A tower stands vertically on the ground. From a point on the g... | Mathematics MCQ | Examoogle

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A tower stands vertically on the ground. From a point on the ground, 30 m away from the foot of the tower, the angle of elevation of the top of the tower is 30°. What is the height (in metres) of the tower?

$30\sqrt{3}$

$15\sqrt{3}$

$20\sqrt{3}$

$10\sqrt{3}$

Correct Answer

📐 MA • Math Trigonometry RatioMathematicsTrigonometry

Option D

$10\sqrt{3}$

Quick Summary:

In a 30°-60°-90° triangle, the sides are in the ratio $1: \sqrt{3}: 2$ . Since distance is the side opposite to 60°, we have $h \sqrt{3} = 30$ , so $h = \frac{30}{\sqrt{3}} = 10\sqrt{3}$ .

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📐MAMath SolutionTrigonometry Ratio

📋 Given

Distance from the foot of the tower d = 30 m, Angle of elevation θ = 30°

🔢 Formula Used

tan⁡(θ)=heightdistance\tan(\theta) = \frac{\text{height}}{\text{distance}}tan(θ)=distanceheight​

⚡ Exam Hall Shortcut / Speed Trick

In a 30°-60°-90° triangle, the sides are in the ratio $1: \sqrt{3}: 2$ . Since distance is the side opposite to 60°, we have $h \sqrt{3} = 30$ , so $h = \frac{30}{\sqrt{3}} = 10\sqrt{3}$ .

⚠️ Common Student Trap / Pitfall

Students often confuse $\tan(30°) = \frac{1}{\sqrt{3}}$ with $\cot(30°) = \sqrt{3}$ , leading to an incorrect height of $30\sqrt{3}$ instead of $10\sqrt{3}$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Set up the trigonometric ratio

The height of the tower $h$ and the distance from the foot $d$ form a right-angled triangle with the angle of elevation. Using the tangent function:

tan⁡(30°)=h30\tan(30°) = \frac{h}{30}tan(30°)=30h​

Substitute the value of tan(30°)

Knowing that $\tan(30°) = \frac{1}{\sqrt{3}}$ , we substitute this into the equation:

13=h30\frac{1}{\sqrt{3}} = \frac{h}{30}3​1​=30h​

Solve for h

Multiplying both sides by 30 gives $h = \frac{30}{\sqrt{3}}$ . Rationalizing the denominator:

h=30×33×3=3033=103h = \frac{30 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{30\sqrt{3}}{3} = 10\sqrt{3}h=3​×3​30×3​​=3303​​=103​

✅

D is correct because the calculated height of the tower is $10\sqrt{3}$ meters.

Core Concepts Used

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Trigonometry Right-angled Triangle Rationalization

💡 EXAM TIP

This is a fundamental application of heights and distances; always remember the ratios of sides for special triangles ( $30°, 45°, 60°$ ) to save calculation time.

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