A kite is flying at the height of 75m from the ground. The string makes an angle `theta` (where `cot theta` `= 8/15` ) with the level ground. Assuming that there is no slack in the string the new length of the string is equal to :

Options

A. 85 metre
B. 65 metre
C. 75 metre
D. 40 metre

Answer

To find the length of the string of the kite flying at a height of 75m with the angle θ such that cot(θ) = 8/15, we can follow these steps:

### Step 1: Understand the relationship between cotangent and the sides of the triangle.
The cotangent of an angle in a right triangle is defined as the ratio of the adjacent side (base) to the opposite side (height). Thus, we have:

\[ \cot(\theta) = \frac{\text{Base}}{\text{Height}} = \frac{8}{15} \]

### Step 2: Identify the height of the kite.
The height of the kite from the ground is given as 75m. Therefore, we can denote:

\[ \text{Height} = 75m \]

### Step 3: Set up the equation using cotangent.
From the cotangent definition, we can express the base (let's denote it as \( x \)) in terms of the height:

\[ \cot(\theta) = \frac{x}{75} = \frac{8}{15} \]

### Step 4: Cross-multiply to solve for the base.
Cross-multiplying gives us:

\[ 15x = 8 \times 75 \]

Calculating \( 8 \times 75 \):

\[ 8 \times 75 = 600 \]

So, we have:

\[ 15x = 600 \]

### Step 5: Solve for \( x \).
To find \( x \), divide both sides by 15:

\[ x = \frac{600}{15} = 40m \]

### Step 6: Use the Pythagorean theorem to find the length of the string.
In a right triangle, the length of the hypotenuse (the string) can be found using the Pythagorean theorem:

\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2 \]

Substituting the values we have:

\[ \text{Hypotenuse}^2 = 40^2 + 75^2 \]

Calculating \( 40^2 \) and \( 75^2 \):

\[ 40^2 = 1600 \]
\[ 75^2 = 5625 \]

Adding these together:

\[ \text{Hypotenuse}^2 = 1600 + 5625 = 7225 \]

### Step 7: Take the square root to find the length of the string.
Now, we take the square root of both sides:

\[ \text{Hypotenuse} = \sqrt{7225} = 85m \]

### Conclusion:
The length of the string is 85 meters.

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