Question

the radius of a quarter circle is 6 kilometers. what is the quarter circles perimeter? round your answer to the nearest hundredth. kilometers

Explanation:

Step1: Recall perimeter of quarter - circle

The perimeter of a quarter - circle (also known as the circumference of the quarter - circle plus the length of the two radii) is given by the formula \(P=\frac{1}{4}(2\pi r)+2r\), where \(r\) is the radius of the circle. We know that the formula for the circumference of a full circle is \(C = 2\pi r\), so a quarter of the circumference is \(\frac{1}{4}\times2\pi r=\frac{\pi r}{2}\), and then we add the lengths of the two radii (since the quarter - circle has two straight sides which are radii of the circle) with length \(r\) each, so \(2r\).

Step2: Substitute the value of \(r\)

Given that \(r = 6\) km. First, calculate the length of the curved part: \(\frac{\pi r}{2}=\frac{\pi\times6}{2}=3\pi\) km. Then calculate the length of the two radii: \(2r=2\times6 = 12\) km.

Step3: Calculate the perimeter

Now, find the perimeter \(P\) by adding the length of the curved part and the length of the two radii. So \(P=3\pi + 12\). Substitute \(\pi\approx3.14159\), then \(3\pi\approx3\times3.14159 = 9.42477\) km. Then \(P\approx9.42477+12=21.42477\) km.

Step4: Round to the nearest hundredth

Rounding \(21.42477\) to the nearest hundredth (two decimal places), we look at the third decimal place which is \(4\). Since \(4\lt5\), we keep the second decimal place as it is. So \(P\approx21.42\) km.

Answer:

\(21.42\)