An electronics store is deciding how to price one of its products. The equation
P = -25d^2 + 500d predicts the total profit P as a function of the product's price in dollars d. What price will produce the highest total profit?
this is just a quadratic, so the max is reached at
x = -b/2a = -500/-50 = 10
To find the price that will produce the highest total profit, we need to determine the maximum value of the profit function P = -25d^2 + 500d.
There are different approaches to finding the maximum of a quadratic function, but one common method is to use calculus. We can take the derivative of the profit function with respect to d and set it equal to zero to find the critical points, which represent the potential maximum or minimum points of the function.
Let's differentiate the profit function P = -25d^2 + 500d with respect to d:
dP/dd = -50d + 500
Now, set this derivative equal to zero and solve for d:
-50d + 500 = 0
-50d = -500
d = -500 / -50
d = 10
To confirm that this is a maximum point, we can check the second derivative. If the second derivative is negative at this point, it indicates a maximum.
Taking the second derivative of the profit function:
d^2P/dd^2 = -50
Since the second derivative is negative (-50), we can conclude that d = 10 corresponds to a maximum point.
Therefore, the price that will produce the highest total profit is $10.