a) 40 more; b) old factory: 10%, new factory: 16%; c) week 4
explanation:
a) at week 0, we see in the graph that the new factory produces 190 specialty items.
using the function, substituting 0 in for w,
=230(1.1)^w\\\\p(0)=230(1.1)^0\\\\p(0)=230(1)=230)
this is a difference of 230-190 = 40.
b) the function for the old factory is written in the form f(x) = a(1+r)ˣ, where a is the original amount, x is the amount of time and r is the growth rate. the equation for the old factory is =230(1.1)^w)
; 230 is in place of a and is the original amount; w is the amount of time; which means 1+r = 1.1. this means that r must be 0.1 or 10%.
for the new factory, we find the percent of increase to find the growth rate; the formula for this is
between the first two points, the amount of change is 220-190 = 30; this gives us
30/190 = 0.158 ≈ 0.16 = 16%
c) at week 0, the old factory has 230 while the new factory has 190.
at week 1, the old factory has 230(1.1) = 253 while the new factory has 220.
at week 2, the old factory has 230(1.1²) = 278.3 while the new factory has 252.
at week 3, the old factory has 230(1.1³) = 306.13 while the new factory has 290.
at week 4, the old factory has 230(1.1⁴) = 336.743 while the new factory has 337. this is the first week that the new factory surpasses the old.
