If a 30-meter high tower casts a 10
square root of three meter long shadow on the ground, what is the angle of elevation
of the sun?
We have a tower and our tower casts
a shadow. The tower is 30 meters tall and the
shadow 10 square root of three meters long. This would be the angle of
elevation made with the sun. What we should notice now is that
we have an opposite side measure and an adjacent side measure to our angle. In addition to that, we will note
that the tower makes a right angle with its shadow.
Having opposite and adjacent side
lengths in a right triangle tells us that we’ll be looking for a tangent ratio. Tangent of our missing angle is
equal to the opposite side measure, 30, over the adjacent side measure, 10 square
root of three. We noticed that this angle can be
simplified. 30 divided by 10 equals three. We can reduce this to three over
the square root of three.
To simplify a bit further, we want
to take the square root of three and try to get it out of the denominator. To do that, we multiply this
fraction by the square root of three over the square root of three. The numerator of that is three
times the square root of three. And in the denominator, we need to
put the square root of three times the square root of three, which equals three.
And now, we have a fraction that
has three over three. That three over three cancels
out. And we’re left with the square root
of three. The tangent of our missing angle is
the square root of three. And this is a form we should
recognize. We know that the tangent of 60
degrees equals the square root of three. And that means that our missing
angle must measures 60 degrees. The angle of elevation of the sun,
here, is 60 degrees.