You are building a ramp that must cover a horizontal distance of exactly 9 feet. The angle of the ramp from the ground is \( 23^{\circ} \). Determine the length of the ramp, in feet. Round to two decimal places as needed. Show your work here feet

To determine the length of the ramp, we can use trigonometric relations. The ramp forms a right triangle where the horizontal distance (the base) is 9 feet and the angle of elevation is \( 23^{\circ} \). Using the cosine function, which relates the adjacent side (horizontal distance) to the hypotenuse (length of the ramp): \[ \cos(23^{\circ}) = \frac{9}{L} \] where \( L \) is the length of the ramp. Rearranging the equation to solve for \( L \): \[ L = \frac{9}{\cos(23^{\circ})} \] Now, calculating \( \cos(23^{\circ}) \): \[ \cos(23^{\circ}) \approx 0.9205 \] Thus, substituting back, we have: \[ L \approx \frac{9}{0.9205} \approx 9.78 \text{ feet} \] So, the length of the ramp is approximately \( 9.78 \) feet when rounded to two decimal places. The answer is \( 9.78 \) feet. To double-check, if you want to explore different angles or setups in your construction, you can always revisit the basic trigonometric identities or use tools like ramps in playgrounds or accessibility designs to enhance your approach! They often use calculated angle measurements for optimal safety and ease of use. Additionally, remember that calculating lengths with ramps isn't just about math; it’s about ensuring everyone can use the space! This initiative promotes inclusivity and thoughtful design in our environments, so keep that spirit alive in your project!