Question

A factory manufactures $120000$ pencils daily. The pencils are cylindrical in shape each of length $25$cm and circumference of base as $1.5$cm. Determine the cost of coloring the curved surfaces of the pencils manufactured in one day at Rs $0.05$ per $d{{m}^{2}}$.

Answer 1

Hint: In this problem we need to calculate the cost for coloring the curved surface of the pencils manufactured in a day by given dimensions. In the problem they have mentioned the circumference of the base which is in circular shape and represented by $2\pi r$. We also have the length of the cylindrical shaped pencil which is represented by $h$. From these two values we will calculate the curved surface area of the pencil which is given by $2\pi rh$. Now we will convert the surface area which is $c{{m}^{2}}$ to $d{{m}^{2}}$ by using the relation $1dm=10cm$. After having the surface in $d{{m}^{2}}$ we will calculate the total surface area to be colored in a day by multiplying the calculated surface area of one pencil with the number of pencils manufactured in a day. Now we need to multiply the cost to color one $d{{m}^{2}}$ with the total surface area to be colored in a day in $d{{m}^{2}}$.

Complete step-by-step answer:
Given that, circumference of the cylindrical shaped base is $1.5$cm.
We know that the circumference of the circle is given by $2\pi r$. So, we can write that $2\pi r=1.5$.
In the problem we have the length of the cylindrical pencil as $25$ cm which is represented by $h$.
Now the curved surface area of the pencil which is cylindrical is given by
$A=2\pi rh$
Substitute all the values we have in the above equation, then we will get
$\begin{align}
  & A=1.5\times 25 \\
 & \Rightarrow A=37.5c{{m}^{2}} \\
\end{align}$
We have the curved surface area in $c{{m}^{2}}$. Using the relation $1dm=10cm$ to convert the area into $d{{m}^{2}}$, then we will get
$A=37.5\times {{\left( 10 \right)}^{-2}}d{{m}^{2}}$
Simplifying the above value by substituting ${{10}^{-2}}=0.01$, then we will have
$\begin{align}
  & A=37.5\times 0.01d{{m}^{2}} \\
 & \Rightarrow A=0.375d{{m}^{2}} \\
\end{align}$
Now the surface area of the pencils to be colored is calculated by multiplying the above area with the total number of pencils manufactured in a day which is $120000$, then we will get
$\begin{align}
  & A=120000\times 0.375 \\
 & \Rightarrow A=45000d{{m}^{2}} \\
\end{align}$
The cost required to color one $d{{m}^{2}}$ is Rs $0.05$. Now the cost required to color $45000d{{m}^{2}}$ is calculated by multiplying this area with $0.05$, then we will get
$\begin{align}
  & C=0.05\times 45000 \\
 & \Rightarrow C=2250 \\
\end{align}$
Hence the cost required to color the pencils manufactured in a day is given by Rs. $2250$.

Note: In this problem they have directly mentioned the circumference of the base which is in the circular shape, so we have directly used this value in further calculations. Sometimes they only mention the radius or diameter of the base, then we can simply substitute those values in the curved surface area of the cylinder which is $A=2\pi rh$.