Answer

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**Hint:**A closed cylinder has three faces on its surface, the curved surface of the cylindrical portion, and the two circular planes closing the two open sides of the curved surface. We can calculate the total surface area by adding the area of these individual faces.

**Complete Step by Step Solution:**

Let us calculate the areas of the individual faces of the closed cylinder of radius 7 m and height 3 m.

The curved surface area ${{A}_{1}}$ of the cylinder will be equal to height times the circumference of the curvature. If r is the radius and h is the height of the cylinder, the curved surface area will be given by the below equation.

$\Rightarrow {{A}_{1}}=\text{circumference}\times \text{height}$

We know that the circumference of a circle is equal to $2\pi r$.

$\Rightarrow {{A}_{1}}=2\pi r\times h=2\pi rh$

Now let us substitute the radius and height in the above equation.

$\Rightarrow {{A}_{1}}=2\pi rh=2\pi \times \text{7 m}\times \text{3 m}$

$\Rightarrow {{A}_{1}}=2\pi \times 21=131.9\text{5 }{{\text{m}}^{2}}$

The area of the two circular planes covering the open ends of the curved surface will be equal to

$\Rightarrow {{A}_{2}}=2\times \pi {{r}^{2}}$

Substituting the value of radius in the above equation, we get

$\Rightarrow {{A}_{2}}=2\times \pi \times {{\left( \text{7 m} \right)}^{2}}=307.8\text{8 }{{\text{m}}^{\text{2}}}$

The total surface area, A of the closed cylinder will be

$\Rightarrow A={{A}_{1}}+{{A}_{2}}$

$\Rightarrow A=131.9\text{5 }{{\text{m}}^{2}}+307.8\text{8 }{{\text{m}}^{\text{2}}}=439.8\text{3 }{{\text{m}}^{\text{2}}}$

**Hence the area of the metal sheet required to make a closed cylinder of radius 7 m and height 3 m is**

$\Rightarrow A=439.8\text{3 }{{\text{m}}^{\text{2}}}\simeq 44\text{0 }{{\text{m}}^{\text{2}}}$.

$\Rightarrow A=439.8\text{3 }{{\text{m}}^{\text{2}}}\simeq 44\text{0 }{{\text{m}}^{\text{2}}}$.

**Note:**

In this problem, they have mentioned that the given cylinder is a closed cylinder. Therefore we have to consider the circular areas covering the open ends. But if the surface area of an open cylinder is asked, we should calculate only the area of the curved surface of the cylinder which is $A=2\pi rh$, given r is the radius and h is the height of the cylinder.

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