You operate a bakery. You buy cakes from a supplier who charges you $5 per cake.
Government regulations dictate that you can only charge a price between $10 and $20
per cake. Your research shows that if you were to charge $x per cake, you would sell
125 - x2 cakes per week.
a) In order to maximize profi�t, what price should you sell the cake for?
b) How much would you charge if there were no government regulations?
a) To maximize profit, we need to determine the price that generates the highest revenue. Revenue is calculated by multiplying the price per cake by the number of cakes sold.
We are given the equation for the number of cakes sold per week:
Quantity (Q) = 125 - x^2
And the price per cake (P) must lie between $10 and $20.
Profit (π) is calculated as:
Profit (π) = Revenue - Cost
The revenue can be calculated as:
Revenue = Price (P) * Quantity (Q) = P * (125 - x^2)
The cost per cake is given as $5 per cake.
To maximize the profit, we need to find the price that results in the highest value for π.
We can write the profit equation as:
Profit (π) = P * (125 - x^2) - 5 * (125 - x^2)
= P * (125 - x^2) - 5 *125 + 5 * x^2
= P * 125 - P * x^2 - 5 * 125 + 5 * x^2
= 125P - Px^2 - 625 + 5x^2
To find the price that maximizes profit, we differentiate the profit equation with respect to x and set it equal to zero:
dπ/dx = 0 - 2Px + 10x = -2Px + 10x = 0
Factor out x:
x(-2P + 10) = 0
From this equation, we can see that either x = 0 or 2P - 10 = 0.
If x = 0, then the number of cakes sold is 125, which is the maximum number of cakes that can be sold.
If 2P - 10 = 0, then P = 5, which is the minimum price allowed.
Given that the price must be between $10 and $20, it is clear that the maximum profit is obtained when x = 0.
Therefore, to maximize profit, you should sell the cake for a price of $0.
b) If there were no government regulations, we would have more flexibility in pricing. In this case, we could charge any price between $0 and infinity.
However, it is important to consider the demand for the cakes. The demand equation is given as:
Quantity (Q) = 125 - x^2
If we set the demand equation equal to zero, we can find the price at which no cakes would be sold:
125 - x^2 = 0
x^2 = 125
x = ±√125
x ≈ ±11.18
This means that if the price is set too high (above $11.18), the quantity demanded will be zero, resulting in no sales.
Therefore, if there were no government regulations, it would be optimal to charge a price that is less than or equal to $11.18 to ensure sales.
To maximize profit, we need to determine the price at which the revenue generated from selling cakes is maximized. The revenue is given by the equation: R = P * Q, where R is the revenue, P is the price per cake, and Q is the quantity sold.
Given that if you charge $x per cake, you would sell 125 - x^2 cakes per week, we can express the quantity sold as a function of the price: Q = 125 - x^2.
a) To find the price that maximizes profit, we need to find the revenue function and then determine the price that maximizes it. The revenue function is obtained by substituting the quantity function into the revenue equation:
R = P * (125 - x^2)
To find the price that maximizes profit, we need to find the value of x that maximizes the revenue, which is equivalent to finding the vertex of the parabola (125 - x^2). The x-coordinate of the vertex is given by -b/2a, where a and b are coefficients from the quadratic equation. In this case, a = -1 and b = 0, so:
x-coordinate of vertex = -0/(2*(-1)) = 0
Therefore, to maximize profit, you should sell the cake for $0 (free). However, this may not be realistic or desirable, so you might want to set a price that is reasonable based on the given constraints.
b) If there were no government regulations, you could charge any price you want. In this case, you would want to set the price that maximizes profit, which we found to be $0. However, as mentioned earlier, setting the price at $0 may not be practical or profitable. Without the government regulation, you could potentially charge a price that maximizes profit within the range of $10 to $20 per cake based on market demand and other business considerations.