We are given the following information:

1. The sum of the square of one number and a second number is 27. This translates to the equation: $x^2 + y = 27$ where $x$ is the first number and $y$ is the second number.

2. We want to maximize the product of these two numbers, $P = x \cdot y$.

Step 1: Express $y$ in terms of $x$

From the equation $x^2 + y = 27$, solve for $y$: $y = 27 - x^2$

Step 2: Write the product function $P$

Now substitute $y = 27 - x^2$ into the product function $P = x \cdot y$: $P(x) = x \cdot (27 - x^2) = 27x - x^3$ This is the function we want to maximize.

Step 3: Find the critical points by taking the derivative

To maximize $P(x)$, we first find its derivative with respect to $x$: $\frac{dP}{dx} = 27 - 3x^2$

Set the derivative equal to zero to find the critical points: $27 - 3x^2 = 0$ $3x^2 = 27$ $x^2 = 9$ $x = \pm 3$

Step 4: Determine which critical point maximizes the product

We now check the second derivative to determine whether these points correspond to a maximum or minimum: $\frac{d^2P}{dx^2} = -6x$ For $x = 3$, we have: $\frac{d^2P}{dx^2} = -6(3) = -18 \quad (\text{which is negative, indicating a maximum})$ For $x = -3$, we have: $\frac{d^2P}{dx^2} = -6(-3) = 18 \quad (\text{which is positive, indicating a minimum})$

Thus, the value $x = 3$ gives a maximum.

Step 5: Find $y$

When $x = 3$, substitute into the equation $x^2 + y = 27$: $3^2 + y = 27$ $9 + y = 27$ $y = 18$

Conclusion

The values of the two numbers that maximize the product are $x = 3$ and $y = 18$. The maximum product is: $P = 3 \cdot 18 = 54$

Do you need any further details or clarifications?

Here are some additional questions you might find interesting:

1. How would the product change if we modified the sum condition to something different, like 30 instead of 27?
2. Can the product be maximized for negative values of $x$?
3. What would the product be if we tried maximizing the sum of the two numbers instead of their product?
4. How does the solution change if we restrict $x$ and $y$ to integer values only?
5. How could we graph the function $P(x) = 27x - x^3$ to visualize the maximum point?

Tip: Always check the second derivative to confirm whether a critical point corresponds to a maximum or a minimum.