Question

miguel 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. 6 ft 22 ft 24 ft 89 ft

Explanation:

Step1: Use sine function

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, here $\theta = 75^{\circ}$ and opposite side is 23 ft. Let the length of the ladder be $l$. So $\sin75^{\circ}=\frac{23}{l}$.

Step2: Solve for $l$

$l=\frac{23}{\sin75^{\circ}}$. Since $\sin75^{\circ}=\sin(45^{\circ} + 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$. Then $l=\frac{23}{0.9659}\approx24$ ft.

Answer:

24 ft