Using $\pi = \frac{22}{7}$, find the total surface area of a c... | Mathematics MCQ

MathematicsPrevious Year Paper 2024

Using $\pi = \frac{22}{7}$ , find the total surface area of a closed cylinder having a base radius of $70\text{ m}$ and height of $110\text{ m}$ .

79,200 m²

82,400 m²

82,500 m²

78,500 m²

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option A

79,200 m²

Quick Summary:

Factor out $2\pi r$ as $2 \times \frac{22}{7} \times 70 = 440$ , then multiply by $(r + h) = 70 + 110 = 180$ . Calculation: $440 \times 180 = 79,200\text{ m}^2$ .

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📐MAMath SolutionDirect Formula

📋 Given

Base radius $r = 70\text{ m}$ , Height $h = 110\text{ m}$ , $\pi = \frac{22}{7}$

🔢 Formula Used

TSA=2πr(r+h)TSA = 2\pi r(r + h)TSA=2πr(r+h)

⚡ Exam Hall Shortcut / Speed Trick

Factor out $2\pi r$ as $2 \times \frac{22}{7} \times 70 = 440$ , then multiply by $(r + h) = 70 + 110 = 180$ . Calculation: $440 \times 180 = 79,200\text{ m}^2$ .

⚠️ Common Student Trap / Pitfall

Confusing the total surface area formula with the curved surface area formula ( $2\pi rh$ ) by forgetting to add the areas of the two circular bases ( $2\pi r^2$ ).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Identify Given Parameters

The given cylinder is a closed cylinder, meaning it includes both circular top and bottom bases along with the curved surface area.

r=70 m,  h=110 m,  π=227r = 70\text{ m},\; h = 110\text{ m},\; \pi = \frac{22}{7}r=70 m,h=110 m,π=722​

State the Total Surface Area Formula

The total surface area ( $TSA$ ) of a closed cylinder is the sum of the curved surface area and the areas of the two circular bases.

TSA=2πrh+2πr2=2πr(r+h)TSA = 2\pi rh + 2\pi r^2 = 2\pi r(r + h)TSA=2πrh+2πr2=2πr(r+h)

Substitute Values into the Formula

Substitute $r = 70$ , $h = 110$ , and $\pi = \frac{22}{7}$ into the expression.

TSA=2×227×70×(70+110)TSA = 2 \times \frac{22}{7} \times 70 \times (70 + 110)TSA=2×722​×70×(70+110)

Simplify and Calculate

Perform the arithmetic multiplication and addition to find the final surface area.

TSA=2×227×70×180=440×180=79,200 m2TSA = 2 \times \frac{22}{7} \times 70 \times 180 = 440 \times 180 = 79,200\text{ m}^2TSA=2×722​×70×180=440×180=79,200 m2

✅

A is correct because the calculated total surface area of the closed cylinder is $79,200\text{ m}^2$ .

Core Concepts Used

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Total Surface Area of Cylinder Curved Surface Area Area of Circular Bases

💡 EXAM TIP

Always verify whether a cylinder is 'closed' (includes two bases) or 'open' (one base or hollow) before applying surface area formulas in mensuration problems.

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