(a)
The model of the profit y (in dollars) as a function of the number of cooktops produced per month is:
y = 10x + 0.5x² - 0.001x³ - 4000, for 0 ≤ x ≤ 450
We can see the intersection of this graph and the axis:
For the y axis: x = 0
y(x = 0) = 10*0 + 0.5*0² - 0.001*0³ - 4000 = -4000
For the x axis: y = 0
0 = 10x + 0.5x² - 0.001x³ - 4000
Solving for x: x = {-91.1503, 87.0538, 504.096}
From these values, only x = 87.0538 lies in 0 ≤ x ≤ 450.
Finally, we calculate the final value: x = 450
y = 10*450 + 0.5*450² - 0.001*450³ - 4000 = 10 625
From these results, the graph is:
(b)
To begin generating a profit, we must find an x value such that y(x) > 0.
From item (a), we found that such value is x = 87.0538 or x = 88 (to the bigger whole number). That is, values beyond 88 for the number of cooktops will generate a profit, as we can see from the graph, where for x ≥ 88 the function is positive.
Answer: 88
(c)
We need to solve the inequality:
y > 15 000
10x + 0.5x² - 0.001x³ - 4000 > 15 000
If we solve the equality first:
10x + 0.5x² - 0.001x³ - 4000 = 15 000
We get: x = {-175.3, 262.64, 412.66}
Only 262.64 and 412.66 lie in 0 ≤ x ≤ 450. So, rounding to the nearest whole number, the interval is:
x ∈ [263, 412]
*x is a whole number*
