Answered

Because the formula for the area of a circle is $\pi r^2$, you need to first find the radius of the lawn. Since the formula for circumference is $2 \pi r$, and the circumference of the lawn is $132$ feet, you know that $2 \pi$ times $r$ equals $132$. Use this equation to help you find the radius of the lawn.



You can also think about this as $132$ divided by $2 \pi$ equals $r$. Consider writing your answer to the hundredths place.



$\begin{aligned}

132 & = 2 \cdot \pi \cdot r \\

r & \approx \square \text{ feet}

\end{aligned}$

Verified Answer :

  • Utilize the circumference formula \(C = 2\pi r\) and substitute the known circumference \(132\) feet.
  • Isolate the radius \(r\) by dividing the circumference by \(2\pi\), resulting in \(r = \frac{66}{\pi}\).
  • Calculate the numerical value of \(r\) and round it to the hundredths place to obtain the final radius.
  • The radius of the lawn is approximately \(\boxed{21.01 \text{ feet}}\).

Explanation

  1. Analysis and problem setup

The problem asks us to find the radius of a circular lawn given its circumference. The circumference \(C\) of a circle is related to its radius \(r\) by the formula \(C = 2 \pi r\). We are given that the circumference \(C = 132\) feet. Our objective is to solve for \(r\) and round the result to the nearest hundredth.

  1. Setting up the equation to solve for the radius

Given the formula \(C = 2 \pi r\), we can substitute the known value \(C = 132\) into the equation:
\[132 = 2 \cdot \pi \cdot r\]
To solve for \(r\), we divide both sides of the equation by \(2\pi\):
\[r = \frac{132}{2\pi}\]
Simplifying the fraction, we get:
\[r = \frac{66}{\pi}\]

  1. Numerical calculation and rounding

Using the approximate value of \(\pi \approx 3.14159\), we calculate the value of \(r\):
\[r = \frac{66}{\pi} \approx 21.00845\dots\]
Rounding this result to the nearest hundredth, we look at the thousandths digit, which is 8. Since 8 is greater than 5, we round up the hundredths digit (0 becomes 1). Thus, the radius is approximately \(21.01\) feet.

Examples

Calculating the radius of a circular object is a common task in landscaping, engineering, and design. For example, if you are designing a circular garden and know the total length of the fencing required for the perimeter (the circumference), you can use the relationship \(r = \frac{C}{2\pi}\) to determine the distance from the center to the edge. This ensures you can accurately place sprinklers, plan flower beds, or determine the necessary materials for paving. Understanding this relationship helps in creating balanced and precise layouts for various outdoor projects.

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