The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, manufactured by Phonola Record Industries, is related to the price/compact disc. The equation
p=-0.00051x+8 (0¡Üx¡Ü12,000)
where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by
C(x)=600+2x-0.00004x^2 (0¡Üx20,000)
To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is R(x) = px, and the profit is P(x) = R(x) - C(x). (Round your answer to the nearest whole number.)
? discs/month
so, use the hint:
R(x) = x*p(x) = -0.00051x^2+8x
P(x) = R(x) - C(x)
= 2.04*10^-8 x^4 - .00134x^3 + 15.694x^2 + 4800x
P'(x) = 0 at x = 9929 or so.
To maximize profits, we need to find the number of copies that Phonola should produce each month. This can be determined by finding the value of x that maximizes the profit function P(x), where P(x) = R(x) - C(x).
First, we need to calculate the revenue function R(x), which represents the total income generated from selling x copies. Since the unit price p is given by p = -0.00051x + 8, the revenue function can be expressed as R(x) = px. Substituting the value of p, we get:
R(x) = (-0.00051x + 8) * x = -0.00051x^2 + 8x
Next, we calculate the cost function C(x), which represents the total cost of producing x copies. The cost function is given as C(x) = 600 + 2x - 0.00004x^2.
Now, we can determine the profit function P(x) by subtracting the cost function C(x) from the revenue function R(x):
P(x) = R(x) - C(x)
P(x) = (-0.00051x^2 + 8x) - (600 + 2x - 0.00004x^2)
P(x) = 0.00003x^2 + 6x - 600
To find the number of copies that maximizes the profit, we need to find the value of x that maximizes the profit function P(x). This can be done by analyzing the graph of the parabolic function P(x) = 0.00003x^2 + 6x - 600. The x-coordinate of the vertex of this parabola will give us the value of x that maximizes the profit.
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. In our case, a = 0.00003 and b = 6.
x = -6/(2 * 0.00003)
x ≈ 100000
Since the number of copies demanded must be between 0 and 12,000 (as per the given demand equation), the optimal number of copies Phonola should produce each month is approximately 12,000 discs.
Therefore, Phonola should produce 12,000 copies of the Walter Serkin recording of Beethoven's Moonlight Sonata each month to maximize its profits.