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Chapter 1: Problem 88

Write the area \(A\) of a circle as a function of its circumference \(C\).

Short Answer

Expert verified
The area 'A' of a circle as a function of its circumference 'C' can be expressed as \( A = \frac{C^2}{4\pi} \).

Step by step solution

01

Recall the formulas

The formula for the circumference of a circle is \( C = 2\pi r \) where 'r' is the radius of the circle and \(\pi\) is a constant approximately equal to 3.14159. The formula for the area of a circle is \( A = \pi r^2 \). We know 'C' and we want to find 'A' expressed in terms of 'C'.
02

Express radius in terms of circumference

We can rearrange the circumference formula to express 'r' (the radius) in terms of 'C' (the circumference). This gives us \( r = \frac{C}{2\pi} \). It's always a good strategy to express the variable we want to eliminate (in this case, the radius) in terms of the remaining variable.
03

Substitute the expression for radius into the area formula

Now we substitute \( r = \frac{C}{2\pi} \) into \( A = \pi r^2 \) to express 'A' in terms of 'C'. This gives us \( A = \pi \left(\frac{C}{2\pi}\right)^2 \).
04

Simplify the equation

Upon simplifying the above equation, we get: \( A = \frac{C^2}{4\pi} \).

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Most popular questions from this chapter

Consider the functions given by \(f(x)=x+2\) and \(f^{-1}(x)=x-2 .\) Evaluate \(f\left(f^{-1}(x)\right)\) and \(f^{-1}(f(x))\) for the indicated values of \(x .\) What can you conclude about the functions? $$ \begin{array}{|l|l|l|l|l|} \hline x & -10 & 0 & 7 & 45 \\ \hline f\left(f^{-1}(x)\right) & & & & \\ \hline f^{-1}(f(x)) & & & & \\ \hline \end{array} $$

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Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\frac{1}{x^{2}} $$

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