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 trigonometric relation

The ladder forms a right triangle with the wall and ground. We use the cosine function, which relates the adjacent side (horizontal distance, $x$), hypotenuse (ladder length, 30 ft), and the angle with the ground:
$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Step2: Substitute known values

Plug in $\theta = 74^\circ$ and hypotenuse = 30 ft:
$\cos(74^\circ) = \frac{x}{30}$

Step3: Solve for horizontal distance

Rearrange to solve for $x$:
$x = 30 \times \cos(74^\circ)$
Calculate using a calculator (ensure it is in degree mode):
$x \approx 30 \times 0.2756 = 8.268$

Step4: Round to nearest hundredth

Round the result to two decimal places:
$x \approx 8.27$

Answer:

8.27 feet