60% of the canvas used to make a closed cylinder with a radius... | Mathematics MCQ | Examoogle

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60% of the canvas used to make a closed cylinder with a radius of 5 cm and a height of 15 cm is reused to make a conical hat with a radius of 8 cm. What is the slant height (in cm) of the conical hat?

Correct Answer

📐 MA • Math Direct FormulaMathematicsMensuration

Option A

Quick Summary:

Calculate $60\%$ of the cylinder's total area $2\pi r(r+h)$ and equate it to the lateral surface area of the cone $\pi R l$ . Cancel $\pi$ immediately from both sides to simplify calculation.

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📐MAMath SolutionDirect Formula

📋 Given

Cylinder radius r = 5 cm, height h = 15 cm. Conical hat radius R = 8 cm. Reuse percentage = 60%.

🔢 Formula Used

TSAcyl=2πr(r+h),CSAcone=πRlTSA_{\text{cyl}} = 2\pi r(r + h), \quad \text{CSA}_{\text{cone}} = \pi R lTSAcyl​=2πr(r+h),CSAcone​=πRl

⚡ Exam Hall Shortcut / Speed Trick

Calculate $60\%$ of the cylinder's total area $2\pi r(r+h)$ and equate it to the lateral surface area of the cone $\pi R l$ . Cancel $\pi$ immediately from both sides to simplify calculation.

⚠️ Common Student Trap / Pitfall

Many students mistake the closed cylinder surface area for just the lateral area ( $2\pi rh$ ) or forget to include the top and bottom circles ( $2\pi r^2$ ).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate Total Surface Area of Cylinder

The total surface area of a closed cylinder is given by the formula $2\pi r(r+h)$ . Substituting $r=5$ and $h=15$ gives $2 \times \pi \times 5 \times (5+15) = 10\pi \times 20$ .

2π×5×20=200π2\pi \times 5 \times 20 = 200\pi2π×5×20=200π

Determine Available Material

Only $60\%$ of the cylinder's surface area is used to make the conical hat.

Available Area=0.60×200π=120π\text{Available Area} = 0.60 \times 200\pi = 120\piAvailable Area=0.60×200π=120π

Calculate Slant Height of Cone

The conical hat material corresponds to the lateral surface area of a cone: $\pi R l$ . Setting $\pi \times 8 \times l = 120\pi$ , we solve for $l$ .

l=120π8π=15l = \frac{120\pi}{8\pi} = 15l=8π120π​=15

✅

A is correct because calculation of the available surface area $120\pi$ divided by the cone radius $8\pi$ yields a slant height of 15 cm.

Core Concepts Used

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Mensuration Total Surface Area of Cylinder Lateral Surface Area of Cone

💡 EXAM TIP

This concept of 'material transformation' is common in 3D geometry problems; always identify if the surface area or volume is being conserved before proceeding.

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