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

Explanation:

Step1: Set up a right - triangle model

We have a right - triangle where the height (opposite side to the given angle) is $h = 23$ feet and the angle between the ladder (hypotenuse) and the ground is $\theta=75^{\circ}$. We use the sine function $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Solve for the hypotenuse

We know that $\sin\theta=\sin(75^{\circ})=\frac{23}{l}$, where $l$ is the length of the ladder. First, find $\sin(75^{\circ})=\sin(45^{\circ} + 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$. Then, $l=\frac{23}{\sin(75^{\circ})}=\frac{23}{0.9659}\approx23.81$.

Answer:

$24$ ft