Question

the solid is melted and recast to form a solid cylinder with a radius of 25.5 cm. if the volume of the solid is 72000cm^3 , find the height of the cylinder and the surface area of the cylinder (take the height of the cylinder as 35cm​

Answer 1

To Find :-

• The Height of the Cylinder.

• The Surface Area of the Cylinder.

Given :-

• Radius = 25.5 cm.

• Volume = 72000 cm³.

We Know :-

Volume of a Cylinder :-

$\underline{\boxed{\bf{V = \pi r^{2}h}}}$

Where :-

• r = Radius of the Cylinder.

• h = Height of the Cylinder.

Curved surface area of the Cylinder :-

$\underline{\boxed{\bf{CSA = 2\pi rh}}}$

Where :-

• r = Radius of the Cylinder.

• h = Height of the Cylinder.

Total Surface area of a Cylinder :-

$\underline{\boxed{\bf{TSA = 2\pi r(h + r)}}}$

Where :-

• r = Radius of the Cylinder.

• h = Height of the Cylinder.

Concept :-

According to the question , the cylinder was formed by melting the solid.

Hence , the Volume of the Cylinder will be equal to the Volume of the Solid.,i.e,

$\bf{V_{(Cylinder)} = 72000 cm^{3}}$.

From this information , we can find the height of the Cylinder.

Solution :-

To Find the Height of the Cylinder :-

Given :-

• Radius = 25.5 cm

• Volume = 72000 cm³

Using the formula and substituting the values in it ,we get :-

$:\implies \bf{V = \pi r^{2}h} \\ \\ \\ :\implies \bf{72000 = \dfrac{22}{7} \times 25.5^{2}h} \\ \\ \\ :\implies \bf{72000 \times 7 = 22 \times 25.5^{2}h} \\ \\ \\ :\implies \bf{72000 \times 7 = 22 \times 25.5 \times 25.5 \times h} \\ \\ \\ :\implies \bf{\dfrac{72000 \times 7}{22} = 25.5 \times 25.5 \times h} \\ \\ \\ :\implies \bf{\dfrac{72000 \times 7}{22} = 650.25 h} \\ \\ \\ :\implies \bf{\dfrac{72000 \times 7}{22 \times 650.25} = h} \\ \\ \\ :\implies \bf{\dfrac{504000}{14305.5} = h} \\ \\ \\ :\implies \bf{35.2 (approx.) cm = h} \\ \\ \\ \therefore \purple{\bf{h = 35 cm}}$

Hence, the height of the Cylinder is 35 cm.

To Find the Surface Area of the Cylinder :-

Curved surface Area :-

Given :-

• Height = 35 cm

• Radius = 25.5 cm

Using the formula and substituting the values in it , we get :-

$:\implies \bf{CSA = 2\pi rh} \\ \\ \\ :\implies \bf{CSA = 2 \times \dfrac{22}{7} \times 25.5 \times 35} \\ \\ \\ :\implies \bf{CSA = 2 \times 22 \times 25.5 \times 5} \\ \\ \\ :\implies \bf{CSA = 44 \times 25.5 \times 5} \\ \\ \\ :\implies \bf{CSA = 44 \times 127.5} \\ \\ \\ :\implies \bf{CSA = 5610 cm^{2}} \\ \\ \\ \therefore \purple{\bf{CSA = 5610 cm^{2}}}$.

Hence, the Curved Surface Area of the Cylinder is 5610 cm².

Total Surface Area :-

Given :-

• Height = 35 cm

• Radius = 25.5 cm

Using the formula and substituting the values in it , we get :-

$:\implies \bf{TSA = 2\pi r(h + r)} \\ \\ \\ :\implies \bf{TSA = 2 \times \dfrac{22}{7} \times 25.5 \times (35 + 25.5)} \\ \\ \\ :\implies \bf{TSA = 2 \times \dfrac{22}{7} \times 25.5 \times 60.5} \\ \\ \\ :\implies \bf{TSA = 2 \times \dfrac{22}{7} \times 1542.75} \\ \\ \\ :\implies \bf{TSA = \dfrac{44}{7} \times 1542.75} \\ \\ \\ :\implies \bf{TSA = \dfrac{67881}{7}} \\ \\ \\ :\implies \bf{TSA = 9697.3} \\ \\ \\ \therefore \purple{\bf{TSA = 9697.3}}$

Hence, the total Surface Area of the Cylinder is 9697.3 cm².

Answer 2

➛GIVEN:-

① Radius r = 25.5 cm

② Volume V = 72000cm³

➛FIND:-

➊ Surface Area of Cylinder = ?

➋ Height of Cylinder = ?

➛SOLUTION:-

Here, in Question it is written that by melting a solid the Cylinder is formed.

So,

↦VOLUME of Solid = VOLUME of Cylinder

now, we know that

$\sf⇾Volume \: of \: Cylinder = \pi {r}^{2} h$

where,

‣ r = 25.5 cm

‣ V = 72000 cm³

substitute these values in formula

we have,

$\sf⇾Volume \: of \: Cylinder = \frac{22}{7} \times {(25.5)}^{2} \times h \\ \tt↦72000 = \frac{22}{7} \times 650.25 \times h \\ \tt ↦ \frac{72000 \times 7}{22 \times 650.25} = h \\ \tt ↦ \frac{ \cancel{504000}}{ \cancel{14305.5}} = h \\ \tt ↦h = 35.23(approx.) \\ \tt ⇀ So, h = 35cm$

now, we know that

$\tt ➛ Total \: Surface \: Area = 2 \pi r(r + h)$

where,

where,‣ r = 25.5 cm

where,‣ r = 25.5 cm‣ h = 35 cm

substitute these values in formula

substitute these values in formulawe have,

$\tt ➛ Total \: Surface \: Area = 2 \times \frac{22}{7} \times 25.5(25.5 + 35) \\ \tt \implies2 \times \frac{22}{7} \times 25.5(60.5) \\ \tt \implies2 \times \frac{22}{7} \times 1542.75 \\ \tt \implies \frac{ \cancel{67881}}{ \cancel7} = 9697.28cm(approx.) \\ \tt → So, T.S.A = 9697.3 {cm}^{</strong><strong>2</strong><strong>}$

$\tt ➨ Hence, \boxed{ \tt h = 35cm} and \boxed{ \tt T.S.A =9697.3 {cm}^{</strong><strong>2</strong><strong>} }$