The equation of the old factory is
[tex]p(w)=230(1.1)^w[/tex]The form of the exponential equation is
[tex]f(x)=a(1+r)^x[/tex]Where:
a is the initial amount (value f(x) at x = 0)
r is the growth rate in decimal
Compare between the 2 equations, then
a = 230 (the value of p(w) at w = 0)
(1 + r) = 1.1
Then the value of r = 1.1 - 1 = 0.1
The rate of growth is 0.1
For the new factory, we will use the graph to make its equation
The points (0, 190) and (1, 220) lie on the graph
Then substitute them in the form of the equation to find a and r
[tex]p(w)=190,w=0[/tex]Then a = 190
[tex]p(w)=190(1+r)^w[/tex][tex]\begin{gathered} w=1,p(w)=220^{} \\ 220=190(1+r)^1 \\ 220=190(1+r) \end{gathered}[/tex]Divide both sides by 190
[tex]\begin{gathered} \frac{220}{190}=\frac{190(1+r)}{190} \\ \frac{22}{19}=1+r \end{gathered}[/tex]Subtract 1 from both sides to find r
[tex]\begin{gathered} \frac{22}{19}-1=1-1+r \\ \frac{3}{19}=r \\ r=\frac{3}{19} \end{gathered}[/tex]The initial amount of the new factory is 190
The rate of growth is 3/19 (0.158)
[tex]p(w)=190(\frac{22}{19})^w[/tex]Now, let us answer the questions
a) During week zero, means w = 0
At w = 0 the old factory has 230 and the new factory has 190, then
230 - 190 = 40, then
The old factory has 40 more specialty items than the new factory ------ answer of a
b) The growth factor of the old factory is 0.1
The growth factor of the new factory is 22/19 (about 0.158)
Then the growth factor of the new factory is greater than the growth factor of the old factory -------- answer of b
c) we need to find the week that the new factory has more items than the old factory
Since at w = 0:
The old factory has 230 and the new factory has 190
Since at w = 1:
[tex]p_o(1)=230(1.1)^1=253[/tex]From the graph, the new factory has 220 at w = 1
Since w = 2:
[tex]p_o(2)=230(1.1)^2=278.3[/tex]From the graph, the new factory has 252 at w = 2
Since w = 3
[tex]p_o(3)=230(1.1)^3=306.13[/tex]From the graph, the new factory has 290
Since at w = 4
[tex]p_o(4)=230(1.1)^4=336.743[/tex]From the graph, the new factory has 337 which is greater than the old factory, then
In week 4 the new factory will have more items than the old factory
Week 4 ------ the answer of c