Question

The volume of a solid cylinder is 96228cm³ and the ratio of its radius to its height is 9:14 . Find the total surface area of cylinder.
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Answer 1

Solution

Given :-

• Volume of solid cylinder = 96,228 cm³ .
• Ratio of radius & height = 9:14

Find :-

• Total surface area of cylinder

Explanation

using Formula

★ Volume of solid cylinder = πr²h

★ Total surface area of cylinder = 2πrh + 2πr².

So, Now

Let

• Radius of cylinder = 9x
• Height of cylinder = 14x

keep all values

==> Volume of solid cylinder = πr²h

==> 96,228 = 22/7 × (9x)² × 14x

==> 96228 × 7/22 = 81×14 × x³

==> (96228 × 7)/(22×81×14) = x³

==> x³ = 96228/(22×2×81)

==> x³ = 96228/3564

==> x³ = 27

==> x³ = 3³

==> x = 3

Since,

• Radius of cylinder be = 9x = 9×3 = 27 cm
• Height of cylinder be = 14x = 14×3 = 42 cm

________________________

Now, Calculate total surface area

Keep all required values

==> Total surface area of cylinder = 2πrh + 2πr².

==> Total surface area of cylinder =( 2 × 22/7 × 27 × 42) +( 2×22/7 × 27²)

==> Total surface area of cylinder = (2×22×27×6)+(44×27×27)/7

==> Total surface area of cylinder = 7128 + (32076)/7

==> Total surface area of cylinder = 7128+ 4542.29

==> Total surface area of cylinder = 11670.29 cm²

___________________

Answer 2

$\red{\large\dag{\bold{\underline{\underline{Given:-}}}}}$

• Volume of cylinder = 96228 cm³
• Ratio of radius to height = 9:14

$\red{\large\dag{\bold{\underline{\underline{To\:find:-}}}}}$

• Total surface area

$\red{\large\dag{\bold{\underline{\underline{Formulas \:used:-}}}}}$

• $\bold \pink{\boxed{ \text{Total surface area = Curved surface area + 2( Area of cross-section)}}}$

where,

• Curved surface area = 2πrh
• Area of cross-section = πr²

again where,

• π = 22/7
• r = radius
• h = height

$\red{\large\dag{\bold{\underline{\underline{Step \:by\: step \:explaination:-}}}}}$

★ As it is given the ratio of its radius to its height . So, let us assume the height be 14y cm and radius be 9y cm.

• radius = 9y cm
• height = 14y cm

★ Now evaluating values in the given formula.

$\implies \dfrac{22}{7} \times (9y) {}^{2} \times 14y = {96228cm {}^{3}}$

★ Bringing the L.H.S. into R.H.S. in order to calculate.

Therefore,

$\implies \: \text{y} {}^{3} = \dfrac{96228 \times 7}{22 \times 9 \times 9 \times 14}$

★ Now, after reducing them to their lowest terms.

$\implies \: y {}^{3} = 27$

$\implies \: y = \sqrt{27}$

y = 3

Thus,

Radius = 9y → 9×3 → 27 cm

Height = 14y → 14×3 → 42cm

★ Here, we have to insert all the values as got now in the given formula of total surface area.

$\implies \: 2 \times \dfrac{22}{7} \times 27(42 + 27)$

$\implies \: 2 \times \dfrac{22}{7} \times 27 \times 69$

$\implies\:11710.3$

$\red{\large\dag{\bold{\underline{\underline{Answer:-}}}}}$

Total surface area is 11710.3cm²

$\red{\large\dag{\bold{\underline{\underline{Additional\: information:-}}}}}$

• A solid which has uniform circular cross-section is called cylinder.
• Volume of cylinder = Area of cross-section × height
• Perimeter = 2πr
• Area of cross-section = πr²