Question

kylie leans a 30 - foot ladder against a wall so that it forms an angle of 74° 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 ($c = 30$ feet), the angle with the ground is $\theta=74^{\circ}$, and we want to find the adjacent side ($x$) to the angle $\theta$. The cosine function is defined as $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, so $\cos\theta=\frac{x}{c}$.

Step2: Solve for $x$

Substitute $c = 30$ and $\theta = 74^{\circ}$ into the formula $x = c\cos\theta$.
We know that $\cos(74^{\circ})\approx0.2756$.
Then $x=30\times\cos(74^{\circ})$.
$x = 30\times0.2756=8.268$.

Answer:

$8.27$ feet