19 செ.மீ உயரமுள்ள ஒரு நேர்வட்டக் கூம்பு (Right circular cone) ... | Mathematics MCQ (தமிழ்) | Examoogle

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19 செ.மீ உயரமுள்ள ஒரு நேர்வட்டக் கூம்பு (Right circular cone) வடிவக் குடுவை முழுமையாக நீரால் நிரப்பப்பட்டுள்ளது. இந்த நீர், கூம்பின் அடிப்பக்க ஆரத்தில் மூன்றில் ஒரு பங்கு ஆரத்தைக் கொண்ட ஒரு நேர்வட்ட உருளை (Right circular cylinder) வடிவக் குடுவையில் ஊற்றப்படுகிறது. எனில், உருளைக் குடுவையில் உள்ள நீரின் உயரம் என்ன?

83 செ.மீ

57 செ.மீ

53 செ.மீ

73 செ.மீ

Correct Answer

📐 MA • Math Direct FormulaMathematicsMensuration

57 செ.மீ

Quick Summary:

Since $V_{cone} = V_{cyl}$ , equate $\frac{1}{3} \pi r_c^2 (19) = \pi (\frac{r_c}{3})^2 h_y$ . The $\pi$ and $r_c^2$ terms cancel, leaving $h_y = 19 \times 3 = 57$ cm.

📐MAMath SolutionDirect Formula

📋 Given

Cone height $h_{c}$ = 19 cm. Cylinder radius $r_{y}$ = (1/3) * Cone radius $r_{c}$ . The cone is filled with water.

🔢 Formula Used

Vcone=13πrc2hcV_{cone} = \frac{1}{3} \pi r_c^2 h_cVcone​=31​πrc2​hc​ and Vcyl=πry2hyV_{cyl} = \pi r_y^2 h_yVcyl​=πry2​hy​

⚡ Exam Hall Shortcut / Speed Trick

Since $V_{cone} = V_{cyl}$ , equate $\frac{1}{3} \pi r_c^2 (19) = \pi (\frac{r_c}{3})^2 h_y$ . The $\pi$ and $r_c^2$ terms cancel, leaving $h_y = 19 \times 3 = 57$ cm.

⚠️ Common Student Trap / Pitfall

Students often forget to square the ratio $1/3$ when substituting the radius, leading to $h_y = 19 \times 3 = 57$ rather than $19 \times 3^2 = 171$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Equating Volumes

Since the water volume is constant, the volume of the cone equals the volume of the cylinder: $V_{cone} = V_{cyl}$

13πrc2hc=πry2hy\frac{1}{3} \pi r_c^2 h_c = \pi r_y^2 h_y31​πrc2​hc​=πry2​hy​

Substitute Given Values

Substitute $h_c = 19$ and $r_y = \frac{r_c}{3}$ into the equation: $\frac{1}{3} \pi r_c^2 (19) = \pi (\frac{r_c}{3})^2 h_y$

193rc2=rc29hy\frac{19}{3} r_c^2 = \frac{r_c^2}{9} h_y319​rc2​=9rc2​​hy​

Solve for Height

Cancel $r_c^2$ from both sides and solve for $h_y$ : $h_y = \frac{19}{3} \times 9 = 19 \times 3 = 57$

hy=57 cmh_y = 57 \text{ cm}hy​=57 cm

✅

B is correct because the conservation of volume formula yields a height of 57 cm for the cylinder.

Core Concepts Used

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Volume of Cone Volume of Cylinder Conservation of Volume

💡 EXAM TIP

This concept is a precursor to fluid dynamics and surface area problems often appearing in mechanical engineering aptitude sections.

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