Question

1. a 32 - foot ladder is leaning against the side of a building. if the ladder makes an angle of 22°44 with the side of the building, how far is the bottom of the ladder from the base of the building?

Explanation:

Step1: Convert the angle from degrees and minutes to decimal degrees

We know that \(1^{\circ}=60'\), so \(44'=\frac{44}{60}\approx0.733^{\circ}\). Then \(22^{\circ}44'\approx22.733^{\circ}\).

Step2: Use the sine function

Let \(x\) be the distance from the bottom of the ladder to the base of the building. The length of the ladder \(L = 32\) feet.
We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 22.733^{\circ}\) and the hypotenuse is the length of the ladder \(L\), and the opposite side is \(x\).
So \(x = L\sin\theta\).
Substitute \(L = 32\) and \(\theta=22.733^{\circ}\) into the formula: \(x = 32\times\sin(22.733^{\circ})\).
Since \(\sin(22.733^{\circ})\approx0.386\), then \(x=32\times0.386 = 12.352\) feet.

Answer:

The bottom of the ladder is approximately \(12.35\) feet from the base of the building.