A company manufactures microchips. Use the revenue function R(x) = x (75 - 3 x)
ID: 3029341 • Letter: A
Question
A company manufactures microchips. Use the revenue function R(x) = x (75 - 3 x) and the cost function C(x) = 123 + 13 x 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 less than or equals x less than or equals 20.
(A) Form a profit function P, and graph R, C, and P in the same rectangular coordinate system.
P(x) equals what?
Show what graph below shows R(x), C(x), and P(x)? The blue, dashed curve represents R; the pink line represents C; and the green, solid curve represents P.
(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 either equal, greater compared, 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.
D.Yes, because the maximum profit occurs at the same y-value as 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 what in million dollars which occurs when how many million chips are produced. (Round to three decimal places as needed.)
The maximum revenue is what in million dollars.
(Round to three decimal places as needed.)
The x-intercepts of P occur at x equals how many million chips.
(Use a comma to separate answers as needed. Round to three decimal places as needed.)
Explanation / Answer
R(x) = x ( 75 - 3x ) = 75 x - 3x^2
C(x) = 123 + 13x
a) profit = revenue - cost
P(x) = 75x - 3x^2 - ( 123 + 13x ) = 75x - 3x^2 - 123 - 13x = 62x - 3x^2 - 123
b) at breakeven point revenue is equal to cost function
75 x - 3x^2 = 123 + 13x
subtract 13 x from both sides
75x - 13x = 123 + 3x^2
62x = 123 + 3x^2
3x^2 + 62x - 123 = 0
solve quadratic equation :
x =1.82 million of chips
The x intercepts of P(x) and the point of intersection points of R(x) and C(x) are same.
c) P(x) = 62x - 3x^2 - 123
x intercepts: 62x - 3x^2 - 123 =0
x = 1.82 , -22.48
Neglect -ve value : x = 1.823 million chips
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