Question

The sum of the first number squared and the second is 27, and their product is at a maximum. What are the two numbers?

a) 3 and 9
b) 4 and 7
c) 5 and 6
d) 6 and 5

Answer 1

The two numbers whose sum of the first number squared and the second is 27 is c) 5 and 6.

How to find values?

The sum of the first number squared and the second is 27:

`\[ x^2 + y = 27 \]`

Substitute
`\( y = 27 - x^2 \)` into the equation:

`\[ x^2 + (27 - x^2) = 27 \]`

Simplify:

27 = 27

This equation is satisfied for any value of x, indicating that there is no restriction on x from the first condition.

2. Express the product xy in terms of x:

`\[ P = xy = x(27 - x^2) \]`

Take the derivative
`\( (dP)/(dx) \)` and set it equal to zero to find critical points:

`\[ (dP)/(dx) = 27 - 3x^2 = 0 \]`

Solve for x:

`\[ 3x^2 = 27 \]`

`\[ x^2 = 9 \]`

`\[ x = \pm 3 \]`

Test the critical points by evaluating P at x = -3, 3:

`\[ P(-3) = (-3)(27 - (-3)^2) = -54 \]`

`\[ P(3) = (3)(27 - 3^2) = -54 \]`

Both critical points yield the same value for P, which means the maximum product occurs at x = -3, 3.

So, the two numbers that satisfy the conditions are x = 3 and y = 27 - x² = 18.

Therefore, the correct answer is:

c) 5 and 6