Question

applying trigonometry to solve real - world problems
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: Identify the trigonometric relationship

We have a right - triangle where the height (opposite side to the given angle) is 23 feet and we want to find the hypotenuse (length of the ladder). We use the sine function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 75^{\circ}$ and the opposite side $y = 23$ feet, and the hypotenuse is $l$ (length of the ladder). So, $\sin75^{\circ}=\frac{23}{l}$.

Step2: Solve for the hypotenuse

We know that $\sin75^{\circ}=\sin(45^{\circ}+ 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$. From $\sin75^{\circ}=\frac{23}{l}$, we can rewrite it as $l=\frac{23}{\sin75^{\circ}}$.

Step3: Calculate the value of $l$

$l=\frac{23}{0.9659}\approx23.8$. Rounding to the nearest whole number, $l = 24$ feet.

Answer:

C. 24 ft