To determine how far up the building the ladder reaches, let's follow these steps:
1. Understand the Geometry: The ladder forms a right triangle with the building and the ground. Here, the ladder is the hypotenuse of this right triangle, and the angle formed between the ladder and the ground is 45 degrees.
2. Right Triangle Properties: In a right triangle with a 45-degree angle, the two non-hypotenuse sides (often called the legs) are of equal length. Since the ladder is the hypotenuse, we need to find the length of the vertical leg (height up the building) using trigonometric relationships.
3. Trigonometric Function: For a right triangle, we use the sine function which relates the angle to the ratio of the opposite side (height up the building) to the hypotenuse (length of the ladder).
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
4. Apply the Sine Function: Here, [tex]\(\theta = 45\)[/tex] degrees and the hypotenuse (ladder length) is 14 feet. We need to solve for the opposite side (height up the building).
[tex]\[
\sin(45^\circ) = \frac{\text{height}}{14}
\][/tex]
5. Value of [tex]\(\sin(45^\circ)\)[/tex]: The sine of 45 degrees is [tex]\(\frac{\sqrt{2}}{2}\)[/tex] or approximately 0.7071.
[tex]\[
\frac{\sqrt{2}}{2} = \frac{\text{height}}{14}
\][/tex]
6. Solve for Height:
[tex]\[
\text{height} = 14 \times \frac{\sqrt{2}}{2} = 14 \times 0.7071 \approx 9.899
\][/tex]
7. Final Comparison with Choices: From the given multiple-choice options:
- A. [tex]\(28 \sqrt{2}\)[/tex] feet ≈ 39.6 feet
- B. [tex]\(14 \sqrt{2}\)[/tex] feet ≈ 19.8 feet
- C. [tex]\(7 \sqrt{2}\)[/tex] feet ≈ 9.9 feet
- D. 7 feet
Since [tex]\(9.899 \approx 9.9\)[/tex] feet, the correct choice is:
C. [tex]\(7 \sqrt{2}\)[/tex] feet
