Problem 79 - Algebra and Trigonometry

Question

As a weather balloon is inflated, the thickness T of its rubber skin is related to the radius of the balloon by $$T(r)=\frac{0.5}{r^{2}}$$ where \(T\) and \(r\) are measured in centimeters. Graph the function \(T\) for values of \(r\) between 10 and \(100 .\)

Step-by-Step Solution

Answer

The function \( T(r) = \frac{0.5}{r^2} \) is a downward curve that decreases as \( r \) increases from 10 to 100.

Step 1: Understand the Function

The function given is a rational function, which can be understood as the thickness of the rubber T decreasing as the radius r of the balloon increases. This is expressed mathematically as \( T(r) = \frac{0.5}{r^2} \).

Step 2: Identify the Domain

In this problem, we are asked to graph the function for values of \( r \) between 10 and 100. Therefore, the domain for graphing purposes is \( 10 \leq r \leq 100 \).

Step 3: Analyze the Behavior of the Function

As \( r \) increases within this domain, \( T(r) \) will decrease because the denominator \( r^2 \) in the function \( \frac{0.5}{r^2} \) becomes larger. This indicates an inverse relationship between \( T \) and \( r \).

Step 4: Calculate Specific Values

To provide reference points for graphing, calculate several values: \( T(10) = \frac{0.5}{10^2} = 0.005 \) cm, \( T(50) = \frac{0.5}{50^2} = 0.0002 \) cm, and \( T(100) = \frac{0.5}{100^2} = 0.00005 \) cm.

Step 5: Plot the Function

Create a graph by plotting the calculated points and observing the curve behavior. Since \( T(r) \) decreases as \( r \) increases, the graph will be a downward curve starting from \( T(10) \) and approaching close to the x-axis as \( r \) approaches 100.

Step 6: Sketch the Graph

On a graph with \( r \) on the x-axis and \( T(r) \) on the y-axis, draw a smooth curve starting from the point at \( r = 10 \) and \( T = 0.005 \) gently decreasing toward \( r = 100 \) where \( T \) approaches 0.00005.

Key Concepts

Inverse Relationship

In the context of rational functions, an inverse relationship describes a situation where one variable increases, another decreases. In our balloon example, as the radius of the balloon ( ") gets larger, the thickness of its rubber (T(r)) gets thinner. This inverse equation: \( T(r) = \frac{0.5}{r^2} \), showcases a classic inverse relationship because the variable \( r^2 \) is in the denominator.
The direct implications mean that for every increase in \( r \), the thickness equation results in a smaller value. Consider some practical applications or real-world scenarios; it is often used in educational settings to teach about proportional relationships.
Ultimately, understanding this inverse dynamic is helpful across math and physics because it applies to varying scenarios where quantities are inversely related.

Rational Function Graph

Rational functions often produce interesting graphs due to their characteristics. In simple terms, the function \( T(r) = \frac{0.5}{r^2} \) indicates how different values of \( r \) impact \( T \).
Here’s what you should look for when graphing such functions:

• The function will always approach, but never quite reach the x-axis.
• It creates a curved line that gets lower as you move right (increase \( r \)).
• Each point on the graph relates directly to calculated values, showing the drop in thickness as the balloon's radius increases.

Understanding these patterns leads to a deeper grasp of how rational functions behave. With parameters extending \( r \) values from 10 to 100, we see the curve sharply falling at first, then leveling off as \( r \) grows.

Function Behavior Analysis

Analyzing the behavior of \( T(r) = \frac{0.5}{r^2} \) helps us predict how the balloon's thickness changes at varying radii. Given that it's a decreasing function, it's crucial to observe:

• How quickly \( T \) decreases, especially noticeable at smaller \( r \) values.
• Rates of change: For example, from \( r = 10 \) to \( r = 50 \), thickness drops swiftly.
• The more r increases, the slower \( T \) seems to decrease, indicating diminishing returns.

This rate and behavior analysis supports broad mathematical understanding, allowing learners to predict without recalculating every instance. Comprehending these elements connects to practically estimating and fine-tuning processes in physics and engineering fields.

Domain of a Function

The domain of a function tells us what input values we can use. For \( T(r) = \frac{0.5}{r^2} \), the problem specifies the domain as \( 10 \leq r \leq 100 \). This means \( r \) can be any real number within this range.
Why is the domain important? Knowing it:

• Sets limits: Defines where the graph is applicable.
• Ensures reasonable values, given \( r \) can't realistically be zero (as it would make thickness undefined).
• Helps anticipate function behavior within these boundaries.

By confining \( r \) between 10 and 100, we focus on realistic scenarios where the balloon's radius doesn't reach extreme sizes that would disrupt practical or mathematical insights.