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Chapter 1: Problem 488

The circumference \(C\) of a circle is a function of its radius given by \(C(r)=2 \pi r\) . Express the radius of a circle as a function of its circumference. Call this function \(r(C) .\) Find \(r(36 \pi)\) and interpret its meaning.

Short Answer

Expert verified
The radius of a circle with circumference \( 36\pi \) is 18 units.

Step by step solution

01

Write the Given Function

The problem provides us with the circumference function of a circle in terms of its radius: \( C(r) = 2\pi r \). Our task is to express the radius \( r \) as a function of the circumference \( C \).
02

Solve for Radius as a Function of Circumference

To express the radius in terms of the circumference, we need to solve for \( r \) in terms of \( C \). Start by rewriting the equation \( C = 2\pi r \). Divide both sides by \( 2\pi \) to isolate \( r \): \( r = \frac{C}{2\pi} \). Therefore, the function for radius becomes \( r(C) = \frac{C}{2\pi} \).
03

Substitute the Given Circumference into the Function

Now we need to find \( r(36\pi) \). Substitute \( C = 36\pi \) into the function: \( r(36\pi) = \frac{36\pi}{2\pi} \).
04

Simplify the Expression

Simplify the expression \( \frac{36\pi}{2\pi} \). The \( \pi \) terms cancel each other out, simplifying to \( \frac{36}{2} \), which equals 18. Thus, \( r(36\pi) = 18 \).
05

Interpret the Result

The result \( r(36\pi) = 18 \) signifies that a circle with a circumference of \( 36\pi \) units has a radius of 18 units. This means that for a circle with this given circumference, the distance from its center to any point on the circle is 18 units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circumference of a Circle
The circumference of a circle refers to the total distance around the circle. It is a crucial concept in geometry and is used to define circular boundaries.
  • The circumference is calculated using the formula: \[ C = 2\pi r \] where \( C \) is the circumference, \( r \) is the radius, and \( \pi \) (pi) is a constant approximately equal to 3.14159.
  • This formula indicates that the circumference is directly proportional to the radius, meaning that if the radius increases, the circumference also increases.
  • The formula \( C = 2\pi r \) is derived from the properties of circles, where \( 2\pi \) is essentially the ratio of the circumference to the radius. This ratio remains constant for all circles.
By understanding this relationship, you can determine the circumference if you know the radius, and vice versa. Calculating the circumference helps in tasks like finding the boundary length of circular fields or designing circular objects. As seen in our example, we find the radius when the circumference is given by rearranging the formula.
Radius of a Circle
The radius of a circle is the distance from the center of the circle to any point on its boundary. Understanding the radius is essential because it is a fundamental property that informs several other circular calculations.
  • The radius is typically denoted by \( r \).
  • As shown before, once the circumference \( C \) is known, we can find the radius using the formula: \[ r = \frac{C}{2\pi} \]
  • In our problem, we calculated that when \( C = 36\pi \), the radius \( r = 18 \).
  • The radius is particularly significant because it acts as a scaling factor for all other parts of the circle; whether it's diameter \( 2r \), area \( \pi r^2 \), or circumference \( 2\pi r \).
Thus, knowing the radius gives a complete understanding of the size and scale of the entire circle. In geometric problems, it's common to use the radius to navigate between diameter, circumference, and area efficiently.
Functions in Math
Functions in mathematics are a way to describe the relationship between two or more variables. A function defines how one variable depends on another.
  • A mathematical function is usually represented as \( f(x) \), where \( f \) names the function and \( x \) is the variable input of the function.
  • For circles, we encountered \( C(r) = 2\pi r \), meaning the circumference \( C \) changes with the radius \( r \).
  • To express the radius in terms of the circumference, we rearrange the function to get \( r(C) = \frac{C}{2\pi} \).
  • This conversion between \( C(r) \) to \( r(C) \) exemplifies how functions can be manipulated to express different relationships between variables.
Understanding these functional relationships allows you to switch perspectives: knowing one variable helps determine others. This property is particularly powerful in solving geometric problems as it interprets direct and inverse relationships between measurements.

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Most popular questions from this chapter

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\) $$ h(x)=\sqrt[4]{\frac{3 x-2}{x+5}} $$

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\) $$ f(x)=4 x+8, g(x)=7-x^{2} $$

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x)+5\)

Write a formula for the function obtained when the graph of \(f(x)=\frac{1}{x}\) is shifted down 4 units and to the right 3 units.

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions. $$ k(x)=(x-2)^{3}-1 $$
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