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: Identify the trigonometric relationship

We know the height (opposite side) and need to find the length of the ladder (hypotenuse). We use the sine - function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 75^{\circ}$ and the opposite side $y = 23$ feet.

Step2: Set up the equation

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

Step3: Solve for $x$

From $\sin(75^{\circ})=\frac{23}{x}$, we can rewrite it as $x=\frac{23}{\sin(75^{\circ})}$. Substituting the value of $\sin(75^{\circ})\approx0.9659$, we get $x=\frac{23}{0.9659}\approx23.81$.

Step4: Round to the nearest whole number

Rounding $23.81$ to the nearest whole number gives $24$.

Answer:

24 ft