Question

the circle below has center k, and its radius is 4 in. given that m∠lkm=70°, find the length of the major arc \\(\overarc{lnm}\\). give an exact answer in terms of π, and be sure to include the correct unit in your answer. length of major arc \\(\overarc{lnm}\\) :

Explanation:

Step1: Find the central angle of the major arc

The total degrees in a circle is \(360^\circ\). The central angle for the minor arc \(LM\) is \(70^\circ\), so the central angle for the major arc \(LNM\) is \(360^\circ - 70^\circ = 290^\circ\).

Step2: Recall the arc length formula

The formula for the length of an arc is \(s=\frac{\theta}{360^\circ}\times2\pi r\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.

Step3: Substitute the values into the formula

Here, \(\theta = 290^\circ\) and \(r = 4\) in. Plugging these into the formula: \(s=\frac{290^\circ}{360^\circ}\times2\pi\times4\). Simplify the fraction \(\frac{290}{360}=\frac{29}{36}\), and then multiply by \(8\pi\) (since \(2\times4 = 8\)): \(\frac{29}{36}\times8\pi=\frac{232\pi}{36}=\frac{58\pi}{9}\). Wait, no, wait: \(2\pi r\) when \(r = 4\) is \(8\pi\). Then \(\frac{290}{360}\times8\pi=\frac{290\times8\pi}{360}=\frac{2320\pi}{360}=\frac{58\pi}{9}\)? Wait, no, 2908=2320, 2320/360=58/9? Wait, 2320 divided by 40 is 58, 360 divided by 40 is 9. Yes. Wait, but let's check again. Wait, the central angle for the major arc: 360 - 70 = 290 degrees. The arc length formula is \(\frac{\theta}{360}\times2\pi r\). So \(\theta = 290\), \(r = 4\). So \(\frac{290}{360}\times2\pi\times4\). Let's compute that. 24=8, so \(\frac{290}{360}\times8\pi\). Simplify 290/360 to 29/36, then 29/36 8π= (298π)/36= (232π)/36= (58π)/9? Wait, no, 232 divided by 4 is 58, 36 divided by 4 is 9. Yes. Wait, but maybe I made a mistake. Wait, 360 - 70 is 290, correct. Radius is 4, so circumference is \(2\pi r = 8\pi\). The major arc is a fraction of the circumference: \(\frac{360 - 70}{360}=\frac{290}{360}=\frac{29}{36}\). Then arc length is \(\frac{29}{36}\times8\pi=\frac{232\pi}{36}=\frac{58\pi}{9}\) inches? Wait, no, 8π is 8π, times 29/36: 829=232, 232/36=58/9. So 58π/9 inches? Wait, but let's check the calculation again. Wait, 360 - 70 = 290. So the arc length is (290/360)2π4. Let's compute 2π4=8π. Then 290/360 8π= (2908π)/360= (2320π)/360= divide numerator and denominator by 40: 58π/9. Yes. Wait, but maybe I messed up the central angle. Wait, the angle given is ∠LKM=70°, which is the central angle for arc LM. So the major arc LNM would be the rest of the circle, so 360 - 70 = 290 degrees. So the arc length is (290/360)2π4. Let's compute that: 290/360 simplifies to 29/36. 2π4 is 8π. So 29/36 8π= (298π)/36= 232π/36= 58π/9. So the length of the major arc LNM is \(\frac{58\pi}{9}\) inches. Wait, but let's check with another approach. The circumference is \(2\pi r = 8\pi\). The minor arc length is (70/360)8π= (70*8π)/360= 560π/360= 14π/9. Then major arc length is circumference minus minor arc: 8π - 14π/9= (72π -14π)/9= 58π/9. Yes, that matches. So that's correct.

Answer:

\(\frac{58}{9}\pi\) inches (or \(\frac{58\pi}{9}\) inches)