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A wheelchair ramp is placed over a set of stairs so that one end is 2 feet off the ground. The other end is at a point that is a horizontal distance of 28 feet away, as shown in the diagram. What is t

Mathematics

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Answer

4∘\text{}4^\circ\text{}

Step-by-step solution

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Okay, wild, let's solve this ramp problem.

A wheelchair ramp is placed over a set of stairs so that one end is 2 feet off the ground. The other end is at a point that is a horizontal distance of 28 feet away, as shown in the diagram. What is the angle of elevation to the nearest degree?

  1. Identify the known values from the diagram and problem description.

    The height of the ramp (opposite side to the angle of elevation) is 2 ft2 \text{ ft}.
    The horizontal distance of the ramp (adjacent side to the angle of elevation) is 28 ft28 \text{ ft}.
    We need to find the angle of elevation, θ\theta.

  2. Choose the appropriate trigonometric ratio.

    Since we have the opposite side and the adjacent side relative to the angle θ\theta, we use the tangent function:

    tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

  3. Substitute the known values into the formula.

    tan⁡(θ)=228\tan(\theta) = \frac{2}{28} tan⁡(θ)=114\tan(\theta) = \frac{1}{14}

  4. Solve for θ\theta using the inverse tangent function.

    θ=arctan⁡(114)\theta = \arctan\left(\frac{1}{14}\right)

  5. Calculate the value and round to the nearest degree.

    Using a calculator:

    θ≈4.085∘\theta \approx 4.085^\circ

    Rounding to the nearest degree:

    θ≈4∘\theta \approx 4^\circ

    The angle of elevation to the nearest degree is 4∘\text{}4^\circ\text{}.

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