To solve this problem, we need to determine the height of the kite using the given length of the string and the horizontal distance between the boy and the point on the ground directly below the kite. We can solve this using the Pythagorean theorem.
The problem involves a right triangle where:
- The hypotenuse (the length of the string) is 20 meters.
- The base (the horizontal distance) is 15 meters.
- The height of the kite is the vertical side we are trying to find.
According to the Pythagorean theorem:
[tex]\[ \text{(Hypotenuse)}^2 = \text{(Base)}^2 + \text{(Height)}^2 \][/tex]
Filling in the known values:
[tex]\[ 20^2 = 15^2 + \text{(Height)}^2 \][/tex]
Calculate the squares:
[tex]\[ 400 = 225 + \text{(Height)}^2 \][/tex]
Now, isolate the height squared:
[tex]\[ \text{(Height)}^2 = 400 - 225 \][/tex]
[tex]\[ \text{(Height)}^2 = 175 \][/tex]
We are given multiple choice options for the height, and we need to match [tex]\(\text{(Height)}^2\)[/tex] with one of the provided options, which are [tex]\(\sqrt{165} m\)[/tex], [tex]\(\sqrt{179} m\)[/tex], [tex]\(\sqrt{178} m\)[/tex], and [tex]\(\sqrt{175} m\)[/tex].
Since we have [tex]\(\text{(Height)}^2 = 175\)[/tex], the height of the kite from the ground is:
[tex]\[ \sqrt{175} \, \text{m} \][/tex]
Thus, the answer is:
[tex]\[ \boxed{\sqrt{175} \, \text{m}} \][/tex]
Therefore, the correct option is D) [tex]\(\sqrt{175} \, \text{m}\)[/tex].
