To determine how far up the building the 14-foot ladder reaches when it makes a 45-degree angle with the building, we can use trigonometry. Specifically, we will use the sine function, which relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse.
We are given:
- The length of the ladder (hypotenuse) = 14 feet
- The angle between the ladder and the building = 45 degrees
Our goal is to find the vertical height (opposite side) that the ladder reaches on the building.
First, we convert the angle from degrees to radians since trigonometric functions typically use radians in mathematical calculations:
[tex]\[ \text{Angle in radians} = 45^\circ \times \left( \frac{\pi}{180^\circ} \right) = \frac{\pi}{4} \approx 0.7854 \text{ radians} \][/tex]
Next, we use the sine function:
[tex]\[ \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
Rearranging to solve for the opposite side (height):
[tex]\[ \text{opposite} = \sin(\text{angle}) \times \text{hypotenuse} \][/tex]
[tex]\[ \text{height} = \sin\left( \frac{\pi}{4} \right) \times 14 \][/tex]
Since [tex]\(\sin(45^\circ) = \sin\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[ \text{height} = \frac{\sqrt{2}}{2} \times 14 \][/tex]
Simplifying this:
[tex]\[ \text{height} = 14 \times \frac{\sqrt{2}}{2} = 7\sqrt{2} \][/tex]
Therefore, the ladder reaches a height of [tex]\( 7\sqrt{2} \)[/tex] feet up the building.
So, the correct answer is:
C. [tex]\( 7\sqrt{2} \)[/tex] feet
