 Hello and welcome to the session, let us understand the following question which says, a spiral is made up of consecutive semicircles with centers alternatively at A and B, starting with center at A of radii, 0.5 centimeter, 1 centimeter, 1.5 centimeter, 2 centimeter and so on, as shown in figure 5.4. What is the total length of such a spiral made up of 13 consecutive semicircles? Take pi is equal to 22 by 7. Here is the given figure. With center A and radius 0.5 centimeter, we have a semicircle of length L1 and again with center B and radius 1 centimeter, we have a semicircle of length L2 and similarly, we have a semicircle of length L3, L4, so on till L13 because it is given to us that spiral is made up of 13 consecutive semicircles and these spiral is made up of radii 0.5 centimeter, 1 centimeter, 1.5 centimeter, so on. Now let us write the solution. Let L1, L2, L3, so on till L13 be the lengths or circumferences of semicircles of radii. R1 is equal to 0.5 centimeter, R2 is equal to 1 centimeter, R3 is equal to 1.5 centimeter, so on respectively. Then L1 is equal to pi R1. L1 is the circumference of the semicircle, so circumference of the semicircle is equal to pi R, where R is the radius of the semicircle. Now this is equal to pi multiplied by R1 is equal to 0.5 centimeter, which is equal to pi multiplied by 5 by 10. This gets cut by 2 which is equal to pi by 2 centimeter. Now similarly L2 is equal to pi R2 which is equal to pi multiplied by R2 is equal to 1 centimeter which is equal to 2 multiplied by pi by 2 which is equal to 2 pi by 2 centimeter and L3 is equal to pi R3 which is equal to pi multiplied by R3 is equal to 1.5 centimeter which is equal to pi multiplied by 1.5. This decimal guess goes off and here we get 10. This gets cut off 5 3s are 15 and 5 2s are 10, so it is equal to 3 pi by 2 centimeter. Similarly L4 is equal to 4 pi by 2, L5 is equal to 5 pi by 2 and so on. Therefore L13 is equal to pi R13 which is equal to pi multiplied by 13 by 2 which is equal to 13 pi by 2 centimeter. Now total length of the spiral is equal to L1 plus L2 plus L3 so on till L13 that is the sum of all the circumference which is equal to pi by 2 plus 2 pi by 2 plus 3 pi by 2 so on till 13 pi by 2 centimeter. Taking pi by 2 common so it is equal to pi by 2 multiplied by 1 plus 2 plus so on till 13 centimeter which is equal to pi by 2 multiplied by n by 2 multiplied by a plus L centimeter where S L that is the sum of first n terms is equal to n by 2 a plus l where n is the number of terms a is the first term and l is the last term. So it is equal to pi by 2 multiplied by n is equal to 13 by 2 multiplied by a is equal to 1 plus l is equal to last term which is again 13 centimeter which is equal to pi by 2 multiplied by 13 by 2 multiplied by 14 centimeter. Now 14 gets cancelled by 2 and we get here 7 so it is equal to pi by 2 multiplied by 13 multiplied by 7 centimeter. Given to us this pi is equal to 22 by 7 so substituting the value of pi we get here 22 by 7 multiplied by 2 multiplied by 13 multiplied by 7 centimeter now here 7 gets cut off and 2 gets cut off with 22 so we get here 11 which is equal to 11 multiplied by 13 which is equal to 143 centimeter. Hence required total length is equal to 143 centimeter. I hope you understood the question that's all for the session bye and have a nice day.