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Sagot :
To solve this problem, we need to determine how high up the building a 10-foot ladder reaches when it makes a 45-degree angle with the building. We can approach this using trigonometry, specifically the sine function.
Here's a step-by-step solution:
1. Understand the given information and visualize the problem:
- We have a right-angled triangle formed by the ladder, the building, and the ground.
- The ladder acts as the hypotenuse (the longest side) of the right-angled triangle. Its length is 10 feet.
- The angle between the ladder and the ground is given as 45 degrees.
2. Identify the sides of the triangle:
- Since we need to find out how far up the building the ladder reaches, we are looking for the length of the side opposite the 45-degree angle (the vertical side).
3. Recall the sine function in a right-angled triangle:
- The sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the hypotenuse.
- Mathematically, this can be written as:
[tex]\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \][/tex]
where:
- [tex]\(\theta\)[/tex] is the angle,
- The opposite side is the vertical height up the building,
- The hypotenuse is the length of the ladder.
4. Substitute the known values:
- Here, [tex]\(\theta\)[/tex] is 45 degrees and the hypotenuse is 10 feet.
- We need to find the opposite side (height).
5. Use the sine function:
[tex]\[ \sin(45^\circ) = \frac{\text{Height}}{10} \][/tex]
6. Determine [tex]\(\sin(45^\circ)\)[/tex]:
- [tex]\(\sin(45^\circ)\)[/tex] is a well-known standard value and equals [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
7. Set up the equation and solve for the height:
[tex]\[ \frac{\sqrt{2}}{2} = \frac{\text{Height}}{10} \][/tex]
8. Isolate the height (multiply both sides by 10):
[tex]\[ \text{Height} = 10 \times \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ \text{Height} = 5\sqrt{2} \][/tex]
So, the distance up the building that the ladder reaches is [tex]\(5\sqrt{2}\)[/tex] feet.
Therefore, the correct answer is:
A. [tex]\(5 \sqrt{2}\)[/tex] feet
Here's a step-by-step solution:
1. Understand the given information and visualize the problem:
- We have a right-angled triangle formed by the ladder, the building, and the ground.
- The ladder acts as the hypotenuse (the longest side) of the right-angled triangle. Its length is 10 feet.
- The angle between the ladder and the ground is given as 45 degrees.
2. Identify the sides of the triangle:
- Since we need to find out how far up the building the ladder reaches, we are looking for the length of the side opposite the 45-degree angle (the vertical side).
3. Recall the sine function in a right-angled triangle:
- The sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the hypotenuse.
- Mathematically, this can be written as:
[tex]\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \][/tex]
where:
- [tex]\(\theta\)[/tex] is the angle,
- The opposite side is the vertical height up the building,
- The hypotenuse is the length of the ladder.
4. Substitute the known values:
- Here, [tex]\(\theta\)[/tex] is 45 degrees and the hypotenuse is 10 feet.
- We need to find the opposite side (height).
5. Use the sine function:
[tex]\[ \sin(45^\circ) = \frac{\text{Height}}{10} \][/tex]
6. Determine [tex]\(\sin(45^\circ)\)[/tex]:
- [tex]\(\sin(45^\circ)\)[/tex] is a well-known standard value and equals [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
7. Set up the equation and solve for the height:
[tex]\[ \frac{\sqrt{2}}{2} = \frac{\text{Height}}{10} \][/tex]
8. Isolate the height (multiply both sides by 10):
[tex]\[ \text{Height} = 10 \times \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ \text{Height} = 5\sqrt{2} \][/tex]
So, the distance up the building that the ladder reaches is [tex]\(5\sqrt{2}\)[/tex] feet.
Therefore, the correct answer is:
A. [tex]\(5 \sqrt{2}\)[/tex] feet
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