The research department in a company that manufactures AM/FM clock radios established the following cost, and revenue functions:
R(x) = x(50-1.25x)
C(x) = 160 + 10x
where 0 < x < 20.
a) Determine when R = C (to the nearest thousand units. e.g. 6.247)
Quantity x = Blank 1 thousand units.
b) Determine the maximum profit (to the nearest thousand dollars).
Profit P = Blank 2 thosand dollars.
My answers don't seem to be correct (x = 16 and P = 160) Please help!
By the way, "blank" is where the answer is supposed to be.
To determine when revenue (R) equals cost (C), we need to set the two equations equal to each other and solve for x.
The function R(x) represents the revenue, which is given by the equation R(x) = x(50 - 1.25x).
The function C(x) represents the cost, which is given by the equation C(x) = 160 + 10x.
a) Set R(x) equal to C(x):
x(50 - 1.25x) = 160 + 10x
To solve this equation, we can start by simplifying it:
50x - 1.25x^2 = 160 + 10x
Next, rearrange the equation to bring all the terms to one side:
1.25x^2 - 40x + 160 = 0
Now, we have a quadratic equation. We can either factor it or use the quadratic formula to solve for x. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1.25, b = -40, and c = 160. Substituting these values, we get:
x = (-(-40) ± √((-40)^2 - 4(1.25)(160))) / (2(1.25))
x = (40 ± √(1600 - 800)) / 2.5
x = (40 ± √(800)) / 2.5
x = (40 ± 28.28) / 2.5
Now, solve for x:
x = (40 + 28.28) / 2.5 ≈ 16.91
x = (40 - 28.28) / 2.5 ≈ 4.29
Since we are given that 0 < x < 20, the nearest thousand units for "x" when R = C is 5 thousand units.
Therefore, "x = 5" is the correct answer for Blank 1.
b) To find the maximum profit (P), we need to subtract the cost from the revenue:
P(x) = R(x) - C(x)
P(x) = x(50 - 1.25x) - (160 + 10x)
Simplify the expression:
P(x) = 50x - 1.25x^2 - 160 - 10x
Combine like terms:
P(x) = -1.25x^2 + 40x - 160
To find the maximum profit, we need to find the vertex of this quadratic equation. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
For our equation, a = -1.25 and b = 40. Substituting these values, we get:
x = -40 / (2(-1.25))
x = -40 / (-2.5)
x = 16
Now, substitute x = 16 into the profit equation:
P(16) = -1.25(16)^2 + 40(16) - 160
P(16) ≈ 160
The nearest thousand dollars for the maximum profit (P) is $160.
Therefore, "P = 160" is the correct answer for Blank 2.