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
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
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
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.