Answer:
40; 0.1 and 0.16
Step-by-step explanation:
a. We know that p(w)=230(1.1)^W represents the number of specialty items produced at the old factory.
This equation is in the form of an exponential growth function:
f(x)=a(1+r)^x
where a is the initial value, r is the growth rate, and x is the number of time intervals.
That means that for p(w)=230(1.1)^W:
230 is the initial value, and 0.1 is the growth rate. (1.1-1)
Therefore, for week 0, there were 230 specialty items produced at the old factory.
We can also plug in 0 for w (0 time intervals) to get the same answer
That would look like:
p(0)=230(1.1)^0
p(0)=230(1)
p(0)=230
On the other hand, for the new factory, if you go find 0 on the x-axis, you'll notice that 190 specialty items were produced in week 0.
230-190=40
This means that our answer for a is that 40 more specialty items were produced at the old factory than the new factory.
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b. Our first step for this question would be to find the equation for the graph.
We can immediately recognize that this is an exponential growth function, similar to the other equation.
As a reminder, exponential growth functions are in the form: f(x)=a(1+r)^x
We can put it in terms of production and weeks:
p(w)=a(1+r)^w
We know that the initial value is 190 by the graph.
p(w)=190(1+r)^w
Now we just need to find the growth rate:
We can do this by choosing a value (other than the first point) and plugging it in. Let's use (1, 220) which is in the form (w, p)
Plug it in to the equation:
220=190(1+r)^1
220=190(1+r)
Distribute
220=190+190r
Subtract 190 from both sides
30=190r
Divide 190 from both sides
r=0.1578947368 (round how your teacher asks)
Next, all we have to do is find the growth rate for the old factory:
p(w)=230(1.1)^W
1.1=1+r
Therefore, r=0.1
The growth rate at the old factory is 0.1.
When comparing them, we'll realize that the growth rate at the new factory is about 0.06 higher.
