A flask in the shape of a right circular cone of height 19 cm ... | Mathematics MCQ | Examoogle

MathematicsMensuration

A flask in the shape of a right circular cone of height 19 cm is filled with water. The water is poured in the right circular cylindrical flask whose radius is one-third of the radius of the base of the circular cone. Then, what is the height of the water in the cylindrical flask?

83 cm

57 cm

53 cm

73 cm

Correct Answer

📐 MA • Math Direct FormulaMathematicsMensuration

Option B

57 cm

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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