Acompany manufactures microchips. use the revenue function r(x)equals=x(75−5x) and the cost function c(x)=121+13x to answer parts (a)−(e), where x is in millions of chips and r(x) and c(x) are in millions of dollars. both functions have domain 1≤x≤20.
(a) form a profit function p, and graph r, c, and p in the same rectangular coordinate system.
p(x)=
graph showing r(x), c(x), and p(x)
(b) discuss the relationship between the intersection points of the graphs of r and c and the x intercepts of p.
the x-values of the intersection points of r and c are (equal, greater compared, or smaller compared) to the x-intercepts of p.
(c) find the x intercepts of p to the nearest thousand chips. find the break-even points to the nearest thousand chips.
the x-intercepts of p occur at x= ? million chips. (use a comma to separate answers as needed. round to three decimal places as needed.)
since the break-even point is the point at which the profit equals zero, do the x-intercepts of the profit function indicate the break-even points? (yes or no)
(d) refer to the graph drawn in part (a).
does the maximum profit appear to occur at the same output level as the maximum revenue?
a. yes, because the maximum profit occurs at the same y-value as the maximum revenue.
b. yes, because the x-value of the maximum profit is the same as the x-value of the maximum revenue.
c. no, because the x-value of the maximum profit is different than the x-value of the maximum revenue.
d. no, because the maximum profit occurs at a different y-value than the maximum revenue.
are the maximum profit and the maximum revenue equal?
a. yes, because the x-value of the maximum profit is the same as the x-value of the maximum revenue.
b. yes, because the maximum profit occurs at the same y-value as the maximum revenue.
c. no, because the x-value of the maximum profit is different than the x-value of the maximum revenue.
d. no, because the maximum profit occurs at a different y-value than the maximum revenue.
(e) verify your conclusion in part (d) by finding the output (to the nearest thousand chips) that produces the maximum profit. find the maximum profit (to the nearest thousand dollars). compare this to the maximum revenue.
the maximum profit is ? million dollars which occurs when ? million chips are produced. (round to three decimal places as needed.)
the maximum revenue is ? million dollars. (round to three decimal places as needed.)