NCERT Class X Chapter 9: Some Application Of Trigonometry Exercise 9.1 Question 5

NCERT Class X Chapter 9: Some Applications of Trigonometry

Question:

A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.

Given:

Height of the kite above ground = 60 m
Inclination of string with ground = 60°

To Find:

Length of the string

Formula:

\[ \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]

Solution:

Step 1: Let the length of the string be \( l \) meters.

Step 2: In the right triangle formed, the height of the kite is the perpendicular, and the string is the hypotenuse. Using the sine ratio:

$$ \sin 60^\circ = \frac{60}{l} $$

Step 3: We know that \( \sin 60^\circ = \frac{\sqrt{3}}{2} \).

$$ \frac{\sqrt{3}}{2} = \frac{60}{l} $$

Step 4: Cross-multiplying to solve for \( l \):

$$ l = \frac{60 \times 2}{\sqrt{3}} = \frac{120}{\sqrt{3}} $$

Step 5: Rationalizing the denominator and calculating the approximate value:

$$ l = \frac{120}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{120\sqrt{3}}{3} = 40\sqrt{3} $$ $$ l \approx 40 \times 1.732 = 69.28 \text{ m} $$

Result:

The length of the string is approximately \( 69.28 \) meters.

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