Question Video: Using Trigonometry to Solve Real-World Problems | Nagwa Question Video: Using Trigonometry to Solve Real-World Problems | Nagwa

Question Video: Using Trigonometry to Solve Real-World Problems Mathematics • First Year of Secondary School

Daniel wants to find the height of a tower. He decides he needs to make a clinometer in order to measure the angle of elevation. He uses a straw, a protractor, some string, and a bit of Blu-Tack as a weight. Daniel stands at a perpendicular distance of 100 ft from the base of the tower and measures the angle on his clinometer to be 59°, as seen in the diagram. Work out the angle of elevation. Given that Daniel’s eyeline is 6 ft from the ground, work out the height of the tower to the nearest foot.

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Video Transcript

Daniel wants to find the height of a tower. He decides he needs to make a clinometer in order to measure the angle of elevation. He uses a straw, a protractor, some string, and a bit of Blu-Tack as a weight. Daniel stands at a perpendicular distance of 100 feet from the base of the tower and measures the angle on his clinometer to be 59 degrees, as seen in the diagram. Work out the angle of elevation. Given that Daniel’s eyeline is six foot from the ground, work out the height of the tower to the nearest foot.

So, we’ve been given a picture of this tower and Daniel and the clinometer that he’s built. A clinometer is a useful tool for measuring angles of elevation and depression, and you could make your own at home using the same equipment as Daniel. We’re told that Daniel is standing at a perpendicular distance of 100 feet from the base of the tower, so let’s add that information onto the diagram.

Daniel measures the angle on his clinometer to be 59 degrees, and that’s the same as this angle here on the figure. It’s the angle between Daniel’s line of sight and the vertical. We’re asked to find the angle of elevation. Now, this is a different type of angle. It’s an angle measured from the horizontal to the line of sight when we look up towards an object. On the diagram, it’s this angle here. Now we can work this out because this angle is in a triangle and we know the measures of the other two angles. There’s one angle, which is a right angle, so has a measure of 90 degrees and one angle, which we know to have a measure of 59 degrees.

Using the fact that the angles sum in any triangle is 180 degrees, we can work this third angle out to be 31 degrees. So, the angle of elevation — that’s the angle between the horizontal and Daniel’s line of sight as he looks up to the top of this tower — is 31 degrees.

In the second part of the question, we’re asked to work out the height of the tower to the nearest foot. That’s the length of this line here in the diagram, which we can call 𝑥 feet. Now, we have a right triangle that we can use to help with this, but we need to be a little careful. The height of the building is not the same as the vertical length in the triangle. We also need to remember that Daniel is standing on the ground, and he’s holding his clinometer at his eye height. Daniel’s eye height is six feet from the ground. So, the total height of the building will be the length of the vertical side in the triangle plus Daniel’s height of six foot.

We must make sure we remember this, but let’s consider how we can use the right triangle to determine the length 𝑦. We know the length of one of the other sides; the horizontal side in this triangle is 100 feet long. And we know the measures of all of the angles. Choosing the angle of 31 degrees to be our reference angle, the side of 100 feet is the adjacent side and the side we want to calculate, the vertical side, is the opposite.

Recalling the acronym SOH CAH TOA, we can use right triangle trigonometry and the tangent ratio to find this unknown length. Through an angle 𝜃 in a right triangle, the tan of angle 𝜃 is defined to be equal to the length of the opposite side divided by the length of the adjacent. So, in this triangle where 𝜃 is 31 degrees and the adjacent is 100 feet, we have that tan of 31 degrees is equal to 𝑦 over 100. Multiplying both sides of this equation by 100, we have that 𝑦 is equal to 100 tan of 31 degrees, which we can evaluate on a calculator, ensuring that it is in degree mode. It gives 60.086 continuing.

Remember, though, that this isn’t the total height of the building because we need to add on the six foot for the distance between Daniel’s eye line and the ground. That gives 66.086 continuing. And then, we recall that the question asked us to round this value to the nearest foot. To the nearest integer, this value is 66. So, we found that the angle of elevation is 31 degrees and the height of the tower to the nearest foot, found using right triangle trigonometry, is 66 foot.

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