A mountain is 8.78 kilometres tall
from the ground. The angle of elevation of the top
of the mountain from a point on the ground is 53 degrees. Find the distance between the point
on the ground and the top of the mountain, giving the answer to the nearest
It’s always good to begin a
question like this with a diagram. So we have our mountain first of
all, which is 8.78 kilometres tall. We’re then told that the angle of
elevation of the top of the mountain from a point on the ground is 53 degrees.
Now an angle of elevation is an
angle measured from the horizontal to the line of sight when we look up towards an
object. So on our diagram, that’s this
angle here. We’re then asked to find the
distance between the point on the ground and the top of the mountain, thus this
distance here, which I’ve marked as 𝑑.
We can see that these three lines —
so that’s the vertical height of the mountain, the horizontal ground, and the
distance between the point on the ground and the top of the mountain — form a
right-angled triangle. We know the length of one side in
this triangle, the size of one of the other angles. And we’re looking to calculate the
length of a second side, which means we can apply right angle trigonometry to this
We begin by labeling the three
sides of this triangle. The longest side of a right-angled
triangle, which is the side opposite the right angle, is always the hypotenuse. The side diagonally opposite the
other known angle — so that’s the angle of 53 degrees — is called the opposite. And the side between the known
angle and the right angle is called the adjacent.
To help us decide which
trigonometric ratio we need in this question, we can recall the acronym SOHCAHTOA,
where S, C, and T stand for sin, cos, and tan and O, A, and H stand for opposite,
adjacent, and hypotenuse. The side we know is the opposite,
and the side we want to calculate is the hypotenuse. So we’re going to be using SOH,
which is the sine ratio. The definition of the sine ratio is
that sin of an angle 𝜃 is equal to the length of the opposite side divided by the
length of the hypotenuse.
Before we start substituting into
this formula though, notice that we’ve been asked to give our answer to the nearest
metre, whereas the length we’ve been given for the opposite is in kilometres. We need to convert this measurement
first. There are 1000 metres in a
kilometre. So multiplying by 1000, we see that
8.78 kilometres is equivalent to 8780 metres.
Now we can substitute into the sine
ratio. And we have that sin of 53 degrees
is equal to 8780 over 𝑑. To solve this equation for 𝑑, we
must first bring it out of the denominator on the right, which we do by multiplying
both sides of the equation by 𝑑, giving 𝑑 sin 53 degrees is equal to 8780.
Next, we need to divide both sides
of the equation by sin of 53 degrees, which we can do with no problem as sin of 53
degrees is just a number. At this point, we can use our
calculator to evaluate this. But we must make sure that our
calculator is in degree mode first. 8780 divided by sin of 53 degrees
Remember, we need to give our
answer to the nearest metre. And as the deciding digit — so in
this case, that’s the first number after the decimal point — is a seven, we round
up. We have then the distance between
the point on the ground and the top of the mountain to the nearest metre is 10994
Remember, we converted that
measurement of 8.78 kilometres into metres before we did our trigonometry. However, we could’ve done
trigonometry using that measurement in kilometres and then converted our answer into
metres at the end before rounding by multiplying by 1000.