Question

Please help me with this problem so I can help my son better understand.

Answer 1

The equation of the old factory is

`p(w)=230(1.1)^w`

The form of the exponential equation is

`f(x)=a(1+r)^x`

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

`p(w)=190,w=0`

Then a = 190

`p(w)=190(1+r)^w`

`\begin{gathered} w=1,p(w)=220^{} \\ 220=190(1+r)^1 \\ 220=190(1+r) \end{gathered}`

Divide both sides by 190

`\begin{gathered} (220)/(190)=(190(1+r))/(190) \\ (22)/(19)=1+r \end{gathered}`

Subtract 1 from both sides to find r

`\begin{gathered} (22)/(19)-1=1-1+r \\ (3)/(19)=r \\ r=(3)/(19) \end{gathered}`

The initial amount of the new factory is 190

The rate of growth is 3/19 (0.158)

`p(w)=190((22)/(19))^w`

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:

`p_o(1)=230(1.1)^1=253`

From the graph, the new factory has 220 at w = 1

Since w = 2:

`p_o(2)=230(1.1)^2=278.3`

From the graph, the new factory has 252 at w = 2

Since w = 3

`p_o(3)=230(1.1)^3=306.13`

From the graph, the new factory has 290

Since at w = 4

`p_o(4)=230(1.1)^4=336.743`

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