एका मनोऱ्याच्या वरून, जमिनीवर असलेल्या दोन वस्तूंचे अवनमन कोन ... | Mathematics MCQ (मराठी) | Examoogle

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एका मनोऱ्याच्या वरून, जमिनीवर असलेल्या दोन वस्तूंचे अवनमन कोन 60° आणि 30° असल्याचे निरीक्षण केले जाते. त्या वस्तू मनोऱ्याच्या एकाच बाजूला आहेत आणि मनोऱ्याची उंची 450 मी आहे. वस्तूंमधील अंतर (मी मध्ये) किती आहे?

$450\sqrt{3}$

$300\sqrt{3}$

$150\sqrt{3}$

$100\sqrt{3}$

Correct Answer

📐 MA • Math Direct FormulaMathematicsTrigonometry

$450\sqrt{3}$

Quick Summary:

For a tower of height h and angles 60° and 30°, the distance between objects is h × ( $\sqrt{3}$ - 1/ $\sqrt{3}$ ), which simplifies to (2h/ $\sqrt{3}$ ) or (2h $\sqrt{3}$ )/3.

📐MAMath SolutionDirect Formula

📋 Given

Tower height h = 450 m, Angle of depression 1 = 60°, Angle of depression 2 = 30°.

🔢 Formula Used

d=h(cot⁡θ2−cot⁡θ1)d = h \left( \cot \theta_2 - \cot \theta_1 \right)d=h(cotθ2​−cotθ1​)

⚡ Exam Hall Shortcut / Speed Trick

For a tower of height h and angles 60° and 30°, the distance between objects is h × ( $\sqrt{3}$ - 1/ $\sqrt{3}$ ), which simplifies to (2h/ $\sqrt{3}$ ) or (2h $\sqrt{3}$ )/3.

⚠️ Common Student Trap / Pitfall

Many students confuse angle of elevation and depression, but since they are complementary, they result in the same triangle geometry; do not mistake the 30° angle for the one closer to the tower.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Define Geometric Components

Let height of tower AB = 450 m. Let the two objects be at C (60°) and D (30°). In $\triangle ABC$ , $\tan 60° = AB/BC$ . In $\triangle ABD$ , $\tan 30° = AB/BD$ .

BC=4503=1503,BD=4503BC = \frac{450}{\sqrt{3}} = 150\sqrt{3}, \quad BD = 450\sqrt{3}BC=3​450​=1503​,BD=4503​

Calculate Distance between objects

The distance between the two objects is $CD = BD - BC$ .

CD=4503−1503=3003CD = 450\sqrt{3} - 150\sqrt{3} = 300\sqrt{3}CD=4503​−1503​=3003​

✅

B is correct because the calculated distance between the two objects is 300 $\sqrt{3}$ m.

Core Concepts Used

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Trigonometry Angles of Depression Right-angled Triangles

💡 EXAM TIP

This concept of distance between two points based on tower height is a staple in civil engineering surveying and RPF/ALP technical exam sections.

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