Answer :
Certainly! Let's walk through the steps to solve this problem.
1. Identify the given values:
- The length of the ladder ([tex]\( L \)[/tex]) is 14 feet.
- The angle ([tex]\( \theta \)[/tex]) between the ladder and the building is 45 degrees.
2. Recognize the trigonometric relationship:
To find out how far up the building the ladder reaches, we need to determine the vertical height from the base to the top of the ladder. The angle provided is between the ladder and the building, and we can use the sine function in trigonometry, which relates an angle of a right triangle to the ratio of the opposite side to the hypotenuse.
The sine function is defined as:
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
Here:
- [tex]\(\theta = 45\)[/tex] degrees
- The hypotenuse is the ladder length, which is 14 feet.
- The opposite side is the height, which we need to find.
3. Apply the value of the sine function:
For [tex]\(\theta = 45\)[/tex] degrees, the sine value is known (or can be found using a calculator or unit circle):
[tex]\[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \][/tex]
4. Substitute the known values into the sine formula:
[tex]\[ \sin(45^\circ) = \frac{h}{14} \][/tex]
So,
[tex]\[ \frac{\sqrt{2}}{2} = \frac{h}{14} \][/tex]
5. Solve for [tex]\(h\)[/tex]:
[tex]\[ h = 14 \cdot \frac{\sqrt{2}}{2} \][/tex]
Simplifying further:
[tex]\[ h = 14 \cdot 0.70710678118 = 9.899494936611664 \ \text{feet} \][/tex]
Thus, the height up the building the ladder reaches is approximately 9.9 feet.
Therefore, the correct choice from the options provided would be:
[tex]\[ \boxed{9.9 \ \text{feet}} \][/tex]
1. Identify the given values:
- The length of the ladder ([tex]\( L \)[/tex]) is 14 feet.
- The angle ([tex]\( \theta \)[/tex]) between the ladder and the building is 45 degrees.
2. Recognize the trigonometric relationship:
To find out how far up the building the ladder reaches, we need to determine the vertical height from the base to the top of the ladder. The angle provided is between the ladder and the building, and we can use the sine function in trigonometry, which relates an angle of a right triangle to the ratio of the opposite side to the hypotenuse.
The sine function is defined as:
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
Here:
- [tex]\(\theta = 45\)[/tex] degrees
- The hypotenuse is the ladder length, which is 14 feet.
- The opposite side is the height, which we need to find.
3. Apply the value of the sine function:
For [tex]\(\theta = 45\)[/tex] degrees, the sine value is known (or can be found using a calculator or unit circle):
[tex]\[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \][/tex]
4. Substitute the known values into the sine formula:
[tex]\[ \sin(45^\circ) = \frac{h}{14} \][/tex]
So,
[tex]\[ \frac{\sqrt{2}}{2} = \frac{h}{14} \][/tex]
5. Solve for [tex]\(h\)[/tex]:
[tex]\[ h = 14 \cdot \frac{\sqrt{2}}{2} \][/tex]
Simplifying further:
[tex]\[ h = 14 \cdot 0.70710678118 = 9.899494936611664 \ \text{feet} \][/tex]
Thus, the height up the building the ladder reaches is approximately 9.9 feet.
Therefore, the correct choice from the options provided would be:
[tex]\[ \boxed{9.9 \ \text{feet}} \][/tex]
