To solve how far up the building a 10-foot ladder reaches when it makes a 45-degree angle with the building, we will use trigonometry. Specifically, we will look at the cosine function, which relates the angle to the adjacent side (height up the building) and the hypotenuse (length of the ladder).
Consider the right triangle formed by the wall, the ground and the ladder leaning against the building. Here are the steps:
1. Identify the components:
- The hypotenuse is the length of the ladder, which is 10 feet.
- The angle between the ladder and the building is 45 degrees.
- We need to find the length of the side opposite the 45-degree angle, i.e., the height reached by the ladder up the building.
2. Use the cosine function:
- Cosine of an angle in a right triangle is defined as the ratio of the adjacent side (height in this case, [tex]\( h \)[/tex]) to the hypotenuse (the ladder length, [tex]\( L \)[/tex]).
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
For our problem:
[tex]\[
\cos(45^\circ) = \frac{h}{10}
\][/tex]
3. Calculate [tex]\(\cos(45^\circ)\)[/tex]:
- The cosine of 45 degrees is a well-known trigonometric value:
[tex]\[
\cos(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
4. Set up the equation:
[tex]\[
\frac{\sqrt{2}}{2} = \frac{h}{10}
\][/tex]
5. Solve for [tex]\( h \)[/tex]:
[tex]\[
h = 10 \times \frac{\sqrt{2}}{2}
\][/tex]
6. Simplify the expression:
[tex]\[
h = 10 \times \frac{1}{\sqrt{2}} = 10 \times \frac{\sqrt{2}}{2}
\][/tex]
7. Calculate the height:
[tex]\[
h = 5\sqrt{2}
\][/tex]
Thus, the height that the ladder reaches up the building is [tex]\( 5\sqrt{2} \)[/tex] feet.
Therefore, the correct option is:
C. [tex]\( 5 \sqrt{2} \)[/tex] feet.
