The length of a string between kite and a point on the ground is 90 m. If the string makes an angle with the level ground and `sin alpha=(3)/(5)`. Find the height of the kite. There is no slack in the string.

Answer

To find the height of the kite, we can use the information provided about the string length and the angle it makes with the ground. Here are the steps to solve the problem:

### Step 1: Understand the relationship
We know that the sine of an angle in a right triangle is defined as the ratio of the length of the opposite side (height of the kite) to the length of the hypotenuse (length of the string).

### Step 2: Write down the given values
- Length of the string (hypotenuse) = 90 m
- \( \sin \alpha = \frac{3}{5} \)

### Step 3: Set up the equation using the sine definition
From the definition of sine:
\[ \sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{x}{90} \]
where \( x \) is the height of the kite.

### Step 4: Substitute the value of \( \sin \alpha \)
We can substitute the value of \( \sin \alpha \) into the equation:
\[ \frac{3}{5} = \frac{x}{90} \]

### Step 5: Cross-multiply to solve for \( x \)
Cross-multiplying gives us:
\[ 3 \times 90 = 5 \times x \]
\[ 270 = 5x \]

### Step 6: Solve for \( x \)
Now, divide both sides by 5:
\[ x = \frac{270}{5} = 54 \]

### Conclusion
The height of the kite is \( 54 \) meters.

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