Video Transcript
Determine the length of arc 𝐴𝐵𝐶
to the nearest hundredth, given that the circle has radius nine.
We are asked then to calculate the
length of this arc which we recall is part of the circumference of the full
circle. The formula for calculating our arc
length is 𝜃 over 360 multiplied by 𝜋𝑑, where 𝜃 is the central angle of the
sector. What we’re doing here is
calculating the circumference of the full circle and then multiplying by the
fraction of the circle that we have.
We’re told that the radius of this
circle is nine units, and this in turn means that the diameter of the circle, which
is always twice the radius, will be 18 units. The central angle of this sector is
given as 72 degrees. So, substituting 72 for 𝜃 and 18
for 𝑑, we have that the length of arc 𝐴𝐵𝐶 is 72 over 360 multiplied by 18𝜋. Now, we can actually simplify this
because 72 is a factor of 360 and the fraction simplifies to one-fifth. So if we were looking to give an
exact answer to this problem, we could give our answer as 18 over five multiplied by
𝜋 or eighteen-fifths of 𝜋.
However, the question asks us to
give our answer to the nearest hundredth, so we’ll use our calculator to evaluate
this. We get 11.30973. And then rounding to the nearest
hundredth, we have 11.31. There were no units given for the
radius in the questions, so there are no units for our answer. But there would be some kind of
length units, such as centimeters or feet. Our answer to the problem is
11.31.
