Question

amira leans a 16 - foot ladder against a wall so that it forms an angle of 66° with the ground. whats the horizontal distance between the base of the ladder and the bottom of the wall? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 16$ feet), the angle with the ground is $\theta=66^{\circ}$, and we want to find the adjacent side ($x$) to the angle. We use the cosine formula $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Substitute the values into the formula

Substitute $\theta = 66^{\circ}$ and $c = 16$ into $\cos\theta=\frac{x}{c}$. So, $\cos(66^{\circ})=\frac{x}{16}$.

Step3: Solve for $x$

Multiply both sides by 16: $x = 16\times\cos(66^{\circ})$.
We know that $\cos(66^{\circ})\approx0.4067$. Then $x = 16\times0.4067$.
$x=6.5072\approx6.51$.

Answer:

$6.51$ feet