Hercules Films is deciding on the price of the video release of its film Bride of the Son of Frankenstein. Marketing estimates that at a price of p dollars, it can sell q = 300,000 − 15,000p copies, but each copy costs $4 to make. What price will give the greatest profit?
Can anyone plzzz help me? :( It would be greatly appreciated! Especially since I've been stuck on this since this morning :/
nnooooo
To find the price that will give the greatest profit, we need to determine the profit function and then find its maximum value.
The profit for each unit sold can be calculated by subtracting the cost from the selling price. In this case, the selling price is p dollars and the cost to make each copy is $4. Therefore, the profit per unit is p - $4.
To find the profit function, we need to multiply the profit per unit by the number of units sold. The number of units sold, q, is given by the equation q = 300,000 - 15,000p. So the profit function (P) can be expressed as:
P = (p - $4)(q)
Substituting the value of q, we have:
P = (p - $4)(300,000 - 15,000p)
Expanding this expression, we get:
P = 300,000p - 15,000p^2 - $1,200,000 + 60,000p
Simplifying further, we have:
P = -15,000p^2 + 360,000p - $1,200,000
Now, we can find the maximum profit by finding the vertex of the parabola defined by the profit function.
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. In this case, a = -15,000, b = 360,000, and c = -1,200,000.
x = -360,000 / (2(-15,000)) = 12
So the price that will give the greatest profit is $12 per copy.
It's important to note that this calculation assumes that the demand equation provided (q = 300,000 - 15,000p) accurately reflects the actual demand for the film at different prices. This may not always be the case in real-world situations, so it's always advisable to conduct further market research and analysis to validate the assumptions and determine an optimal pricing strategy.