Question

The total surface area of a right circular cylinder is $165\pi c{{m}^{2}}$ . if the radius of its base is $5cm$, find its height and volume.

Answer 1

Hint: from the question we have been asked to find the height and volume of the right circular cylinder. As we know that the formula for total surface area of a right circular cylinder is $2\pi r\left( r+h \right)$. from this we will get the height. Then we know that the formula for volume of the right circular cylinder is $\pi {{r}^{2}}h$. From this we will get the volume.

Complete step by step solution:
From the question given the total surface area of a right circular cylinder is
$\Rightarrow 165\pi c{{m}^{2}}$
And also given that the radius of its base is,
$\Rightarrow r=5cm$
Now we have to find both height and volume, to find volume we should have to know height so first we will find height.
We can find height from the total surface area, as we know that the formula for total surface area of a right circular cylinder is
$\Rightarrow 2\pi r\left( r+h \right)=165\pi c{{m}^{2}}$
By solving this we will get height,
$\Rightarrow 2\pi r\left( r+h \right)=165\pi c{{m}^{2}}$
$\Rightarrow 2\pi \times 5\left( 5+h \right)=165\pi c{{m}^{2}}$
$\Rightarrow 2\times 5\left( 5+h \right)=165cm$
By solving further, we will get,
$\Rightarrow \left( 5+h \right)=\dfrac{165cm}{10}$
$\Rightarrow 5+h=16.5$
$\Rightarrow h=16.5-5$
$\Rightarrow h=11.5cm$
Therefore, the height of the cylinder is $\Rightarrow h=11.5cm$
The figure will be,

Now, we will find the volume of the right circular cylinder,
As we know that the formula for volume of the right circular cylinder is
$\Rightarrow v=\pi {{r}^{2}}h$
By substituting the respective values in the formula, we will get the volume,
By substituting we will get,
$\Rightarrow v=\pi {{r}^{2}}h$
$\Rightarrow v=\pi \times {{\left( 5 \right)}^{2}}\times 11.5$
$\Rightarrow v=3.14\times {{\left( 5 \right)}^{2}}\times 11.5$
$\Rightarrow v=902.75c{{m}^{3}}$

Therefore, the volume of the right circular cylinder is $v=902.75c{{m}^{3}}$.

Note: Students should know the formulas of the right circular cylinder like, the formula for total surface area of a right circular cylinder is $2\pi r\left( r+h \right)$. The formula for volume of the right circular cylinder is $\pi {{r}^{2}}h$. Along with these two formulas students should also know the formula for lateral or curved surface area of the cylinder is $2\pi rh$.