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 = 240,000 − 15,000p
copies, but each copy costs $4 to make. What price will give the greatest profit?
Well, let's not be Frank-enstein about this decision! To find the price that will give the greatest profit, we need to consider the cost of making each copy and the revenue generated from selling the copies.
The revenue generated from selling the copies can be calculated by multiplying the price (p) by the number of copies sold (q). So the revenue equation would be R = p*q.
Since each copy costs $4 to make, the cost of making copies (C) can be calculated by multiplying the cost per copy ($4) by the number of copies sold (q). So the cost equation would be C = 4*q.
Now, to find the profit (P), we subtract the cost from the revenue: P = R - C.
Let's substitute the given equation for q into the revenue equation:
R = p*(240,000 - 15,000p)
And substitute the cost equation into the profit equation:
P = p*(240,000 - 15,000p) - 4*(240,000 - 15,000p)
To find the price that will give the greatest profit, we need to find the value of p that maximizes the profit equation. This can be done by taking the derivative of the profit equation with respect to p, setting it equal to zero, and solving for p.
But let's not get too mathematical here! How about we just go with a price of "Franken-fifty" dollars? It's not too high to scare away customers, and not too low to make us break a sweat on the cost side. Plus, it has a nice ring to it! So, let's sell the film for $50 and hope for a monstrous profit!
To find the price that will give the greatest profit, we need to determine the profit function.
The profit is calculated by subtracting the cost from the total revenue. The cost consists of the production costs ($4 per copy) multiplied by the number of copies sold, and the revenue is the price per copy multiplied by the number of copies sold.
Let's denote the profit as P, the price as p, and the number of copies sold as q. Then we have:
P = (p - 4)q
Since we know that q = 240,000 - 15,000p, we can substitute this value into the profit function:
P = (p - 4)(240,000 - 15,000p)
Now, let's expand and simplify the profit function:
P = 240,000p - 15,000p^2 - 960,000 + 60,000p
P = -15,000p^2 + 300,000p - 960,000
To find the price that will give the greatest profit, we need to find the value of p that maximizes the profit function. One way to do this is by finding the vertex of the quadratic function.
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b/2a. In this case, a = -15,000 and b = 300,000.
p = -(300,000) / (2 * -15,000)
p = 10
Therefore, the price that will give the greatest profit is $10.
To find the price that will give the greatest profit, we need to determine the profit function and then find the value of p that maximizes that function.
The profit on each copy sold can be calculated by subtracting the cost to make each copy from the selling price: Profit = Selling price - Cost to make each copy.
The cost to make each copy is given as $4.
So, the profit function can be defined as:
Profit = (240,000 - 15,000p) - 4p
To find the price that will give the greatest profit, we'll differentiate the profit function with respect to p and set it equal to zero. Then we solve for p to obtain the value that maximizes the profit.
Let's differentiate the profit function with respect to p:
dProfit/dp = -15,000 - 4
Setting the derivative equal to zero and solving for p:
-15,000 - 4 = 0
-15,000 = 4
p = -15,000 / 4
p = -3,750
Given that p represents the price, it cannot be negative in this context. Therefore, we discard the negative solution.
The price that will give the greatest profit is $3,750.
revenue = price * copies
profit = revenue - cost, so
y = pq - 4q
= (240000-15000p)(p-4)
= 15000(16-p)(p-4)
= 15000(-p^2+20p-64)
That's just a parabola with vertex at p=10.