To determine how far up the building a 14-foot ladder reaches when it makes a 45-degree angle with the ground, we can use trigonometry.
Here’s the step-by-step solution:
1. Understand the Problem:
We have a right triangle where:
- The ladder serves as the hypotenuse ([tex]\(c\)[/tex]) of the triangle, which is 14 feet.
- The angle between the ladder and the ground is [tex]\(45^\circ\)[/tex].
- We want to find the height ([tex]\(h\)[/tex]) the ladder reaches up the building, which is the side opposite the [tex]\(45^\circ\)[/tex] angle.
2. Identify the Trigonometric Function:
The sine function relates the angle to the length of the opposite side and the hypotenuse. The sine of an angle in a right triangle is defined as:
[tex]\[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
\][/tex]
Here, [tex]\(\theta = 45^\circ\)[/tex], the opposite side is the height ([tex]\(h\)[/tex]), and the hypotenuse ([tex]\(c\)[/tex]) is the ladder length (14 feet).
3. Set Up the Equation:
[tex]\[
\sin(45^\circ) = \frac{h}{14}
\][/tex]
4. Determine the Value of [tex]\(\sin (45^\circ)\)[/tex]:
From trigonometry, we know:
[tex]\[
\sin(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
5. Solving for the Height (h):
Substitute [tex]\(\sin (45^\circ)\)[/tex] into the equation:
[tex]\[
\frac{\sqrt{2}}{2} = \frac{h}{14}
\][/tex]
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 right-hand side:
[tex]\[
h = 14 \times \frac{\sqrt{2}}{2} = 14 \div 2 \times \sqrt{2} = 7 \sqrt{2}
\][/tex]
Therefore, the height up the building that the ladder reaches is [tex]\(7\sqrt{2}\)[/tex] feet.
So, the correct answer is:
C. [tex]\(7 \sqrt{2}\)[/tex] feet
