Question

applying trigonometry to solve real-world problems
ilegui needs to fix a window screen that is 23 feet above the ground. the ladder he uses makes a 75° angle with the ground. what is the shortest possible length of the ladder if the top of it is 23 feet off the ground? round to the nearest whole number.
24 ft
22 ft
6 ft
80 ft

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height (opposite side to the 75° angle) is 22 ft, and we need to find the length of the ladder (hypotenuse). The sine function relates the opposite side and the hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Solve for the hypotenuse (ladder length)

Let $L$ be the length of the ladder. We know $\theta = 75^\circ$ and the opposite side is 22 ft. So, $\sin(75^\circ) = \frac{22}{L}$. Solving for $L$, we get $L = \frac{22}{\sin(75^\circ)}$.

Step3: Calculate the value

First, find $\sin(75^\circ)$. $\sin(75^\circ) \approx 0.9659$. Then, $L = \frac{22}{0.9659} \approx 22.78$. Rounding to the nearest whole number, we get 23? Wait, no, the options are 24, 22, 6, 80. Wait, maybe I miscalculated. Wait, $\sin(75^\circ)$ is approximately 0.9659, so 22 divided by 0.9659 is approximately 22.78, which is closer to 23, but the options have 24. Wait, maybe I made a mistake. Wait, let's check again. Wait, the height is 22 ft, angle with ground is 75°, so $\sin(75°) = 22 / L$, so $L = 22 / \sin(75°)$. Let's calculate $\sin(75°)$ more accurately. $\sin(75°) = \sin(45° + 30°) = \sin45°\cos30° + \cos45°\sin30° = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \approx \frac{2.449 + 1.414}{4} \approx \frac{3.863}{4} \approx 0.9659$. Then 22 / 0.9659 ≈ 22.78, which is approximately 23, but the options have 24. Wait, maybe the height is 23? Wait, the problem says "the tip of it is 23 feet off the ground"? Wait, maybe I misread. Let me check the problem again. "What is the shortest possible length of the ladder if the tip of it is 23 feet off the ground?" Oh! I misread 22 as 23? Wait, no, the user's image: "the tip of it is 23 feet off the ground". So opposite side is 23 ft. Then $L = 23 / \sin(75°) \approx 23 / 0.9659 \approx 23.81$, which rounds to 24. Ah, that's it. I misread the height as 22, but it's 23. So the correct calculation is $L = \frac{23}{\sin(75^\circ)} \approx \frac{23}{0.9659} \approx 23.81 \approx 24$.

Answer:

24 ft