Final answer:
To determine the angle the escalator makes with the first floor, the inverse tangent of the vertical height (27 feet) over the horizontal length (calculated using the Pythagorean theorem) should be found and then rounded to the nearest tenth. The horizontal length is obtained by subtracting the square of the vertical height from the square of the escalator's length, taking the square root, and then using it in the tangent ratio.
Step-by-step explanation:
To find the angle the escalator makes with the first floor, we can use trigonometry. Specifically, we need to calculate the inverse tangent (also known as arctan or tan-1) of the ratio of the vertical height to the horizontal length of the escalator. We have the vertical height (27 feet) and the hypotenuse (61 feet), but we first need the horizontal length, which we can find using the Pythagorean theorem.
Let's call the horizontal length x, the vertical height y, and the length of the escalator h. Then:
- x2 + y2 = h2
- x2 + 272 = 612
- x2 = 612 - 272
- x = sqrt(612 - 272)
After finding x, we then calculate the angle θ:
- tan θ = opposite/adjacent
- tan θ = y/x
- θ = tan-1(27/x)
We can now calculate the angle using a calculator with the tan-1 function. Once the angle is found, we round it to the nearest tenth to obtain our final answer.