Question

Shane is standing at the base of a truck-mounted stairway to board an aircraft. The stairway begins at the ground. The length of the stairway is 18 feet, and the elevation of the top of the stairway from the ground is 13 feet. An image of a right triangle at the staircase is 18 feet in length and height of 13 feet at an x angle. What is the approximate angle made by the stairway with the ground?

1) 54.16°
2) 43.76°
3) 35.84°
4) 46.24°

Answer 1

Using the trigonometric function tangent and the Pythagorean theorem, we calculated that the approximate angle made by the stairway with the ground is 46.24°.

Step-by-step explanation:

To determine the approximate angle (θ) made by the stairway with the ground, we can use the trigonometric function tangent, which relates the opposite side to the adjacent side in a right triangle. Here, the opposite side is the elevation of the stairway (13 feet), and the adjacent side is the horizontal distance (which can be found using the Pythagorean theorem).

First, we calculate the horizontal distance (let's call it base) of the triangle using the Pythagorean theorem:

• The square of the hypotenuse (stairway length) is 18² = 324.
• The square of the opposite side (elevation) is 13² = 169.
• Thus, the square of the base is 324 - 169 = 155.
• Therefore, the base = √155 ≈ 12.45 feet.

Now, we use the tangent function to find the angle θ:

• θ = tan⁻¹(opposite/adjacent) = tan⁻¹(13/12.45).
• Calculating this, θ ≈ 46.24°.

Therefore, the approximate angle made by the stairway with the ground is 46.24°.