(a) During Week 0, how many more specialty items were produced at the old factory than at the new factory? Explain.
Old: p(0) = 240 (1.1)^0 = 240
New: (0,190)
That's 50 more items from the old factory in week 0.
Answer: 50
(b) Find and compare the growth rates in the weekly number of specialty items produced at each factory. Show your work.
The growth rate of the old factory is 1.1, 10% per week.
Let's say the new factor is exponential as well. Matching week 0 we get
q(w) = 190 r^w
ln q = ln 190 + w ln r
w ln r = ln q - ln 190
ln r = (ln q - ln 190)/w
r = e^{(ln q - ln 190)/w}
Let's calculate this for a few different points,
r(1) = e^{(ln 220 - ln 190)/1} = 1.1578947368421058
r(2) = e^{(ln 252 - ln 190)/2} = 1.1516578439248717
r(4) = e^{(ln 337 - ln 190)/4} = 1.1540352511556147
r(7) = e^{(ln 505 - ln 190)/7} = 1.14986870386165
That's pretty consistent. We'll go with r=1.15, 15% increase per per week.
Answer: Old factory 10% growth per week, new factor 15% growth per week
(c) When does the weekly number of specialty items produced at the new factory exceed the weekly number of specialty items produced at the old factory? Explain.
We have p(w) = 240(1.1)^2 and q(w) = 190(1.15)^w
240(1.1)^w = 190(1.15)^w
(240/190) (1.1)^w =(1.15)^w
ln(240/190) + w ln (1.1) =w ln(1.15)
ln(240/190) = w ( ln(1.15) - ln (1.1) )
w = ln(240/190) / ln(1.15/1.1) = 5.255468797423731
Answer: By week 6 the new factory exceeds the output of the old