To determine how far up the building the 14-foot ladder reaches when it makes a 45-degree angle with the ground, we will approach the problem using trigonometry, specifically using the sine function.
Given:
- Ladder length (hypotenuse, [tex]\( h \)[/tex]): 14 feet
- Angle of elevation ([tex]\( \theta \)[/tex]): 45 degrees
We need to find the distance up the building, which we will call [tex]\( y \)[/tex]. The relationship between the angle, the hypotenuse, and the opposite side (which is the height up the building in this case) can be described using the sine function:
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
[tex]\[ \sin(45^\circ) = \frac{y}{14} \][/tex]
We know from trigonometric tables or unit circle values that:
[tex]\[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \][/tex]
We can now solve for [tex]\( y \)[/tex]:
[tex]\[ \frac{\sqrt{2}}{2} = \frac{y}{14} \][/tex]
To isolate [tex]\( y \)[/tex], we multiply both sides of the equation by 14:
[tex]\[ y = 14 \times \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ y = 7 \sqrt{2} \][/tex]
Thus, the distance up the building is:
[tex]\[ y = 7 \sqrt{2} \][/tex]
Therefore, the correct answer is:
A. [tex]\( 7 \sqrt{2} \)[/tex] feet
