Let's approach this problem step-by-step:
1. Understanding the geometry of the problem:
- A 10-foot ladder is standing against a building.
- The ladder makes an angle with the building. This means we're dealing with a right triangle where the ladder is the hypotenuse, one leg is the height up the building, and the other leg is the distance from the base of the building to the bottom of the ladder.
2. Assuming the angle:
- We often use a 45-degree angle for such problems unless stated otherwise. This angle is commonly used to simplify calculations as it ensures that the legs of the triangle are equal in length.
3. Using trigonometric ratios:
- In a right triangle, the height can be found using the sine (sin) of the angle.
- For an angle θ (theta) in a right triangle:
[tex]\[
\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}
\][/tex]
- Here, [tex]\(\theta = 45^\circ\)[/tex], the hypotenuse is the length of the ladder (10 feet), and we want to find the opposite side (height up the building).
4. Correct calculation:
- For θ = 45°:
[tex]\[
\sin(45^\circ) = \frac{\text{height}}{10}
\][/tex]
- The value of [tex]\(\sin(45^\circ)\)[/tex] is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
[tex]\[
\frac{\sqrt{2}}{2} = \frac{\text{height}}{10}
\][/tex]
5. Solving for height:
- Cross-multiplying to solve for height:
[tex]\[
\text{height} = 10 \times \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
\text{height} = 10 \times 0.7071
\][/tex]
[tex]\[
\text{height} = 7.071 \text{ feet}
\][/tex]
The ladder reaches approximately 7.071 feet up the building.
So, the correct answer to the question is not directly provided in the multiple choices, but from the given information and choices, none appear correct based on the standard method and recognized height calculation using a 45-degree angle. It seems there might be another interpretation or additional information required for the given options.
