Question

the radius of a quarter circle is 7 centimeters. what is the quarter circles perimeter? round your answer to the nearest hundredth. centimeters

Explanation:

Step1: Recall the formula for the perimeter of a quarter - circle

The perimeter of a quarter - circle (also known as the circumference of a quarter - circle plus the length of two radii) is given by the formula \(P=\frac{1}{4}(2\pi r)+2r\), where \(r\) is the radius of the circle. We can also factor out \(r\) and write it as \(P = r(\frac{\pi}{2}+2)\) or simplify the first form: \(\frac{1}{4}(2\pi r)=\frac{\pi r}{2}\), so \(P=\frac{\pi r}{2}+2r\).

Given that \(r = 7\) cm.

Step2: Substitute the value of \(r\) into the formula

First, calculate \(\frac{\pi r}{2}\):
Substitute \(r = 7\) into \(\frac{\pi r}{2}\), we get \(\frac{\pi\times7}{2}=\frac{7\pi}{2}\approx\frac{7\times3.14159}{2}=\frac{21.99113}{2} = 10.995565\)

Then, calculate \(2r\):
Substitute \(r = 7\) into \(2r\), we get \(2\times7 = 14\)

Step3: Calculate the perimeter

Now, add the two parts together: \(P=\frac{7\pi}{2}+14\approx10.995565 + 14=24.995565\)

Step4: Round to the nearest hundredth

Rounding \(24.995565\) to the nearest hundredth, we look at the thousandth place, which is \(5\). When the digit in the thousandth place is \(5\) or greater, we round up the hundredth place. So \(24.995565\approx25.00\) (when rounded to the nearest hundredth). Wait, let's check the calculation again. Wait, \(\pi\approx3.1416\)

\(\frac{1}{4}\) of the circumference of the whole circle: The circumference of a whole circle is \(C = 2\pi r\), so a quarter - circumference is \(\frac{1}{4}\times2\pi r=\frac{\pi r}{2}\). For \(r = 7\), \(\frac{\pi\times7}{2}=\frac{7\times3.1416}{2}=11.0\) (approx)

Two radii: \(2\times7 = 14\)

Then the perimeter is \(11.0+14 = 25.0\)? Wait, no, let's do it more accurately.

\(\frac{\pi r}{2}+2r=r(\frac{\pi}{2}+2)\)

Substitute \(r = 7\):

\(\frac{\pi}{2}+2=\frac{3.14159265}{2}+2=1.570796325 + 2=3.570796325\)

Then \(7\times3.570796325 = 24.995574275\)

Rounding to the nearest hundredth: The number is \(24.995574275\). The hundredth digit is \(9\), the thousandth digit is \(5\). So we round up the hundredth digit: \(24.99 + 0.01=25.00\)

Answer:

\(25.00\)