 Hello students! Let's work out the following problem. It says a vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 30 meter. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 40 degrees and the top of the flagstaff is 60 degrees. Determine the height of the tower and the horizontal distance. So let's now move on to the solution. We are given that a tower is surmounted by a flagstaff of height 30 meter and the angle of elevation on the horizontal plane of the bottom of the flagstaff is 45 degrees. That is this much angle is 45 degrees and the top of the tower angle of elevation is 60 degrees. So let BC be the tower, height be X meter. We have to find the horizontal distance. So let AB be the horizontal distance and let it be equal to Y meter. Now in triangle ABC perpendicular upon base is equal to tan 45 degrees. That is BC upon AB is equal to tan 45 degrees. Now BC is the height of the tower which is X meter and AB is the horizontal distance and tan 45 degrees is 1. So from this we have X is equal to, while that means horizontal distance is same as the height of the tower. Now in triangle ABD again perpendicular upon base is equal to tan 60 degrees because this angle of elevation is 60 degrees. So BD upon AB is equal to tan 60 degrees. Now BD is BC plus CD and AB is Y meter and tan 60 degrees is root 3. Now BC is the height of the tower which is X meter and CD is 30 meter upon Y is equal to root 3. Now from 1 we have X is equal to Y. So from this we have X plus 30 is equal to root 3 X as X is equal to Y right. From this we have root 3 X minus X is equal to 30. So this implies root 3 is 1.732 X minus X is equal to 30. So 1.732 minus 1 into X is equal to 30. So from this we have 0.732 X is equal to 30. So this implies X is equal to 30 upon 0.732 and 30 divided by 0.732 is 40.98. So this implies X is equal to 40.98. Therefore the height of the tower is equal to 40.98 meter which is equal to the horizontal distance because we have proved that X is equal to Y 40.98 is the answer. And since it's the distance so we must put the unit. So this completes the question and the session. Bye for now. Take care. Have a good day.