Video: CBSE Class X • Pack 4 • 2015 • Question 27

CBSE Class X • Pack 4 • 2015 • Question 27

04:17

Video Transcript

At a point 𝐴, 20 metres above the water level in a lake, the angle of elevation of a cloud is 30 degrees. The angle of depression of the reflection of the cloud in the lake at 𝐴 is 60 degrees. Find the distance between the cloud and the surface of the lake.

Let’s begin by sketching this out. Remember the angle of elevation of the cloud is the angle between the horizontal — here that’s the dotted line — and the line from the point 𝐴 to the cloud. The angle of depression of the reflection is the angle made between the horizontal — once again, that’s the dotted line — and the line from point 𝐴 to the reflection.

We know that point 𝐴 is 20 metres above the lake. Let’s call the vertical height of the cloud above point 𝐴 ℎ. Since we know that the reflection of the cloud will appear at an equal distance from the surface of the lake, we can call this length ℎ plus 20.

Notice that we have two right-angled triangles with a shared side, for which we know the measure of one of its angles. And we have an expression for the length of one of their sides. We can use right angle trigonometry to help us form an equation for ℎ. Let’s start with this triangle.

The line between 𝐴 and the cloud is the hypotenuse. That’s the longest side of the triangle and it always sits directly opposite the right angle. The side that we called ℎ metres is the opposite side, it’s the side opposite the angle of 30 degrees. And the remaining side is the adjacent. That’s the side next to the angle of 30 degrees.

Let’s use the opposite side of this triangle since we have an expression for that length. And we’ll also use the adjacent since that’s the shared side between the two triangles. We’ll use the tangent ratio, where tan 𝜃 is equal to opposite over adjacent.

Substituting what we know in, we get tan 30 is equal to ℎ over 𝑥. I’ve called the adjacent side in this triangle 𝑥 to prevent confusion. Remember tan of 30 is equal to one over root three or root three over three. Here, we’ll use one over root three for reasons, which will become obvious in a moment.

We want an equation for 𝑥 in terms of ℎ. So we’re going to multiply both sides by 𝑥. That gives us 𝑥 over root three is equal to ℎ. We’re then going to multiply both sides of the equation by the square root of three. And that gives us that 𝑥 is equal to the square root of three multiplied by ℎ.

Now, have we chosen square root of three over three as our value for tan 30, we would have eventually gotten this answer. However, we would have had to rationalize the denominator of our fraction, which would have created more work. Now, we have an expression for the length of the shared side between the triangles: it’s root three multiplied by ℎ metres.

Let’s now consider the second right-angled triangle. Once again, we’ll label this triangle this time with respect to the angle of 60 degrees. We aren’t really interested in the hypotenuse of this triangle. So once again, we’ll use tan of 𝜃.

Substituting what we know into this formula, we get tan of 60 equals 20 plus ℎ plus 20 all over root three ℎ. That simplifies to ℎ plus 40 all over root three ℎ. Since we know that tan of 60 is equal to root three, we can replace tan of 60 in our equation. It becomes root three equals ℎ plus 40 all over root three ℎ.

To solve, we’ll multiply both sides by the square root of three multiplied by ℎ. Root three multiplied by root three is three. So our equation becomes three ℎ equals ℎ plus 40. Next, we’ll substract ℎ from both sides to get two ℎ equals 40. And finally, we’ll divide everything by two. And we get that ℎ is equal to 20.

We are being asked to find the distance of the cloud from the lake. So we can replace ℎ with 20. And we see that the cloud is 20 plus 20 which is 40 metres above the lake.