Question Statement

A pedestrian bridge is to be constructed with a slope of $5^{\circ }$ to meet accessibility requirements. If the length of the bridge (the distance between the two ends along the slope) is $120$ meters, determine the vertical height difference using the inverse sine function.

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Background and Explanation

To solve this problem, we use right-angle trigonometry. The sine of an angle relates the length of the side opposite the angle (vertical height) to the hypotenuse (the length of the bridge).

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Solution

Based on the geometric properties of the bridge:

• Let $BC$ be the vertical height of the bridge, represented by $y$.
• Let $AC$ be the length of the bridge, which is $120\text{ m}$.
• Let $\theta$ be the angle of the slope, which is $5^{\circ }$.

We start with the relationship defined by the inverse sine function as requested:

$\theta ={\sin }^{-1}\left(\frac{BC}{AC}\right)$

To solve for the height $BC$, we take the sine of both sides to isolate the ratio:

$\sin \theta =\frac{BC}{AC}$

Substitute the known values ($\theta =5^{\circ }$ and $AC=120$) into the equation:

$\sin 5^{\circ }=\frac{y}{120}$

Now, solve for $y$ by multiplying both sides by $120$:

$y=120\sin 5^{\circ }$

Using a calculator to find the value of $\sin 5^{\circ }\approx 0.08715$:

$y=120\times 0.08715$ $y=10.46\text{ m}$

The vertical height difference is $10.46$ meters.

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Key Formulas or Methods Used

• Sine Ratio: $\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$
• Inverse Sine Function: $\theta ={\sin }^{-1}\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)$
• Algebraic Rearrangement: Isolating the unknown variable ($y$) by multiplying by the denominator.

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Summary of Steps

1. Identify the given values: angle $\theta =5^{\circ }$ and hypotenuse (bridge length) $=120\text{ m}$.
2. Set up the trigonometric relationship using the inverse sine formula.
3. Rearrange the formula to solve for the vertical height ($y$).
4. Calculate the final value using the sine of the given angle.