Consider the following. θ = 70^∘, d = 18 m (a) Find the...

Consider the following. $\theta = 70^\circ$, $d = 18$ m (a) Find the length of the arc that subtends the given central angle $\theta$ on a circle of diameter $d$. (Round your answer to two decimal places.) 9.77 $\times$ m (b) Find the area $A$ of the sector determined by $\theta$. (Round your answer to two decimal places.) A = 39.11 $\times$ m$^2$

Transcript

00:01 In today's problem, we will be looking at finding the length of the arc that subtenes the given central angle theta on a circle of diameter d and also we'll be finding the area of the sector subtended by the angle theta.

00:18 Now with respect to this given problem here, this long big statement problem is translated into a simple figure.

00:26 So you've got the circle.

00:30 The theta is equal to 80 is the angle that is subtended here right here theta is the central angle which is 80 degrees and the length of the arc is ab so that's the distance ab that they are talking about to find that's the first part of the problem find the length of the arc with respect to the second part of the problem they're asking us to find the area a of the sector so you can see that oab is the sector area and what they've given us is obviously the angle theta 80 degrees and d which is the diameter of the circle to be equal to 12 meters so that's the given information translated into a pictorial representation so let's write down let's look at part a in the first place finding the length of the arc so part a there is a formula to find the length of the arc the length of the arc the length of the arc a b can be provided by so that's length of the arc is normally denoted by the symbol s so s equals to that's the length s equals to arc length is equal to theta divided by 360 that's a standard formula multiplied by two into pi into the radius so what they've given us here is the diameter so obviously it would mean this would imply that radius is down radius is diameter divided by 2 that would be 12 by 2 which is equal to 6 meters so radius is 6 meters for this problem so we'll substitute that s equals to theta is given to be 80 degree we already saw that divided by 360 multiplied by two into pi into radius is six so if you did the calculation in a standard calculator you'd find that to be equal to 8 .38 meters so that's the value of s s equals to 8 .38 meters i'm stopping at two decimal places because they are asking us to rounded off to two decimal places so 8 .38 you'd find that in the calculator as well rounded off to two decimal places would be s being equal to 8 .38 meters.

03:10 The first answer to this question is how you'd find out the arc length.

03:15 The second part of the problem involves finding the area of the arc of the sector.

03:21 Area of the sector, which is called a, that's the area...

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