Answer :
To solve the problem, we need to determine how high up a wall a 24-foot ladder will reach when it makes an angle of 58° with the ground.
### Step-by-Step Solution:
1. Identify the components of the problem:
- Length of the ladder (hypotenuse in the right triangle): 24 feet.
- Angle between the ladder and the ground: 58°.
2. Understand the trigonometric relationship:
We are dealing with a right triangle where:
- The hypotenuse is the ladder.
- The height up the wall is the side opposite the angle.
- The angle given is between the ladder and the ground.
We can use the sine function, which relates the angle to the ratio of the opposite side to the hypotenuse in a right triangle:
[tex]\[ \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
3. Set up the equation using the sine function:
Let [tex]\( h \)[/tex] represent the height up the wall.
[tex]\[ \sin(58°) = \frac{h}{24} \][/tex]
4. Solve for [tex]\( h \)[/tex]:
Multiply both sides of the equation by the length of the ladder:
[tex]\[ h = 24 \times \sin(58°) \][/tex]
5. Calculate the value of [tex]\( h \)[/tex]:
Using a calculator to find [tex]\( \sin(58°) \)[/tex]:
[tex]\[ \sin(58°) \approx 0.848 \][/tex]
Then multiply by the length of the ladder:
[tex]\[ h \approx 24 \times 0.848 = 20.352 \][/tex]
6. Round the answer to the nearest tenth:
The value 20.352 rounded to the nearest tenth is 20.4.
### Conclusion:
The height up the wall that the 24-foot ladder will reach when it makes a 58° angle with the ground is approximately 20.4 feet. Thus, the correct answer is:
[tex]\[ \boxed{B. \, 20.4 \, \text{feet}} \][/tex]
### Step-by-Step Solution:
1. Identify the components of the problem:
- Length of the ladder (hypotenuse in the right triangle): 24 feet.
- Angle between the ladder and the ground: 58°.
2. Understand the trigonometric relationship:
We are dealing with a right triangle where:
- The hypotenuse is the ladder.
- The height up the wall is the side opposite the angle.
- The angle given is between the ladder and the ground.
We can use the sine function, which relates the angle to the ratio of the opposite side to the hypotenuse in a right triangle:
[tex]\[ \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
3. Set up the equation using the sine function:
Let [tex]\( h \)[/tex] represent the height up the wall.
[tex]\[ \sin(58°) = \frac{h}{24} \][/tex]
4. Solve for [tex]\( h \)[/tex]:
Multiply both sides of the equation by the length of the ladder:
[tex]\[ h = 24 \times \sin(58°) \][/tex]
5. Calculate the value of [tex]\( h \)[/tex]:
Using a calculator to find [tex]\( \sin(58°) \)[/tex]:
[tex]\[ \sin(58°) \approx 0.848 \][/tex]
Then multiply by the length of the ladder:
[tex]\[ h \approx 24 \times 0.848 = 20.352 \][/tex]
6. Round the answer to the nearest tenth:
The value 20.352 rounded to the nearest tenth is 20.4.
### Conclusion:
The height up the wall that the 24-foot ladder will reach when it makes a 58° angle with the ground is approximately 20.4 feet. Thus, the correct answer is:
[tex]\[ \boxed{B. \, 20.4 \, \text{feet}} \][/tex]
