression.

The result \(8\pi rh + 4\pi h^2\) shows how much the surface area changes with a small increase \(h\) in the radius. As seen in previous examples, the change in surface area is not constant but varies with the radius. For larger radii, increases are more noticeable, indicating the relationship between radius and surface area is dynamic and influenced by initial size.