Question

A cylindrical vessel, without a lid, has to be tin-coated on both sides. If the radius of the base is $70cm$ and its height is $1.4m$, calculate the cost of tin-coating at the rate of $Rs.3.50$ per $1000c{m^2}$.

Answer 1

Hint: As mentioned in the question, a cylindrical vessel has to be tin-coated on its both sides i.e., a vessel has to be coated inside and outside. That means the area which has to be coated is doubled now. Here, it mentioned that the vessel is without a lid, so we will find the area of base (which is in the shape of a circle) and the lateral surface area of the cylinder. After calculating the area we can find out the cost of tin-coating.

Formula used:
Lateral surface area of cylinder = $2\pi rh$
Area of circle = $\pi {r^2}$

Complete step-by-step solution:
Let us first draw a figure from the information provided in the question.

Radius of the base of a cylindrical vessel $\left( r \right)$ = $70cm$ and height $\left( h \right)$ = $1.4m$ = $140cm$
Total surface area (excluding upper lid) on both sides = 2(Area of lateral surface of cylindrical vessel) + 2(area of base)
Here we are coating both side(inside+outside of the vessel) that’s why we multiplying by 2 with expression.
$ \Rightarrow Area = 2\pi rh \times 2 + \pi {r^2} \times 2$
On multiplication, we get
$= 4\pi rh + 2\pi {r^2}$
Take $\pi r$ as common
$= \pi r\left( {4h + 2r} \right)$
On substituting the values of radius and height, we get
$= \dfrac{{22}}{7} \times 70\left[ {4 \times 140 + 70 \times 2} \right]c{m^2}$
$= 220\left[ {560 + 140} \right]c{m^2}$
On multiplication, we get
$ =220 \times 700c{m^2}$
$= 154000c{m^2}$
Rate of tin coating = \[Rs{\text{ }}3.50{\text{ }}per{\text{ }}1000{\text{ }}cm{\;^2}\]
Total cost = \[Rs{\text{ }}\dfrac{{154000 \times 3.50}}{{1000}}\]
Total cost = \[Rs{\text{ }}154 \times 3.50\]
Total cost = \[Rs{\text{ 539}}\]
Thus, the cost of tin-coating a cylindrical vessel without lid at the rate of $Rs.3.50$ per $1000c{m^2}$ is \[Rs{\text{ 539}}\].

Note: Here, an important point to note is we didn’t apply the total surface area of the cylinder because the cylindrical vessel is without a lid. Students should be careful about the unit. Calculation should be done in the same unit either in $m$ or $cm$. Take care of the calculations to be sure of the final answer.