Which arc has the longer radius, an arc with a length of 16π with central angle 2π/3, or an arc with a length of 12π with central angle 4π/5? Explain.
-The second arc has a longer radius because the ratio of the central angle to the arc length is greater.
-The first arc has a longer radius because it has the greater arc length.
-The first arc has a longer radius because the ratio of the arc length to the central angle is greater.
-The second arc has a longer radius because it has the greater central angle.
s = rθ, so compare
(16π) / (2π/3) and (12π)/(4π/5)
its da 3rd one
To determine which arc has the longer radius, we need to compare the ratio of the arc length to the central angle for each arc. The formula for the length of an arc is given by:
Arc Length = radius * central angle
Let's calculate the ratio for each arc:
1) For the first arc with a length of 16π and a central angle of 2π/3:
Arc Length / Central Angle = (16π) / (2π/3) = (16π) * (3/2π) = 24
2) For the second arc with a length of 12π and a central angle of 4π/5:
Arc Length / Central Angle = (12π) / (4π/5) = (12π) * (5/4π) = 15
From the calculations, we can see that the ratio of the arc length to the central angle is greater for the first arc (24) than the second arc (15). Therefore, the first arc, with a length of 16π and a central angle of 2π/3, has the longer radius.