Answer :
To find the height up the building that the 12-foot ladder reaches when it makes a 45-degree angle with the building, let's go through the step-by-step process.
1. Understand the problem:
- We have a right triangle where:
- The hypotenuse (the ladder) is 12 feet.
- The angle between the ladder and the building is 45 degrees.
- We want to find the height up the building, which is the length of the side opposite the given angle.
2. Identifying the sides:
- In a right triangle, the side opposite the angle of 45 degrees and the hypotenuse are given.
- The formula involving the sine of an angle is [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex].
3. Apply the formula:
- For [tex]\(\theta = 45^\circ\)[/tex], we have:
[tex]\[ \sin(45^\circ) = \frac{\text{height}}{12} \][/tex]
- The value of [tex]\(\sin(45^\circ)\)[/tex] is known to be [tex]\(\frac{\sqrt{2}}{2}\)[/tex] or approximately 0.7071.
4. Set up the equation:
[tex]\[ \frac{\sqrt{2}}{2} = \frac{\text{height}}{12} \][/tex]
5. Solve for the height:
[tex]\[ \text{height} = 12 \times \frac{\sqrt{2}}{2} = 12 \times 0.7071 \approx 8.48528137423857 \][/tex]
Therefore, the height up the building that the ladder reaches is approximately 8.49 feet.
In this case, none of the provided multiple choice options (A, B, C, or D) seem to directly match 8.49 feet. However, the correct answer should be recognized as consistent with our calculations. Since none of the given options align with the solution, it suggests there could be an error in the provided options.
1. Understand the problem:
- We have a right triangle where:
- The hypotenuse (the ladder) is 12 feet.
- The angle between the ladder and the building is 45 degrees.
- We want to find the height up the building, which is the length of the side opposite the given angle.
2. Identifying the sides:
- In a right triangle, the side opposite the angle of 45 degrees and the hypotenuse are given.
- The formula involving the sine of an angle is [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex].
3. Apply the formula:
- For [tex]\(\theta = 45^\circ\)[/tex], we have:
[tex]\[ \sin(45^\circ) = \frac{\text{height}}{12} \][/tex]
- The value of [tex]\(\sin(45^\circ)\)[/tex] is known to be [tex]\(\frac{\sqrt{2}}{2}\)[/tex] or approximately 0.7071.
4. Set up the equation:
[tex]\[ \frac{\sqrt{2}}{2} = \frac{\text{height}}{12} \][/tex]
5. Solve for the height:
[tex]\[ \text{height} = 12 \times \frac{\sqrt{2}}{2} = 12 \times 0.7071 \approx 8.48528137423857 \][/tex]
Therefore, the height up the building that the ladder reaches is approximately 8.49 feet.
In this case, none of the provided multiple choice options (A, B, C, or D) seem to directly match 8.49 feet. However, the correct answer should be recognized as consistent with our calculations. Since none of the given options align with the solution, it suggests there could be an error in the provided options.
