To find out how far up the building the 14-foot ladder reaches when it makes a 45-degree angle with the building, we can use trigonometry. Specifically, we will use the cosine function because we are dealing with the adjacent side of the right triangle formed by the ladder, the building, and the ground.
Here's the step-by-step solution:
1. Understand the problem: We have a right triangle where:
- The hypotenuse is the ladder, which is 14 feet long.
- The angle between the ladder and the building is 45 degrees.
- We need to find the height up the building where the ladder touches it, which corresponds to the adjacent side of the triangle.
2. Set up the trigonometric ratio: For a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse:
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
In this problem:
- \(\theta = 45^\circ\)
- Hypotenuse = 14 feet
- Adjacent (height up the building) is what we need to find.
3. Write the equation using the cosine function:
[tex]\[
\cos(45^\circ) = \frac{\text{adjacent}}{14}
\][/tex]
4. Find the value of \(\cos(45^\circ)\):
[tex]\[
\cos(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
5. Substitute the cosine value into the equation:
[tex]\[
\frac{\sqrt{2}}{2} = \frac{\text{adjacent}}{14}
\][/tex]
6. Solve for the adjacent side (height):
[tex]\[
\text{adjacent} = 14 \times \frac{\sqrt{2}}{2} = 14 \times 0.7071
\][/tex]
[tex]\[
\text{adjacent} \approx 9.899494936611665 \text{ feet}
\][/tex]
Therefore, the correct answer is not listed in the provided options. The exact height up the building that the ladder reaches is approximately:
[tex]\[ \boxed{9.899494936611665 \text{ feet}} \][/tex]
