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Q4:
The angle of elevation of the top of a tower from a point on the ground, which is $30$ m away from the foot of the tower, is $30°$. Find the height of the tower.

Solution :

Given:

  • The distance from the foot of the tower to the point on the ground ($BC$) = $30$ m.
  • The angle of elevation of the top of the tower from the point on the ground ($\angle ACB$) = $30^\circ$.

To Find:

  • The height of the tower ($AB$).
30 m h C B A 30°

Step 1: Defining the Variables and Assumptions

Let $AB$ be the tower of height $h$ meters. Let $C$ be the point on the ground at a distance of $30$ m from the foot of the tower $B$. Thus, $BC = 30$ m. The angle of elevation $\angle ACB = 30^\circ$. We assume the tower stands vertically on the ground, making $\triangle ABC$ a right-angled triangle at $B$.

Step 2: Selecting the Trigonometric Ratio

In the right-angled triangle $\triangle ABC$, we have the side adjacent to the angle $\angle C$ (which is $BC$) and we need to find the side opposite to the angle $\angle C$ (which is $AB$). The trigonometric ratio that relates the opposite side and the adjacent side is the tangent function:

$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC}$

Step 3: Substituting the Values

Substitute $\theta = 30^\circ$, $BC = 30$ m, and $AB = h$ into the formula:

$\tan(30^\circ) = \frac{h}{30}$

Step 4: Solving for $h$

We know from trigonometric standard values that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$.

$\frac{1}{\sqrt{3}} = \frac{h}{30}$

Multiply both sides by $30$ to isolate $h$:

$h = \frac{30}{\sqrt{3}}$

Step 5: Rationalizing the Denominator

To simplify the expression, multiply the numerator and the denominator by $\sqrt{3}$:

$h = \frac{30}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$

$h = \frac{30\sqrt{3}}{3}$

$h = 10\sqrt{3}$

Step 6: Final Calculation

Using the approximate value of $\sqrt{3} \approx 1.732$:

$h = 10 \times 1.732 = 17.32$ m

Final Answer: The height of the tower is $10\sqrt{3}$ m or approximately $17.32$ m.


More Questions from Class 10 Mathematics Applications of Trigonometry EXERCISE 9.1


CBSE Solutions for Class 10 Mathematics Applications of Trigonometry


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