Sure, let's solve the given problems step by step.
### 3. Distance from the Foot of the Ladder to the Fence
Daniel, who is 1.3 meters tall, is standing next to a fence that is 2.8 meters high. We need to find how far the foot of the ladder is from the fence when he can just see the top of the fence over his head.
We can visualize this as a right triangle where:
- The height of the triangle is the combined height of Daniel and the fence minus the height of the fence itself (since he is already standing at the same level).
- The base of the triangle is the distance we need to find.
- The hypotenuse of the triangle would be the combined height of Daniel and the fence.
By using the Pythagorean theorem:
1. Height of Daniel and the fence together: [tex]\(1.3 \, \text{m} + 2.8 \, \text{m} = 4.1 \, \text{m}\)[/tex]
2. Height of the fence: [tex]\(2.8 \, \text{m}\)[/tex]
3. Using the right triangle equation:
[tex]\[
\text{distance}^2 + 2.8^2 = 4.1^2
\][/tex]
[tex]\[
\text{distance}^2 + 7.84 = 16.81
\][/tex]
[tex]\[
\text{distance}^2 = 16.81 - 7.84
\][/tex]
[tex]\[
\text{distance}^2 = 8.97
\][/tex]
[tex]\[
\text{distance} = \sqrt{8.97}
\][/tex]
[tex]\[
\text{distance} \approx 2.995 \, \text{m}
\][/tex]
Thus, the distance from the foot of the ladder to the fence is approximately 2.995 meters.
### 4. Height of the Tree
Here, we are given that a pole 0.8 meters long stands in the shadow of a tree such that the ends of their shadows coincide.
Given:
- Length of the pole: [tex]\(0.8 \, \text{m}\)[/tex]
- Length of the pole’s shadow: [tex]\(2.5 \, \text{m}\)[/tex]
- Length of the tree’s shadow: [tex]\(10.4 \, \text{m}\)[/tex]
The problem utilizes the property of similar triangles. The ratio of the height of the pole to the length of its shadow is the same as the ratio of the height of the tree to the length of its shadow.
Using this ratio:
[tex]\[
\frac{\text{height of pole}}{\text{length of pole's shadow}} = \frac{\text{height of tree}}{\text{length of tree's shadow}}
\][/tex]
Substituting the given values:
[tex]\[
\frac{0.8}{2.5} = \frac{\text{height of tree}}{10.4}
\][/tex]
Solving for the height of the tree:
[tex]\[
\text{height of tree} = \frac{0.8}{2.5} \times 10.4
\][/tex]
[tex]\[
\text{height of tree} = 0.32 \times 10.4
\][/tex]
[tex]\[
\text{height of tree} \approx 3.328 \, \text{m}
\][/tex]
Thus, the height of the tree is approximately 3.328 meters.
Combining both results, we have:
1. The distance from the foot of the ladder to the fence is approximately 2.995 meters.
2. The height of the tree is approximately 3.328 meters.
