We're going to look at some applications of radians. The first one we're gonna look at is arc length, and arc length is simply a segment of a circle. So we're looking for the length of a portion of the circumference of a circle, and we use the letter S to represent arc length. And we have our formula for arc length, which is S equals theta times R where theta is an angle measure in radians and R is the radius of the circle. So it's a very simple formula to use. We simply multiple theta times the radius of the circle.
So here we have an application problem. A fan has a radius of 4.5 inches and an angle of 2Pi over 3 radians when fully opened. And what is the arc length of the circular arc formed by the open fan and at its top? So we're gonna use our formula, which says S equals 2Pi divided by 3 times the radius of 4.5. And we will pull up our calculator to run the calculation so we make sure we do it accurately. Two times Pi equals 6.28. We're gonna divide that by 3, which gets us 2.094 and so on, and then we have to multiply that times 4.5. And we get 9.424 and so on, which we'll just round to 9.42. So we'll call this 9.42 inches.
And here's the second application. The stop board of a shot put circle in a circular arc is 1.22 meters in length. The radius of the circle is 1.06 meters. What is the central angle? Well, again, we can set up and use the formula, but this time we're given S and we're given the radius, and we're looking for the theta. So we're gonna have 1.22 equals theta, which we're trying to find, times the radius of 1.06. So now that we're looking at this, we can see pretty easily that we can find theta by dividing both sides of this equation by 1.06. And again, we will pull out the calculator. 1.22 divided by 1.06 equals 1.15 approximately.
So 1.15 radians equals theta. And we have our angle, our central angle in radians. And these applications all have to use theta in terms of radians. We cannot use degrees because, remember, we've talked about degrees don't get along with other measures such as measures of time and seconds or hours, nor measures of distance in feet or meters and so on. So we have to use radians if we want to make these applications work and make sense. So the second application we're gonna talk about is area of a sector. And the area of a sector is the portion of an area of a circle that is cut off by an angle theta. So now we're looking at an area.
And our formula is, again, not too terribly complicated. Area equals one-half theta times R squared. So here's a problem. A lawn sprinkler can water up to a distance of 65 feet. It turns through an angle of 3Pi over 4 radians. What area can it cover? So again, we're gonna set up our formula. Area equals one-half, which is the same thing as 0.5, times our theta, which is 3Pi divided by 4 times our radius squared. And we'll pull up the calculator and run the calculation. So I'm gonna start with 3Pi over 4 and get that taken care of before I do anything else. So 3 times Pi. That equals 9.42 and so on. We'll then divide that by 4. So we have the 3Pi over 4 equals about 2.35, 2.36. Now we'll multiply that times 0.5 for the one half and that equals that. And then we have to multiply it times 65 squared and that equals 4977.46. 4977.46 meters squared. Remember, area is in squared units.
Now we have a second one that takes that formula and turns it around just a little bit. The face of a flat wedge is a circular sector with area of 38 square centimeters and a diameter of 12.2 centimeters. What is the arc length of the wedge? So this is gonna take us two steps. First, we're going to apply our area formula up here, and this time we're given the area and we're given the diameter. So we know the area, which is 38, equals, again, one half or .5 times theta, which we don't know, we haven't been given that, we need to find it, times the radius. Well, if the diameter is 12.2, the radius is half that or 6.1. And then we have to square that. So pulling up the calculator, again, 6.1 squared equals 37.21 times .5 equals 18.61. So now we have 38 equals 18.61 times theta. So we just simply have to divide both sides by 18.61.
So 38 divided by 18.61 equals 2.04. So 2.04 radians equals theta. But that doesn't answer our question yet 'cause we want the arc length. So we have to go back and use the other formula, S equals theta times R. And now we have a theta that we can use. So S has to equal 2.04 times our radius, which was 6.1. So I'm multiplying our theta, which we have in front of us, times 6.1, the radius, and we get an arc length of about 12.46 centimeters. And that is a couple of application problems dealing with area of a sector using this formula in various ways.
And the last application we're gonna look at it is angular velocity. And the formula for linear – and that angular velocity is part of the formula for linear velocity. And the formula for linear velocity is V equals W times R where V is the linear velocity and W equals angular velocity. So there's the angular velocity part of this. So here's a problem. A person on a hang glider is moving in a horizontal circular arc of radius 90 meters which an angular velocity of 0.125 radians per second. Find the person's linear velocity. So we're just gonna plug into this formula. V equals the angular velocity, which is 0.125 times the radius, which is 90. As simple as that. And .125 times 90 equals 11.25. So our linear velocity is 11.25 meters per second. And it's really as simple as that.
So we have three very – three applications of radian measure. The first being arc length with the formula S equals theta times R. The next one being area of a sector where the formula is area equals one-half theta times R squared. And then angular velocity where the formula for linear velocity is V equals WR where W is the angular velocity and V is the linear velocity.
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