Question

10. The ratio between the curved surface area and the total surface area of a
right circular cylinder is 1:2. Find the volume of the cylinder if its total
surface area is 616 cm?.​

Answer 1

Given :-

The ratio between the curved surface area and the total surface area of a right circular cylinder is 1:2.

To find :-

Find the volume of the cylinder if its total surface area is 616 cm² ?

Explanation :-

★ Curved surface area of cylinder ★

• Circumference of circle × height
• 2πr × h = 2πrh

★ Total surface area of cylinder ★

• Curved surface area + 2 × area of base
• 2πrh + 2πr² = 2πr(h + r)

Solution :-

• The ratio between the curved surface area and the total surface area = 1 : 2

→ 2πrh/2πr(h + r) = 1/2

• Cancel 2πr

→ h/h + r = 1/2

→ 2h = h + r

→ 2h - h = r

→ h = r ----(i)

• Total surface area of cylinder = 616 cm²

→ 2πr(h + r) = 616

→ 2πr(r + r) = 616 (using i)

→ 2 × 22/7 × r × 2r = 616

→ 44/7 × 2r² = 616

→ 88r²/7 = 616

→ 88r² = 616 × 7

→ r² = 616 × 7/88

→ r² = 7 × 7

→ r = √49 = 7 cm

• From (i)

→ h = r

→ h = 7 cm

• Volume of cylinder

→ πr²h

→ 22/7 × (7)² × 49

→ 22/7 × 49 × 7

→ 22 × 49

→ 1078cm³

• •°• Volume of cylinder is 1078 cm³

Answer 2

Answer :-

$\: \: \mapsto \: \: \underline{\boxed{\sf{\blue{Understanding \: the \: concept}}}}$

Here the concept of Total Surface Area and Curved Surface Area of Cylinder has been used. According to this, the Total Surface Area of Cylinder is taken to be the curved surface area of Cylinder along with the area of two base circles.

And curved surface area is just given as the round portioned part of the Cylinder.

$\: \: \huge{\boxed{\rm{\Longrightarrow \: \: T.S.A \: of \: Cylinder \: = \: (2πr^{2}) \: + \: (2πrh)}}}$

$\: \: \huge{\boxed{\rm{\Longrightarrow \: \: C.S.A \: of \: Cylinder \: = \: 2πrh}}}$

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★ Question :-

The ratio between the curved surface area and the total surface area of a right circular cylinder is 1:2. Find the volume of the cylinder if its total surface area is 616 cm² ?

★ Solution :-

Given,

» TSA of the cylinder = 616 cm²

» Ratio of CSA and TSA = 1 : 2

• Let the radius of the cylinder be 'r' cm and let the height of thw cylinder be 'h' cm.

So, using these values, we get,

$\: \: \Longrightarrow \: \: \bf{\dfrac{CSA}{TSA} \: = \: \dfrac{1}{2}}$

⟹ 2(CSA of cylinder) = TSA of cylinder

⟹ 2 × CSA = 616

⟹ CSA = 616/2

⟹ CSA = 308 cm²

Now let us apply the formulas, and then find the value.

$\: \: \Longrightarrow \: \: \bf{\dfrac{2πrh}{2πr(r \: + \: h)} \: = \: \dfrac{1}{2}}$

$\: \: \Longrightarrow \: \: \bf{\dfrac{h}{(h \: + \: r)} \: = \: \dfrac{1}{2}}$

➮ 2h = h + r

➮ 2h - h = r

➮ h = r

We see that, the radius of cylinder is equal to the height of cylinder.

Now,

➠ 2πrh = 308

➠ 2πr × r = 308

➠2πr² = 308

➠ 2 × (22/7) × r² = 308

➠ r² = (308 × 7) / (2 × 22)

➠ r² = 49

➠ r = √49 = 7

➠ r = h = 7 cm

Now by applying this value into the formula of Volume, we get,

➥ Volume of Cylinder = πr²h

➥ Volume of cylinder = (22/7) × 7 × 7 × 7

➥ Volume of Cylinder = 22 × 49

➥ Volume of Cylinder = 1078 cm³

$\: \: \underline{\boxed{\rm{\red{Hence, \: the \: Volume \: of \: Cylinder \: is \: \underline{1078 \: cm^{3}}}}}}$

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$\: \: \boxed{\rm{More \: to \: know }}$

• Volume of cube = (side)³

• Volume of Cuboid = Length × Breadth × Height

• Volume of Cone = ⅓ × πr²h