Question

if a 12 foot ladder is leaned against a wall at a 65° angle with the ground, how high above the ground is the top of the ladder?
round your answer to the nearest hundredth.

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 12$ feet), the height above the ground is the side opposite the given angle ($\theta=65^{\circ}$). We use the sine function: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Substitute the values into the formula

Let $h$ be the height. Then $\sin(65^{\circ})=\frac{h}{12}$.

Step3: Solve for $h$

Multiply both sides by 12: $h = 12\times\sin(65^{\circ})$.
Using a calculator, $\sin(65^{\circ})\approx0.9063$.
So $h = 12\times0.9063=10.8756$.

Answer:

$10.88$