Angle of Elevation Quiz: Angle of Elevation Application

height = 100 times tan(20 degrees) = 100 times 0.3640 = 36.4 m. tan(20 degrees) = 0.3640 is a standard calculator value. Option A gives 35 m, requiring tan(theta) = 0.35, corresponding to approximately 19.3 degrees. Option C gives 40 m, requiring tan(theta) = 0.4, corresponding to approximately 21.8 degrees. Option D gives 45 m, requiring tan(theta) = 0.45, corresponding to approximately 24.2 degrees.

height = 100 times tan(10 degrees) = 100 times 0.1763 = 17.63 m, which rounds to 17.6 m. tan(10 degrees) is a small value because 10 degrees is a shallow angle, producing a height much less than the horizontal distance. Option B gives 18 m, requiring tan(theta) = 0.18, corresponding to approximately 10.2 degrees. Option C gives 20 m, requiring tan(theta) = 0.2, corresponding to approximately 11.3 degrees. Option D gives 15 m, requiring tan(theta) = 0.15, corresponding to approximately 8.5 degrees.

The shadow is the horizontal adjacent side and the tree height is the vertical opposite side. height = 20 times tan(40 degrees) = 20 times 0.8391 = 16.782 m, which rounds to 16.8 m. Option B gives 17 m, which is a less precise rounding that does not match the exact calculation as closely. Option C gives 15 m, requiring tan(theta) = 0.75, corresponding to approximately 36.9 degrees. Option D gives 14 m, requiring tan(theta) = 0.7, corresponding to approximately 34.9 degrees.

Tangent compares the opposite side (height) to the adjacent side (horizontal distance), confirming A. When the angle and horizontal distance are known, height = distance times tan(theta) is applied directly, confirming B. Trigonometric ratios including tangent are defined only for right triangles, confirming C. Option D states tan(theta) = distance divided by height, which is the cotangent ratio, making it false.

height = 200 times tan(30 degrees) = 200 times 0.5774 = 115.48 m, which rounds to 115.5 m. tan(30 degrees) = 1 divided by sqrt(3) approximately equals 0.5774. Option B gives 100 m, requiring tan(theta) = 0.5, corresponding to approximately 26.6 degrees. Option C gives 110 m, requiring tan(theta) = 0.55, corresponding to approximately 28.8 degrees. Option D gives 105 m, requiring tan(theta) = 0.525, corresponding to approximately 27.7 degrees.

The answer is True. At exactly 45 degrees, tan(45 degrees) = 1 because the opposite and adjacent sides are equal. For angles below 45 degrees, the opposite side is shorter than the adjacent side, making the ratio less than 1. For example, tan(30 degrees) = 0.577 and tan(20 degrees) = 0.364, both less than 1. This means that for shallow angles of elevation, the calculated height is always less than the horizontal distance.

height = 15 times tan(60 degrees) = 15 times 1.7321 = 25.98 m, which rounds to 25.9 m. tan(60 degrees) = sqrt(3) approximately equals 1.732. Option B gives 26 m, which is a slightly less precise rounding of the same calculation and does not match the exact computed value as closely as 25.9 m. Option C gives 24 m, requiring tan(theta) = 1.6, corresponding to approximately 58 degrees. Option D gives 20 m, requiring tan(theta) = 1.333, corresponding to approximately 53.1 degrees.

height = 30 times tan(75 degrees) = 30 times 3.7321 = 111.96 m, which rounds to 112.1 m. tan(75 degrees) is approximately 3.732, a large value because 75 degrees is a steep angle close to 90 degrees. Option A gives 110 m, requiring tan(theta) = 3.667, corresponding to approximately 74.7 degrees. Option C gives 95 m, requiring tan(theta) = 3.167, corresponding to approximately 72.5 degrees. Option D gives 130.5 m, requiring tan(theta) = 4.35, corresponding to approximately 77 degrees.

By definition, tan(theta) = opposite divided by adjacent. The opposite side is the one directly across from the angle in the right triangle, and the adjacent side is the one next to the angle that is not the hypotenuse. Option A gives the hypotenuse, which is used in sine and cosine ratios but not in tangent. Option B names a trigonometric function rather than a triangle side. Option C gives the adjacent side, which would produce a ratio of 1 and has no general use.

The answer is True. height = distance times tan(theta). When the angle is constant, tan(theta) is a fixed multiplier. Doubling the distance multiplies the entire right side of the equation by 2, doubling the height. For example, at 30 degrees with a distance of 50 m, height = 50 times 0.5774 = 28.87 m. At 100 m the height becomes 100 times 0.5774 = 57.74 m, which is exactly double. The relationship between height and distance is directly proportional at a fixed angle.

height = distance times tan(angle) = 50 times tan(30 degrees) = 50 times 0.5774 = 28.87 m, which rounds to 28.9 m. The horizontal distance is the adjacent side and the height is the opposite side, so tangent is the correct ratio to use. Option B gives 25 m, requiring tan(theta) = 0.5, corresponding to approximately 26.6 degrees. Option C gives 26.5 m, requiring tan(theta) = 0.53, and option D gives 30 m, requiring tan(theta) = 0.6.

Options A and B are two equivalent forms of the same tangent relationship — tan(theta) = height divided by distance, and rearranging gives height = distance times tan(theta). Option C is obtained by dividing both sides by tan(theta) and is a valid rearrangement for finding the horizontal distance. Option D inverts the tangent ratio, giving cotangent rather than tangent, which is incorrect.

height = 70 times tan(45 degrees) = 70 times 1 = 70 m. At 45 degrees, tangent equals exactly 1 because the opposite and adjacent sides are equal. This means the height equals the horizontal distance exactly. Option B gives 65 m, which would require tan(theta) to be less than 1, corresponding to an angle below 45 degrees. Options C and D give values greater than 70 m, which would require tan(theta) greater than 1, meaning angles above 45 degrees.

height = 100 times tan(25 degrees) = 100 times 0.4663 = 46.63 m, which rounds to 47 m. Option A gives 45 m, requiring tan(theta) = 0.45, corresponding to approximately 24.2 degrees. Option C gives 49 m, requiring tan(theta) = 0.49, corresponding to approximately 26.1 degrees. Option D gives 50 m, requiring tan(theta) = 0.5, which corresponds to approximately 26.6 degrees, not 25 degrees.

The answer is True. height = distance times tan(theta). When the distance is fixed, tan(theta) acts as the only variable multiplier. As the angle increases from 0 to 90 degrees, tan(theta) increases from 0 toward infinity, so the calculated height increases as well. For example, at 30 degrees the height is distance times 0.577, while at 60 degrees it is distance times 1.732, which is a much larger value for the same distance.

height = 25 times tan(37 degrees) = 25 times 0.7536 = 18.84 m, which rounds to 18.9 m. tan(37 degrees) approximately equals 0.7536. Option A gives 15 m, requiring tan(theta) = 0.6, corresponding to approximately 31 degrees. Option C gives 19 m, which is slightly above the precise calculation of 18.84 m and does not match the correct rounding. Option D gives 20 m, requiring tan(theta) = 0.8, corresponding to approximately 38.7 degrees.

height = 60 times tan(60 degrees) = 60 times 1.7321 = 103.92 m, which rounds to 103.9 m. tan(60 degrees) = sqrt(3) approximately equals 1.732, a standard exact value. Option B gives 80 m, requiring tan(theta) = 1.333, corresponding to approximately 53.1 degrees. Option C gives 100 m, requiring tan(theta) = 1.667, corresponding to about 59 degrees but not exactly 60. Option D gives 90 m, requiring tan(theta) = 1.5, corresponding to approximately 56.3 degrees.

In an angle of elevation problem, height is the opposite side and horizontal distance is the adjacent side. Since tan(theta) = opposite divided by adjacent = height divided by distance, rearranging gives height = distance times tan(theta). Option A uses sine, which relates opposite to hypotenuse and would require knowing the line-of-sight distance, not the horizontal distance. Option B uses cosine, which gives the adjacent side. Option D is the cotangent, which would give the distance when multiplied by height, not the other way around.

The answer is True. The angle of elevation forms a right triangle where the vertical height is the side opposite the angle and the horizontal distance is the side adjacent to it. By the tangent definition, tan(theta) = opposite divided by adjacent = height divided by distance. Rearranging this gives height = distance times tan(theta), which is the standard formula used to calculate the height of objects from a known horizontal distance.

height = 40 times tan(45 degrees) = 40 times 1 = 40 m. tan(45 degrees) = 1 exactly because the opposite and adjacent sides are equal at 45 degrees. The height equals the horizontal distance, which is 40 m. Option A gives 30 m, requiring tan(theta) = 0.75, corresponding to approximately 36.9 degrees. Option B gives 35 m, requiring tan(theta) = 0.875. Option D gives 45 m, requiring tan(theta) = 1.125, corresponding to approximately 48.4 degrees.