Question Video: Estimating a Building’s Height given the Angle of Elevation from a Certain Point to the Top of the Building

A person is trying to estimate the height of the Eiffel Tower. He measured a distance of 250 m from the base of the tower. From that point, he measured the angle of elevation to the top of the tower to be 52°. Use these measurements to approximate the height of the tower to the nearest meter.

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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.

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