Question Statement

A ramp is to be constructed to a platform 2 meters high. If the ramp is 5 meters long, calculate the angle of elevation of the ramp.

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Background and Explanation

To solve this problem, we use basic trigonometry in a right-angled triangle. The angle of elevation is the angle formed between the horizontal ground and the ramp as it rises toward the platform.

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Solution

Based on the provided information, we can visualize the ramp as the hypotenuse of a right-angled triangle where the platform height is the perpendicular side.

1. Identify the given values: From the figure and the problem description: $\begin{array}{rl}\text{Height of platform }(BC)& =2\text{ m}\\ \text{Length of ramp }(AC)& =5\text{ m}\\ \text{Angle of elevation }(\angle BAC)& =\theta \end{array}$

2. Choose the correct trigonometric ratio: In a right-angled triangle, the sine of an angle is the ratio of the length of the side opposite the angle (perpendicular) to the length of the longest side (hypotenuse). $\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}}=\frac{BC}{AC}$

3. Substitute the values and solve for $\theta$: Substitute the known lengths into the equation: $\sin \theta =\frac{2}{5}$

To find the value of the angle $\theta$, we take the inverse sine (${\sin }^{-1}$) of the result: $\theta ={\sin }^{-1}\left(\frac{2}{5}\right)$

Using a calculator to find the degree measure: $\theta =23^{\circ }35^{\prime }$

The angle of elevation of the ramp is approximately $23^{\circ }35^{\prime }$.

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Key Formulas or Methods Used

• Sine Ratio: $\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$
• Inverse Trigonometry: Used to find an unknown angle when two sides are known ($\theta ={\sin }^{-1}(x)$).

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Summary of Steps

1. Identify the height of the platform as the perpendicular side ($2\text{ m}$) and the ramp length as the hypotenuse ($5\text{ m}$).
2. Set up the sine ratio formula: $\sin \theta =\frac{\text{Perp}}{\text{Hyp}}$.
3. Divide the perpendicular by the hypotenuse to get $0.4$.
4. Calculate the inverse sine of the result to find the angle of elevation.