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URL: https://nakafa.com/en/exercises/high-school/snbt/mathematical-reasoning/try-out/2026/set-1/9

Exercises: Try Out 2026 - Set 1: Real exam simulation to sharpen your skills and build confidence. - Question 9

---

## Exercise 9

### Question

export const metadata = {
  title: "Question 9",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "11/23/2025",
};

A cuboid-shaped cardboard box has a base perimeter of <InlineMath math="6x \text{ cm}" /> and a height of <InlineMath math="\left(5 - \frac{1}{3}x\right) \text{ cm}" /> where <InlineMath math="x" /> is a real number.

If the base of the box is rectangular with a width of <InlineMath math="(4x - 5) \text{ cm}" /> and <InlineMath math="A" /> is the set containing all real numbers <InlineMath math="x" /> that satisfy the dimensions of the cuboid, the correct set <InlineMath math="A" /> is ....



### Choices

- [ ] $$\{x \mid x < 5, x \in \mathbb{R}\}$$
- [ ] $$\{x \mid x > \frac{5}{4}, x \in \mathbb{R}\}$$
- [x] $$\{x \mid \frac{5}{4} < x < 5, x \in \mathbb{R}\}$$
- [ ] $$\{x \mid x < 15, x \in \mathbb{R}\}$$
- [ ] $$\{x \mid 5 < x < 15, x \in \mathbb{R}\}$$

### Answer & Explanation

import { NumberLine } from "@repo/design-system/components/contents/mathematics/number-line";

export const metadata = {
  title: "Solution for Question 9",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "11/23/2025",
};

We will determine the set of values for <InlineMath math="x" /> that satisfies the condition for the cuboid to have real (positive) dimensions.

#### Determining Cuboid Dimensions

Based on the previous question and the given information:
- Length (<InlineMath math="p" />) is <InlineMath math="5 - x" />
- Width (<InlineMath math="l" />) is <InlineMath math="4x - 5" />
- Height (<InlineMath math="t" />) is <InlineMath math="5 - \frac{1}{3}x" />

#### Positive Dimension Conditions

For the cuboid to be real, all its dimensions must be positive (<InlineMath math="> 0" />).

**Condition <InlineMath math="1" />: <InlineMath math="p > 0" />**

<MathContainer>
  <BlockMath math="p > 0" />
  <BlockMath math="5 - x > 0" />
  <BlockMath math="5 > x" />
  <BlockMath math="x < 5" />
</MathContainer>

**Condition <InlineMath math="2" />: <InlineMath math="l > 0" />**

<MathContainer>
  <BlockMath math="l > 0" />
  <BlockMath math="4x - 5 > 0" />
  <BlockMath math="4x > 5" />
  <BlockMath math="x > \frac{5}{4}" />
</MathContainer>

**Condition <InlineMath math="3" />: <InlineMath math="t > 0" />**

<MathContainer>
  <BlockMath math="t > 0" />
  <BlockMath math="5 - \frac{1}{3}x > 0" />
  <BlockMath math="5 > \frac{1}{3}x" />
  <BlockMath math="15 > x" />
  <BlockMath math="x < 15" />
</MathContainer>

#### Determining Set Intersection

We find the intersection of these three conditions:
1. <InlineMath math="x < 5" />
2. <InlineMath math="x > \frac{5}{4}" /> (or <InlineMath math="1.25" />)
3. <InlineMath math="x < 15" />

From conditions <InlineMath math="1" /> and <InlineMath math="3" />, we take the stricter one, which is <InlineMath math="x < 5" />. Combined with condition <InlineMath math="2" />, we get the range:

<BlockMath math="\frac{5}{4} < x < 5" />

<NumberLine
  title="Solution Set A"
  description={<>The intersection of all positive dimension conditions is <InlineMath math="\frac{5}{4} < x < 5" />.</>}
  segments={[
    {
      start: Number.NEGATIVE_INFINITY,
      end: 1.25,
      label: <InlineMath math="-" />,
      backgroundColor: "transparent"
    },
    {
      start: 1.25,
      end: 5,
      label: <InlineMath math="A" />,
      startLabel: <InlineMath math="\frac{5}{4}" />,
      endLabel: <InlineMath math="5" />,
      startInclusive: false,
      endInclusive: false
    },
    {
      start: 5,
      end: Number.POSITIVE_INFINITY,
      label: <InlineMath math="-" />,
      backgroundColor: "transparent"
    }
  ]}
/>

So, the set <InlineMath math="A" /> is:

<BlockMath math="A = \left\{ x \mid \frac{5}{4} < x < 5, x \in \mathbb{R} \right\}" />


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