To determine how far up the building the ladder reaches when it makes a 45-degree angle with the ground, you can use trigonometry. Specifically, we'll use the sine function because we are dealing with the opposite side (the height reached up the building) and the hypotenuse (the length of the ladder).
### Step-by-Step Solution:
1. Understand the given information:
- Length of the ladder (hypotenuse) = 10 feet
- Angle between the ladder and the ground = 45 degrees
2. Determine the trigonometric relationship:
- In a right triangle, the sine of an angle is equal to the length of the opposite side divided by the hypotenuse. Mathematically, this is expressed as:
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
3. Apply the sine function:
- Here, [tex]\(\theta\)[/tex] is 45 degrees.
- The hypotenuse is 10 feet.
- Let the opposite side (height up the building) be [tex]\( h \)[/tex].
[tex]\[
\sin(45^\circ) = \frac{h}{10}
\][/tex]
4. Resolve the equation for [tex]\( h \)[/tex]:
- First, recall that [tex]\(\sin(45^\circ)\)[/tex] is a well-known trigonometric value:
[tex]\[
\sin(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
- Substitute [tex]\(\sin(45^\circ)\)[/tex] with [tex]\(\frac{\sqrt{2}}{2}\)[/tex] in the equation:
[tex]\[
\frac{\sqrt{2}}{2} = \frac{h}{10}
\][/tex]
5. Solve for [tex]\( h \)[/tex]:
- Multiply both sides of the equation by 10 to isolate [tex]\( h \)[/tex]:
[tex]\[
h = 10 \times \frac{\sqrt{2}}{2}
\][/tex]
- Simplify the right side:
[tex]\[
h = 5\sqrt{2}
\][/tex]
Final result: The ladder reaches [tex]\( 5\sqrt{2} \)[/tex] feet up the building. So, the correct answer is:
B. [tex]\( 5\sqrt{2} \)[/tex] feet.
