Video Transcript
A person is trying to estimate the
height of the Eiffel Tower. He measured a distance of 250
meters from the base of the tower. From that point, he measured the
angle of elevation to the top of the tower to be 52 degrees. Use these measurements to
approximate the height of the tower to the nearest meter.
When presented with a problem like
this, the first thing that we should do is sketch out the information that we were
given. We have an Eiffel Tower, and
someone has measured 250 meters from the base of the tower. From that point, he measured an
angle of elevation to the top of the tower to be 52 degrees. Once we have all of this
information down into a diagram form, we should be able to see a right triangle
forming.
The height of the tower forms a
right angle with the base. The height is our unknown value and
what we’re trying to solve for. So now we need to start at our
angle of elevation and label the sides of the right triangle. The height is opposite to the angle
we know. The 250-meter base is adjacent to
the angle we know. And the other line from the person
to the top of the tower is the hypotenuse.
In this problem, we’re not
interested in the value of the hypotenuse. We’re dealing with the opposite
side and the adjacent side, which means we’ll consider the ratios SOHCAHTOA. Sin of 𝜃 equals the opposite over
the hypotenuse. Cos of 𝜃 equals the adjacent over
the hypotenuse. And tan of 𝜃 equals the opposite
over the adjacent. Based on the information we’re
given, we need the tangent ratio.
If the tan of 𝜃 equals the
opposite over adjacent, we can say the tan of 52 degrees equals ℎ, the height of the
tower, over 250 meters. To get an estimate for ℎ, we’ll
need to solve for ℎ to get ℎ by itself. And so we multiply both sides of
the equation by 250. And then, we’ll see that 250 times
tan of 52 degrees equals ℎ. When we plug that into a
calculator, we get 319.9854 continuing. If we want to round to the nearest
meter, we’ll be rounding to the nearest whole number. So we’ll look to the first decimal
place and see that we should round up. The units we’re measuring in is
meters. And so we would say that an
estimate for the height of the Eiffel Tower based on the given information is 320
meters.
