To determine how far up the building the ladder reaches, we need to use some basic principles of trigonometry.
The ladder forms a right triangle with the building and the ground. The ladder itself acts as the hypotenuse of this triangle, while the height the ladder reaches on the building acts as the opposite side. The angle between the ladder and the building is given as 45 degrees.
We can use the sine function to determine the height the ladder reaches, because sine is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
Let's summarize this in mathematical terms:
1. The length of the ladder (hypotenuse) is 12 feet.
2. The angle between the ladder and the building (theta) is 45 degrees.
The sine function for a given angle is defined as:
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
Here, we need to solve for the opposite side (the height the ladder reaches):
[tex]\[ \sin(45^\circ) = \frac{\text{height}}{12} \][/tex]
The value of [tex]\(\sin(45^\circ)\)[/tex] is [tex]\(\frac{\sqrt{2}}{2}\)[/tex]. Therefore, we can write:
[tex]\[ \frac{\sqrt{2}}{2} = \frac{\text{height}}{12} \][/tex]
To find the height, we need to solve for the variable on the numerator of the right fraction:
[tex]\[ \text{height} = 12 \times \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ \text{height} = 12 \times \frac{1}{2} \times \sqrt{2} \][/tex]
[tex]\[ \text{height} = 6 \times \sqrt{2} \][/tex]
Thus, the height reached by the ladder is:
[tex]\[ 6\sqrt{2} \text{ feet} \][/tex]
Therefore, the correct answer is C. [tex]\(6 \sqrt{2} \text{ feet}\)[/tex].
