Can you find the Blue shaded area? | (Quarter circles) | |

[00:00:00]Welcome to PreMath. In this video, we have got these two quarter circles of different dimensions. This quarter circle with a center A, and this other quarter circle with a center D fully inscribed in a rectangle ABCD as you can see in this given diagram such that this side length BC of this rectangle is nine units, whereas this horizontal segment length EB is eight units, and finally this horizontal FC length is seven units. And now our task is to calculate the area of this blue shaded region. This area and this area combined.

[00:00:54]Please don't forget to give a thumbs up and subscribe. And please keep in mind that this figure may not be 100% true to the scale. And in this video, I'm going to share with you with two different approaches. So therefore, please watch the video till the very end.

[00:01:08]And here's our very first method. And now let's make an observation and focus on this given equation. We can see that this blue shaded area is going to be equal to the sum of these two quarter circle area. This big quarter circle area plus the other smaller quarter circle area as well. So therefore, now our task is to calculate the area of this uh larger quarter circle and the area of this smaller quarter circle uh as well. And here's our next step. Let's focus on this larger quarter circle. I'm going to label its radius as upper case R, then this radius has got to be upper case R as well. And now let's focus on this uh smaller quarter circle. I'm going to label this uh radius as lower case r, then this radius has got to be lower case r as well. And now, let's make an observation. We can see that this uh side uh AD length equal to this side BC length. And we know this uh side AD length is going to be upper case r plus lower case r. So, I can write upper case r plus lower case r, and this length has got to be equal to nine. So, therefore, we can write down upper case r plus lower case r is going to be equal to nine. And I'm going to label this one as our equation number one.

[00:02:43]And here's our next step. I'm going to calculate this side CD length as upper case r plus seven, and this uh AB side length is going to be lower case r plus eight. And since these two side lengths are equal, so therefore, I can equate them. I can write upper case r plus seven is going to be equal to lower case r plus eight. And now, I'm going to move this lower case r to the left-hand side, and at the very same time, I'm going to move this uh seven in the opposite direction, or simply, I can write upper case r minus lower case r equals to simply one. And I'm going to label this one as our equation number two. So, thus, we are ended up with these two equations, equation one and equation two. And now, we are going to add these two equations. If we add them, we can see this negative r and positive r, they are gone. So, upper case r plus upper case r is going to give us two times uppercase R equals to 10 on the right-hand side. And now we are going to divide both sides by two to isolate the uppercase R. So, therefore we can see that our uppercase R radius value turns out to be five units. So, thus our uppercase R value turns out to be five.

[00:04:17]And now we are going to substitute this uppercase R value five over here. And I'm going to subtract five from both sides. And here we can see this five and negative five is gone. So, therefore our lowercase R value turns out to be four units. So, thus our lowercase R value turns out to be four and uppercase R value is five. And now we are going to calculate the area of this larger quarter circle. And now let's recall the area of a circle formula. Area is always equal to pi times R squared, where uppercase R is the radius. Therefore, the big quarter circle area is going to be pi times uppercase R squared and I'm going to divide this one by four.

[00:05:10]And our uppercase R value is five, so I'm going to substitute that value over here. So, that is going to become 25 pi divided by four. And now we are going to calculate the area of this smaller quarter circle. And now let's recall the area of a circle formula once again. Area equals to pi R squared, where lowercase R is the radius. So, therefore the smaller quarter circle area is going to be pi lowercase R squared and we are going to divide that one by four. And our lowercase R value is 4. I'm going to substitute that value over here, and that is going to be equal to 16 pi divided by 4. And now, let's recall this equation once again, the blue shaded area equals to the big quarter circle area plus the area of the smaller quarter circle. And we already figured out our big quarter circle area and the small quarter circle area as well. Let's go ahead and fill in the blanks in this equation. So, our big quarter circle area is 25 pi divided by 4 plus our smaller quarter circle area is 16 pi divided by 4. And now, we can see we have the same common denominator.

[00:06:38]And if we simplify further more, that is going to give us 41 pi divided by 4 square units. The area of this blue shaded area And now, let me share with you the second method as well. And now, let's focus on this smaller quarter circle.

[00:06:59]I'm going to label its radius as lower-case r. Then this radius has got to be lower-case r as well. And now, let's make an observation. We can see that this side AB length is equal to this side CD length of this rectangle.

[00:07:20]And we know this segment length is 8, and this segment length is 7. So, therefore, we conclude that this radius of this larger quarter circle has got to be lower-case r plus 1.

[00:07:38]If this radius is lower-case r plus 1, then this radius has got to be lower case r plus one as well.

[00:07:47]And now, let's make an observation. We can see that this whole side AD length is going to be lower case r plus lower case r plus one. I can write lower case r plus lower case r plus one. We can see that this side AD length equal to this side BC length. We know this length is two times r plus one and this is equal to nine. So, we can equate them. So, therefore, we can write two times lower case r plus one equals to nine.

[00:08:25]Let's subtract one from both side. This is gone. So, two times lower case r turns out to be equal to eight. And now, we are going to divide both sides by two to isolate lower case r. So, therefore, we can see our lower case r value turns out to be four units. So, that's our lower case r value turns out to be four units. And now, let's focus on this larger quarter circle. I'm going to label its radius as upper case r. Then this radius has got to be upper case r as well. And now, we can see our upper case r is going to be equal to lower case r plus one. So, I can write upper case r equal to lower case r plus one.

[00:09:11]Now, we can see our lower case r value is four. We are going to substitute that four value over here. So, that's our lower case r value turns out to be four and upper case r value is five by using the second method.

[00:09:24]So, that's our revolution area turns out to be 41 pi divided by four square units by using both methods.

[00:09:32]And that is going to be equal to 32.2 square units as well.

[00:09:37]And that's our final answer.

[00:09:39]Thanks for watching and please don't forget to subscribe to my channel for more exciting videos. Bye.