Length of the tight thread of a kite from a point on the ground is 85 m. If thread subtends an angle `theta` with the ground such that `tantheta=8/15` then find the height at which kite is flying.

Answer

To solve the problem, we need to find the height at which the kite is flying, given the length of the thread and the tangent of the angle it makes with the ground.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a kite flying at a height, and the length of the thread from a point on the ground to the kite is given as 85 m. The angle θ that the thread makes with the ground is such that \( \tan \theta = \frac{8}{15} \).

2. **Setting Up the Right Triangle**:
We can visualize this situation as a right triangle where:
- The height of the kite (KA) is the opposite side to the angle θ.
- The distance from the point on the ground to the point directly below the kite (AP) is the adjacent side.
- The length of the thread (KP) is the hypotenuse.

3. **Using the Tangent Function**:
The tangent of angle θ is defined as the ratio of the opposite side to the adjacent side:
\[ \tan \theta = \frac{\text{Height (KA)}}{\text{Base (AP)}} \]
Given \( \tan \theta = \frac{8}{15} \), we can write:
\[ \frac{KA}{AP} = \frac{8}{15} \]
This implies that:
\[ KA = \frac{8}{15} \cdot AP \]

4. **Using the Pythagorean Theorem**:
According to the Pythagorean theorem:
\[ KP^2 = KA^2 + AP^2 \]
We know that \( KP = 85 \) m, so:
\[ 85^2 = KA^2 + AP^2 \]

5. **Expressing AP in terms of KA**:
From the earlier step, we can express AP in terms of KA:
\[ AP = \frac{15}{8} \cdot KA \]
Substituting this into the Pythagorean theorem:
\[ 85^2 = KA^2 + \left(\frac{15}{8} \cdot KA\right)^2 \]

6. **Calculating the Values**:
Now, substituting the values:
\[ 85^2 = KA^2 + \frac{225}{64} \cdot KA^2 \]
\[ 7225 = KA^2 \left(1 + \frac{225}{64}\right) \]
\[ 7225 = KA^2 \left(\frac{64 + 225}{64}\right) \]
\[ 7225 = KA^2 \left(\frac{289}{64}\right) \]
\[ KA^2 = 7225 \cdot \frac{64}{289} \]
\[ KA^2 = 1600 \]
\[ KA = \sqrt{1600} = 40 \text{ m} \]

7. **Final Answer**:
Therefore, the height at which the kite is flying is **40 meters**.