We are tasked with maximizing the utility function:

$U(x, y) = x^2 + 2xy + y^2 + 4a^2 + 8a + 10$

subject to the constraint:

$2x + y = 10$

and the non-negativity constraints $x \geq 0$ and $y \geq 0$.

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

Using the constraint $2x + y = 10$, we solve for $y$:

$y = 10 - 2x$

Step 2: Substitute $y$ into the utility function

Now substitute $y = 10 - 2x$ into the utility function $U(x, y)$:

$U(x, 10 - 2x) = x^2 + 2x(10 - 2x) + (10 - 2x)^2 + 4a^2 + 8a + 10$

Step 3: Simplify the utility function

First, simplify each term:

• $2x(10 - 2x) = 20x - 4x^2$
• $(10 - 2x)^2 = 100 - 40x + 4x^2$

Substitute these into $U(x, 10 - 2x)$:

$U(x, 10 - 2x) = x^2 + (20x - 4x^2) + (100 - 40x + 4x^2) + 4a^2 + 8a + 10$

Now combine like terms:

$U(x, 10 - 2x) = (x^2 - 4x^2 + 4x^2) + (20x - 40x) + (100) + 4a^2 + 8a + 10$

$U(x, 10 - 2x) = x^2 - 20x + 110 + 4a^2 + 8a$

Step 4: Maximize the utility function

To maximize $U(x, 10 - 2x)$, we take the derivative with respect to $x$ and set it equal to zero:

$\frac{d}{dx} \left( x^2 - 20x + 110 + 4a^2 + 8a \right) = 2x - 20 = 0$

Solving for $x$:

$2x = 20 \quad \Rightarrow \quad x = 10$

Step 5: Find $y$

Now substitute $x = 10$ into the constraint $y = 10 - 2x$:

$y = 10 - 2(10) = 10 - 20 = -10$

However, $y$ must satisfy the non-negativity constraint $y \geq 0$. Since $y = -10$ is not feasible, we must check the boundary conditions.

Step 6: Boundary Condition

The boundary condition occurs when $y = 0$, so from the constraint $2x + y = 10$, we solve for $x$ when $y = 0$:

$2x = 10 \quad \Rightarrow \quad x = 5$

Step 7: Conclusion

The optimal value of $y$ chosen by Rohit is $\boxed{0}$.

Would you like more details or have any questions?

Here are 5 related questions to expand on this problem:

1. How would the optimal value of $y$ change if the constraint was different?
2. What happens to the utility function if $a = 0$?
3. Can the problem be solved using a different optimization technique?
4. How does the range of $a$ (between 0 and 1) affect the optimal choice of $x$ and $y$?
5. How would the solution change if the non-negativity constraint was removed?

Tip: Always check boundary conditions when maximizing functions subject to constraints!