Problem 117 - A Graphical Approach to Precalculus with Limits

Question

Suppose that an aluminum can is manufactured so that its radius \(r\) can vary from 0.99 inches to 1.01 inches. What range of values is possible for the circumference \(C\) of the can? Express the answer by using a compound inequality. (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

The range of possible circumferences is \( 1.98\pi \leq C \leq 2.02\pi \).

Step 1: Understand the Formula for Circumference

The circumference \( C \) of a circle is given by the formula \( C = 2\pi r \). Here, \( \pi \) (Pi) is a mathematical constant approximately equal to 3.14159, and \( r \) is the radius of the circle.

Step 2: Define the Range of the Radius

According to the problem, the radius \( r \) can vary from 0.99 inches to 1.01 inches, which can be expressed by the inequality \( 0.99 \leq r \leq 1.01 \).

Step 3: Calculate the Circumference for the Minimum Radius

Substitute the minimum value of the radius into the circumference formula: \( C = 2\pi(0.99) \). This gives \( C = 1.98\pi \).

Step 4: Calculate the Circumference for the Maximum Radius

Substitute the maximum value of the radius into the circumference formula: \( C = 2\pi(1.01) \). This gives \( C = 2.02\pi \).

Step 5: Express the Range of Circumference as a Compound Inequality

Using the values from Steps 3 and 4, the range of circumference \( C \) is \( 1.98\pi \leq C \leq 2.02\pi \).

Key Concepts

Radius of a Circle

When dealing with circles, the concept of the radius is fundamental. The radius of a circle is the distance from the center of the circle to any point on its perimeter. This measurement is crucial in different calculations, such as determining the area and circumference of a circle.
For practical understanding, you can think of the radius as half the length of the diameter of the circle. So, if you know the diameter, you simply divide it by two to find the radius.
The formula for the circumference, or the distance around the circle, directly involves the radius:

• Circumference: \( C = 2\pi r \)

Where \( C \) is the circumference, \( \pi \) is a mathematical constant (approximately 3.14159), and \( r \) is the radius. In our exercise, the radius can vary from 0.99 inches to 1.01 inches, showcasing how even small changes in the radius can impact the circumference greatly.

Compound Inequality

Understanding compound inequalities is valuable in mathematics, especially when dealing with ranges of values. A compound inequality is an equation with two or more inequalities joined together by either 'and' or 'or.' In the case of an 'and' inequality, both conditions must be true. In an 'or' inequality, at least one of the conditions must be true.
For example, in our exercise, we examine a range for the radius, expressed as \( 0.99 \leq r \leq 1.01 \). This illustrates a compound inequality where the radius \( r \) lies between 0.99 and 1.01 inches. This form of a compound inequality ensures that the values of \( r \) satisfy both parts of the inequality simultaneously.
Similarly, for the circumference determined by these radius values, the compound inequality is \( 1.98\pi \leq C \leq 2.02\pi \), indicating the acceptable range of circumference values.

Mathematical Constants

In mathematics, constants are fixed values that do not change. One of the most famous constants is \( \pi \), a symbol used to represent the ratio of the circumference of a circle to its diameter. The approximate value of \( \pi \) is 3.14159, though it is an irrational number, meaning it has an infinite number of non-repeating decimals.
Constants like \( \pi \) simplify calculations, making it easier to solve formulas that involve circular shapes. Their consistent values allow us to apply them universally in mathematical equations and scientific calculations.

• Mathematical constant \( \pi \): approximately equal to 3.14159

In our exercise, the constant \( \pi \) is crucial for calculating the circumference of the circle with varying radii. Every step to find the circumference uses \( \pi \) multiplied by the radius, underscoring the importance and presence of mathematical constants in geometry.