To solve this problem, let's understand the relationship between the height of the pole, the length of its shadow, and how these proportions can help us determine the height of the tree.
We know:
- The height of the pole is [tex]\(3\)[/tex] meters.
- The length of the shadow of the pole is [tex]\(5\)[/tex] meters.
- The length of the shadow of the tree is [tex]\(10\)[/tex] meters.
Both the pole and the tree cast shadows due to the same light source, which means we can use the concept of similar triangles to find the height of the tree. In similar triangles, the ratios of corresponding sides are equal.
Given:
- Let the height of the tree be [tex]\(x\)[/tex] meters.
- Therefore, we have the following proportion based on similar triangles:
[tex]\[ \frac{\text{height of the pole}}{\text{length of the pole's shadow}} = \frac{\text{height of the tree}}{\text{length of the tree's shadow}} \][/tex]
Substituting the given values:
[tex]\[ \frac{3}{5} = \frac{x}{10} \][/tex]
Now, we solve this proportion for [tex]\(x\)[/tex]:
1. Cross-multiplying the terms to solve for [tex]\(x\)[/tex]:
[tex]\[ 3 \times 10 = 5 \times x \][/tex]
[tex]\[ 30 = 5x \][/tex]
2. Dividing both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{30}{5} \][/tex]
[tex]\[ x = 6 \][/tex]
Thus, the height of the tree is [tex]\(6\)[/tex] meters.
Regarding the proportions given in the problem:
- The proportion [tex]\[ \frac{3}{10} = \frac{x}{5} \][/tex] is not correct because it does not match the similar triangle relationship described.
- The equation [tex]\[ 3x = \frac{110}{5} \][/tex] is incorrect and does not correspond to the problem's principles.
- The proportion [tex]\[ \frac{x}{3} = \frac{110}{5} \][/tex] is not valid either.
Therefore, the correct proportion to solve for the height of the tree based on the principles of similar triangles is:
[tex]\[ \frac{3}{5} = \frac{x}{10} \][/tex]
The height of the tree is [tex]\(6\)[/tex] meters, which is obtained from solving the correct proportion.
