Question

short response

1. a 12 - ft - long ladder is leaning against a wall and makes an 80° angle with the ground. how high up the wall does the ladder reach, and how far is the base of the ladder from the base of the wall? round to the nearest inch.

Explanation:

Step1: Find height on wall (opposite side)

We use sine: $\sin(80^\circ)=\frac{h}{12}$, so $h = 12\times\sin(80^\circ)$.
$\sin(80^\circ)\approx0.9848$, so $h\approx12\times0.9848 = 11.8176$ ft. Convert to inches: $11.8176\times12\approx141.81$ inches, $\approx142$ inches.

Step2: Find base distance (adjacent side)

We use cosine: $\cos(80^\circ)=\frac{d}{12}$, so $d = 12\times\cos(80^\circ)$.
$\cos(80^\circ)\approx0.1736$, so $d\approx12\times0.1736 = 2.0832$ ft. Convert to inches: $2.0832\times12\approx24.9984$ inches, $\approx25$ inches.

Answer:

Height on wall: $\approx142$ inches, Base distance: $\approx25$ inches.