To determine how far up the building the ladder reaches, we can use trigonometry. Here is a step-by-step solution:
1. Understand the problem:
- We have a ladder leaning against a building.
- The length of the ladder is 10 feet.
- The angle between the ladder and the ground is 45 degrees.
- We need to find the vertical height from the ground to the point where the ladder touches the building.
2. Identify the relevant trigonometric ratio:
- In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side (height in this case) to the hypotenuse (ladder length here).
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
- Here, [tex]\(\theta = 45^\circ\)[/tex], the hypotenuse (ladder) is 10 feet, and the opposite side (height) is what we need to find.
3. Apply the sine function:
- Use the value of sine for the angle of 45 degrees, which is [tex]\(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)[/tex].
[tex]\[
\sin(45^\circ) = \frac{h}{10}
\][/tex]
- Express [tex]\(h\)[/tex] (height reached on the building):
[tex]\[
\frac{\sqrt{2}}{2} = \frac{h}{10}
\][/tex]
4. Solve for height [tex]\(h\)[/tex]:
- Multiply both sides by 10 to isolate [tex]\(h\)[/tex]:
[tex]\[
h = 10 \times \frac{\sqrt{2}}{2}
\][/tex]
- Simplify the right side of the equation:
[tex]\[
h = 10 \times 0.7071
\][/tex]
- Calculate [tex]\(h\)[/tex]:
[tex]\[
h \approx 7.0711 \text{ feet}
\][/tex]
Thus, the ladder reaches approximately 7.071 feet up the building. Therefore, none of the given choices (A. [tex]$20 \sqrt{2}$[/tex] feet, B. [tex]$5 \sqrt{2}$[/tex] feet, C. [tex]$10 \sqrt{2}$[/tex] feet, D. 5 feet) directly match the calculated height, but the correct height is found to be around 7.071 feet.
