To solve this problem, we can use trigonometry. Specifically, we can use the sine function, which relates the angle of a right-angled triangle to the ratio of the length of the side opposite the angle to the hypotenuse. The ladder against the tree forms a right-angled triangle, with the tree and the ground as the two perpendicular sides and the ladder as the hypotenuse.
Let's solve the problem in steps:
1. Calculate the initial height of the ladder on the tree using the initial angle (23.4°).
2. Calculate the new height of the ladder on the tree using the new angle (18°).
3. Determine the difference in height to find out how much higher the boy needs to push the ladder for the angle to be 18°.
**Step 1: Calculate the initial height of the ladder on the tree using the initial angle (23.4°).**
We have:
- Ladder length (hypotenuse) = 2.8 m
- Initial angle with the ground = 23.4°
The height of the ladder on the tree (opposite side) can be found using the sine function:
\[ \text{Height}_{\text{initial}} = \text{Ladder length} \times \sin(\text{Initial angle}) \]
First, let's convert the angle to radians because trigonometric functions in most calculators use radians. But here, in this scenario, you can use the angle in degrees directly, assuming you have a scientific calculator or a tool that allows for such a computation.
Using the sine function with the initial angle:
\[ \text{Height}_{\text{initial}} = 2.8 \times \sin(23.4°) \]
\[ \text{Height}_{\text{initial}} \approx 2.8 \times 0.3987 \] (using a calculator for sin(23.4°))
\[ \text{Height}_{\text{initial}} \approx 1.11636 \]
**Step 2: Calculate the new height of the ladder on the tree using the new angle (18°).**
Now let's use the same procedure to find the new height with the new angle.
\[ \text{Height}_{\text{final}} = \text{Ladder length} \times \sin(\text{Final angle}) \]
\[ \text{Height}_{\text{final}} = 2.8 \times \sin(18°) \]
\[ \text{Height}_{\text{final}} \approx 2.8 \times 0.3090 \] (using a calculator for sin(18°))
\[ \text{Height}_{\text{final}} \approx 0.8652 \]
**Step 3: Determine the difference in height.**
Now we simply subtract the initial height from the final height to find out how much higher the boy needs to push the ladder:
\[ \text{Height difference} = \text{Height}_{\text{final}} - \text{Height}_{\text{initial}} \]
\[ \text{Height difference} \approx 0.8652 - 1.11636 \]
\[ \text{Height difference} \approx -0.25116 \]
The negative value indicates that the final position of the ladder is actually lower on the tree than the initial position. This means the boy doesn't need to push the ladder higher up; instead, he needs to lower it to achieve an angle of 18° with the ground.
In summary, the boy needs to push the ladder down so that it is approximately 0.25116 meters lower on the tree to make an angle of 18° with the ground.
