The problem provides the profit function for the Bluebird Bakery, given by:

$f(x) = -500(x - 0.45)^2 + 400$

This function models the profit in dollars, $f(x)$, where $x$ represents the price of a cookie in dollars. We are asked to find the profit for two specific cookie prices: $0.40 and $0.75.

Step 1: Calculate the profit for selling cookies at $0.40 each

Substitute $x = 0.40$ into the profit function:

$f(0.40) = -500(0.40 - 0.45)^2 + 400$ $f(0.40) = -500(-0.05)^2 + 400$ $f(0.40) = -500(0.0025) + 400$ $f(0.40) = -1.25 + 400 = 398.75$

Step 2: Calculate the profit for selling cookies at $0.75 each

Substitute $x = 0.75$ into the profit function:

$f(0.75) = -500(0.75 - 0.45)^2 + 400$ $f(0.75) = -500(0.30)^2 + 400$ $f(0.75) = -500(0.09) + 400$ $f(0.75) = -45 + 400 = 355$

Final Answers:

• The profit for selling cookies for $0.40 each is $398.75.
• The profit for selling cookies for $0.75 each is $355.00.

Would you like more details or have any other questions?

Here are five related questions you might explore:

1. What is the maximum possible profit, and at what price does it occur?
2. How would the profit change if the price of the cookies was increased to $1.00?
3. What is the significance of the constant term 400 in the profit function?
4. How does the negative coefficient $-500$ affect the shape of the profit function?
5. Can you derive a general strategy for maximizing profit in this scenario?

Tip: The vertex form of a quadratic function, like the one here, makes it easy to identify the maximum or minimum point, which is useful in optimization problems.