A machine wants to make 110 parts, and it initially takes 200 seconds to make each part. Each time 5 parts are made, the time it takes to make a part is reduced by 2 seconds. However, there is a curve to how fast the machine can make parts. As the machine reaches the production curve, less time is saved for each 10 parts. Starting at 70 parts, each 10 parts will only save 1 second. At 80 to 89 parts made, the machine only saves 0.5 seconds of time per part. From 90 to 110 parts made, the machine only saves 0.1 seconds.

How long will it take to make the 110 parts?

Question

A machine wants to make 110 parts, and it initially takes 200 seconds to make each part. Each time 5 parts are made, the time it takes to make a part is reduced by 2 seconds.  However, there is a curve to how fast the machine can make parts. As the machine reaches the production curve, less time is saved for each 10 parts.  Starting at 70 parts, each 10 parts will only save 1 second.  At 80 to 89 parts made, the machine only saves 0.5 seconds of time per part.  From 90 to 110 parts made, the machine only saves 0.1 seconds.

How long will it take to make the 110 parts?

anonymous · Answer

@mathbrz

whpalmer4 · Answer

the problem is a bit unclear.  If parts 66-70 take x seconds, do parts 71-75 take x-2 seconds, x-1 seconds, or x seconds?

anonymous · Answer

I posted a similar problem to this before, and used summation to solve it

whpalmer4 · Answer

Well, it's a straightforward problem, except for understanding exactly where the production rate changes...

anonymous · Answer

$$5\sum_{0}^{69}(200-2x) + 1\sum_{1}^{9}(62-x)+0.5\sum_{1}^{9}(53-x)+0.1\sum_{1}^{20}(44-x)$$

whpalmer4 · Answer

hmm, the answer for that seems to be > 110*200, which should be an upper bound on the answer...

whpalmer4 · Answer

No, I interpreted it differently.

Parts 1-5 take 200 seconds each
Parts 6-10 take 198 seconds each
Parts 11-15 take 196 seconds each
Parts 16-20 take 194 seconds each
Parts 21-25 take 192 seconds
Parts 26-30 take 190 seconds
Parts 31-35 take 188 seconds
Parts 36-40 take 186 seconds
Parts 41-45 take 184 seconds
Parts 46-50 take 182 seconds
Parts 51-55 take 180 seconds
Parts 56-60 take 178 seconds
Parts 61-65 take 176 seconds
Parts 66-70 take 174 seconds
Parts 71-75 take 173 seconds
Parts 76-80 take 173 seconds
parts 81-90 take 172.5 seconds
Parts 91-110 take 172.4 seconds

anonymous · Answer

I mixed up my previous problem with this

whpalmer4 · Answer

Starts out at 200 seconds per part, and after 5 are made the process is improved to take 198 seconds per part.  And so on, but exactly when the improvements slow down is hard to pin down from the description.

anonymous · Answer

Taking you're break down, I can just do it on paper.  I was trying to express is using summation, but I'm not that good at it

anonymous · Answer

*your

whpalmer4 · Answer

As I wasn't under any obligation to do the problem in the "correct" way, I just added it all up without bothering to construct the summation equation:

5*200 + 5*198 + 5*196 + 5*194 + 5*192 + 5*190 + 5*188 + 5*186 + 
 5*184 + 5*182 + 5*180 + 5*178 + 5*176 + 5*174 + 5*173 + 5*173 + 
 10*172.5 + 20*172.4

But, I'm not all that convinced that I interpreted the problem correctly when the enhancements taper off...

My result ended up averaging just a bit over 180 seconds per part, which is around parts 46-55 and thus right in the middle as you would expect with a mostly linear speedup.

anonymous · Answer

I may have mistyped those details.  I think you're right