Question

1. a 20 foot ladder leans against a building and makes an angle of 72° with the ground. find the distance between the foot of the ladder and the building.
2. a straight road to the top of a hill is 2500 feet long and makes an angle of 12° with the horizontal. find the height of the hill.

Explanation:

Step1: Identify the trigonometric relationship for problem 5

We have a right - triangle where the ladder is the hypotenuse ($c = 20$ feet), the angle between the ladder and the ground is $\theta=72^{\circ}$, and we want to find the adjacent side $x$ to the given angle. We use the cosine function $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.
$\cos\theta=\frac{x}{c}$

Step2: Solve for $x$ in problem 5

Substitute $\theta = 72^{\circ}$ and $c = 20$ into the formula:
$x = c\cos\theta=20\cos72^{\circ}$
$x\approx20\times0.3090 = 6.18$ feet

Step3: Identify the trigonometric relationship for problem 6

We have a right - triangle where the road is the hypotenuse ($c = 2500$ feet), the angle between the road and the horizontal is $\theta = 12^{\circ}$, and we want to find the opposite side $y$ to the given angle. We use the sine function $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
$\sin\theta=\frac{y}{c}$

Step4: Solve for $y$ in problem 6

Substitute $\theta = 12^{\circ}$ and $c = 2500$ into the formula:
$y = c\sin\theta=2500\sin12^{\circ}$
$y\approx2500\times0.2079=519.75$ feet

Answer:

1. The distance between the foot of the ladder and the building is approximately $6.18$ feet.
2. The height of the hill is approximately $519.75$ feet.