# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-10/5

Exercises: Try Out - Set 10: Real exam simulation to sharpen your skills and build confidence. - Maximum Value of Objective Function

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## Exercise 5

### Question

export const metadata = {
  title: "Maximum Value of Objective Function",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

If the maximum value of <InlineMath math="x + y" /> on the set <InlineMath math="\{(x,y)|x \ge 0, y \ge 0, x + 5y \le 10, 5x + y \le a^2 - 5\}" /> is <InlineMath math="5" />, then the value of <InlineMath math="a = \dots" />


### Choices

- [ ] $$3$$
- [ ] $$4$$
- [x] $$5$$
- [ ] $$16$$
- [ ] $$25$$

### Answer & Explanation

export const metadata = {
  title: "Maximum Value Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Given the system of inequalities:

<MathContainer>
  <BlockMath math="\begin{cases} x \ge 0 \\ y \ge 0 \\ x + 5y \le 10 \\ 5x + y \le a^2 - 5 \end{cases}" />
</MathContainer>

We can sum the two main inequalities to find the bound for <InlineMath math="x + y" />:

<MathContainer>
  <BlockMath math="(x + 5y) + (5x + y) \le 10 + (a^2 - 5)" />
  <BlockMath math="6x + 6y \le 5 + a^2" />
  <BlockMath math="6(x + y) \le 5 + a^2" />
</MathContainer>

It is given that the maximum value of <InlineMath math="x + y" /> is <InlineMath math="5" />. Therefore, we substitute this maximum value into the inequality:

<MathContainer>
  <BlockMath math="6(5) = 5 + a^2" />
  <BlockMath math="30 = 5 + a^2" />
  <BlockMath math="a^2 = 25" />
</MathContainer>

Thus, the value of <InlineMath math="a" /> is:

<MathContainer>
  <BlockMath math="a = 5" />
</MathContainer>

(We take the positive value since the options are positive).

So, the value of <InlineMath math="a" /> is <InlineMath math="5" />.


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