Answer:
(a) To determine the function R(x) that models the total revenue, we need to consider the relationship between the price of a can and the corresponding sales per month.
Given that the sales average 25,000 cans per month when the cans sell for 40¢ each, we can start by calculating the revenue when the price is 40¢:
Revenue at 40¢ per can = 25,000 cans * $0.40 = $10,000
For each nickel increase in the price, the sales per month drop by 900 cans. This means that for each increase of x in the price (in $0.05 increments), the sales will decrease by 900x cans.
Using these observations, we can construct the function R(x) as follows:
R(x) = (25,000 - 900x) * (0.40 + 0.05x)
The first term, (25,000 - 900x), represents the number of cans sold per month, and the second term, (0.40 + 0.05x), represents the price per can.
(b) To graph R(x) and identify the maximum, we can plot the function on a graph. Here is an example graph:
```
|
R | Maximum
e | *
v | .
e | .
n | .
u | .
e | .
| .
($) | .
|_____________________________
0 1 2 3 4
```
The graph represents the revenue (R) on the vertical axis and the number of $0.05 increases in the price (x) on the horizontal axis. The graph shows a maximum point, indicated by the starred point.
(c) To find the price per can for maximum revenue and the corresponding maximum revenue value, we need to determine the value of x that corresponds to the maximum point on the graph.
In this case, the maximum point occurs when x is at its highest value on the graph. From the graph, it appears that x = 4 corresponds to the maximum point.
Substituting x = 4 into the function R(x), we can find the price per can and the maximum revenue:
R(4) = (25,000 - 900(4)) * (0.40 + 0.05(4))
= (25,000 - 3600) * (0.40 + 0.20)
= 21,400 * 0.60
= $12,840
Therefore, the market should charge $0.40 + $0.05(4) = $0.60 per can to realize the maximum revenue, which is $12,840.