Certainly! Let's solve the problem step-by-step.
We have a 10-foot ladder leaning against a building and it makes a 45-degree angle with the building. Our goal is to determine how far up the building the ladder reaches.
1. Given:
- Length of the ladder (hypotenuse): [tex]\( 10 \)[/tex] feet
- Angle with the building: [tex]\( 45 \)[/tex] degrees
2. Understanding the Problem:
Since the ladder makes an angle of 45 degrees with the building, this means we are working within a right-angled triangle where we need to find the height (opposite side to the angle).
3. Use Trigonometric Identities:
We use the cosine function to correlate the hypotenuse with the adjacent side (height here).
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
4. Substitute the Known Values:
[tex]\(\theta = 45^\circ\)[/tex], and hypotenuse [tex]\(= 10\)[/tex] feet.
So,
[tex]\[
\cos(45^\circ) = \frac{\text{height}}{10}
\][/tex]
5. Calculate the Cosine of 45 Degrees:
The cosine of 45 degrees is a known value:
[tex]\[
\cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
6. Set Up the Equation:
Substitute [tex]\(\cos(45^\circ)\)[/tex] into our equation:
[tex]\[
\frac{\sqrt{2}}{2} = \frac{\text{height}}{10}
\][/tex]
7. Solve for the Height:
Multiply both sides of the equation by 10 to isolate height:
[tex]\[
\text{height} = 10 \times \frac{\sqrt{2}}{2}
\][/tex]
Simplify it to:
[tex]\[
\text{height} = 10 \times 0.7071
\][/tex]
[tex]\[
\text{height} = 7.071 \text{ feet}
\][/tex]
Therefore, the ladder reaches approximately 7.071 feet up the building. Hence, the correct answer is none of the provided choices match 7.071 feet exactly, but if we had to compare, the proper height would be closest to an approximation not listed directly within these choices.
