A construction worker needs to build a ramp for wheelchair access. If the height of the entrance is 3 feet and the ramp must create an angle of \( 30^{\circ} \) with the ground, what is the length of the ramp?

To determine the length of the ramp needed for wheelchair access, we can model the situation using a right triangle: - **Height of the entrance (opposite side)**: 3 feet - **Angle with the ground**: \( 30^{\circ} \) - **Ramp length (hypotenuse)**: \( L \) feet ### Using Trigonometry The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] Plugging in the known values: \[ \sin(30^{\circ}) = \frac{3}{L} \] We know that: \[ \sin(30^{\circ}) = \frac{1}{2} \] So, \[ \frac{1}{2} = \frac{3}{L} \implies L = \frac{3}{\frac{1}{2}} = 6 \text{ feet} \] ### Using the 30-60-90 Triangle Ratio In a 30-60-90 triangle, the sides are in the ratio \( 1 : \sqrt{3} : 2 \), where: - The side opposite \( 30^{\circ} \) is \( 1 \) unit - The hypotenuse is \( 2 \) units Given the opposite side is 3 feet, the hypotenuse (ramp length) is: \[ 2 \times 3 = 6 \text{ feet} \] ### Conclusion **The length of the ramp should be 6 feet.**