Stuck on a homework project in a highschool college algebra question. I'm given the following information:
Tact time is the average time to pick and place one part. Throughput is the number of boards per hour, or placements per hour. Variables include:
$n$: The number of boards per hour
$p$: The number of parts per board
$C$: The tact time
$L$: The load-unload time
$M$: The mark reading time
Throughput is calculated by: $$TH(n) = \frac{n}{Cnp+L+M}$$
Setting: $C=0.2 \mbox{ seconds}$, $L=5 \mbox{ seconds}$, $M=1 \mbox{ second}$, and $p=50 \mbox{ parts per board}$, I get the equation $$TH(n) = \frac{n}{(10n+6)}$$ I'm supposed to find out the minimum number of boards per panel ($n$) that will allow me to have a $TH(n)$ of at least $300$. So I've set $$300=\frac{n}{(10n+6)}$$ and tried to solve for $n$. I get $-0.6002$. This just doesn't make sense to me, that is, having a negative answer for a throughput time seems odd to me. Looking at the equation, and graphing, it seems there is a limit of throughtput of $1/10$…which again looks suspect. Pretty much nothing about this scenario makes sense to me as throughput, from that equation, seems to always be less than $1/10$.
Note: This is from Sullivan and Sullivan, College Algebra 2nd Edition, page 68 Project Motorola, "How Many Cellular Phones Can I Make?" Similar to what is found here: http://wps.prenhall.com/wps/media/objects/1257/1287289/ChapterR/Pm_howma.pdf
Best Answer
You're right that $TH(n) = \frac{n}{10n + 6}$ will get as high as $300$ for positive $n$. However, if you look at the dimensions, the function you have for throughput is in boards per second. So, you can certainly never make $300$ boards per second. (In fact, as you noticed, you can never do more that $0.1$ board per second. The factory can only move so fast, after all.) $300$ boards per hour is possible, however. To find the $n$ that makes this possible, convert $300$ boards per hour to boards per second, and then put that number in for $TH(n)$.