Question

In a section of a timber mill, cylinder logs of wood, all of uniform dimensions, arrive as the input and are cut into smaller cylinder pieces of the same radius using manual and mechanical saws. To operate a mechanized saw two workers are needed. The team of four workers takes two hours to cut a log into two cylindrical pieces using a manual saw, whereas just two workers are needed to do the same work in one hour using a mechanized saw. The time required to make a cut is proportional to the area across which to cut 60 log into four equal pieces each, if they have two mechanical saws and two manual saws?
(a) 40 hours
(b) 80 hours
(c) 120 hours
(d) 60 hours

Answer 1

Hint: In this question, we first need to calculate the total number of cuts to be made for 60 logs into four pieces. Now, as manual saw takes twice the time as that of mechanical saw we can get the number of cuts can be made. Then from that as given there are 2 saws of each kind now on dividing the earlier cuts we got by 2 we get the cuts made by each saw and then multiplying with respective time gives the total time.

Complete step-by-step answer:
Now, as given in the question that the time required to make a cut is proportional to the area across means that the time taken by manual and mechanical saws to make a cut will be constant.
Here, to cut 1 log into 4 cylindrical pieces we need to make 3 cuts on each one of them.
Now, the number of cuts to be made to cut 60 logs is given by
\[\Rightarrow 60\times 3\]
Now, on further simplification we get,
\[\Rightarrow 180\]
Now, we have two manual saws and two mechanical saws which require 2 persons each for the mechanical one and 4 persons each for the manual one.
Now, given in the question that manual saws take 2 hours to cut and mechanical saws take 1 hour to cut.
So, the manual saws take twice the time taken by the mechanical saws to cut.
Now, when these saws are running in parallel then in same time \[\dfrac{1}{3}\]cuts will be made by manual saws and remaining will be made by mechanical saws
Now, let us calculate the number of cuts made by manual saws
\[\Rightarrow \dfrac{1}{3}\times 180\]
Now, on further simplification we get,
\[\Rightarrow 60\]
Thus, 60 cuts will be made by manual saws
Now, let us calculate the cuts made by mechanical saws
\[\Rightarrow \left( 1-\dfrac{1}{3} \right)\times 180\]
Now, on simplifying it further we get,
\[\Rightarrow \dfrac{2}{3}\times 180\]
Now, this can be written in the simplified form as
\[\Rightarrow 120\]
Thus, 120 cuts will be made by 2 mechanical saws
Now, 60 cuts will be made by 2 manual saws
Let us now calculate the number of cuts that will be made by each manual saw
\[\begin{align}
  & \Rightarrow \dfrac{60}{2} \\
 & \Rightarrow 30 \\
\end{align}\]
Now, the time taken to make 30 cuts by manual saw is given by
\[\begin{align}
  & \Rightarrow 30\times 2 \\
 & \Rightarrow 60 \\
\end{align}\]
Thus, manual saw takes 60 hours to cut the given logs
Now, let us calculate the cuts made by each mechanical saw
\[\Rightarrow \dfrac{120}{2}\]
\[\Rightarrow 60\]
Now, the time taken to make 60 cuts by mechanical saw is given by
\[\Rightarrow 60\times 1\]
\[\Rightarrow 60\]
Thus, the mechanical saw takes 60 hours to cut 60 logs
Thus, the process of cutting the logs can be done efficiently in 60 hours.
Hence, the correct option is (d).

Note:
It is important to note that we need to find the number of cuts made by each of the saws when they run in parallel so that we get the time taken by each of them to make the required number of cuts.
It is also to be noted that we need to divide the amount of cuts that can be done based on the time taken by both of them so that we can get the total time taken to make the required number of cuts.
Instead of considering the number of cuts to be 4 in case of 3 then we get the corresponding total cuts to be made incorrect and so the final result.