To determine how far up the building the 14-foot ladder reaches, given that it makes a 45-degree angle with the building, you'll need to use some trigonometry, specifically the sine function.
### Step-by-Step Solution:
1. Understanding the problem:
- The ladder length is 14 feet.
- The angle formed between the ladder and the ground is 45 degrees.
- We need to find the height ("h") up the building where the ladder touches.
2. Trigonometric Functions:
- The trigonometric relationship we will use here is sine, which relates the opposite side to the hypotenuse in a right triangle.
- The formula for sine is:
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
- In this case:
[tex]\[
\sin(45^\circ) = \frac{h}{14}
\][/tex]
3. Sin of 45 Degrees:
- From trigonometry, we know that:
[tex]\[
\sin(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
4. Setting up the Equation:
- Plug in the values into the equation:
[tex]\[
\frac{\sqrt{2}}{2} = \frac{h}{14}
\][/tex]
5. Solving for h:
- To solve for [tex]\( h \)[/tex], multiply both sides of the equation by 14:
[tex]\[
h = 14 \times \frac{\sqrt{2}}{2}
\][/tex]
- Simplify the multiplication:
[tex]\[
h = 14 \times \frac{1}{\sqrt{2}} \times \sqrt{2} = 14 \times \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
h = 14 \div 2 \times \sqrt{2} = 7 \times \sqrt{2}
\][/tex]
Thus, the height up the building that the ladder reaches is:
[tex]\[
h = 7 \sqrt{2}
\][/tex]
Therefore, the correct answer is:
A. [tex]$7 \sqrt{2}$[/tex] feet
