Time Delay Production Problem

Question

three milling machines are to produce a total of 5700 parts. The first one starts and makes 25 parts/hr. Sixteen hours later a second machine starts and makes parts with a rate of 30 parts/hr. Finally, 32 hours later the third machine starts at a rate of 10 parts/hr. If all 3 machines operate until 5700 parts are made, how long will the three machine be in operation?

Answer 1

To solve this, we need to account for the lead time the first machine has over the second and third, as well as the lead the second machine has over the third.

 

The first machine makes parts at a rate of

 

25t

 

Where t is in hours.

 

However, it has a 16 hour head start over the second machine, and an additional 10 hour head start on the third. So the rate it makes parts can be expressed as

 

25(t+16+10)

 

The second machine makes parts at a rate of

 

30t

 

With a 10 hour head start on the third

 

30(t+10)

 

The third machine has no head start, and makes parts at a rate of  

 

10t

 

We add the rate of all the machines together, and set it equal to our goal

 

25(t+16+10) + 30(t+10) + 10t = 5700

 

Solve for t

 

65t+950 = 5700

t = 73.077

 

So it will take about 73 hours so the three machines to make 5700 parts.