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Let's analyze the problem step by step based on the given conditions.

### Given:
1. A cuboid with a square base of length \( 3 \text{ cm} \), meaning the base dimensions are \( 3 \text{ cm} \times 3 \text{ cm} \).
2. The height of the cuboid is \( x \text{ cm} \).
3. Total surface area of the cuboid is not greater than \( 210 \text{ cm}^2 \).
4. Volume of the cuboid is not less than \( 90 \text{ cm}^3 \).

### Part (a): Finding the range of values for \( x \).

#### Step 1: Surface Area Constraint
The surface area of a cuboid is given by:
\[
\text{Surface Area} = 2(lw + lh + wh)
\]
where \( l \), \( w \), and \( h \) are the length, width, and height of the cuboid, respectively. In this problem, \( l = w = 3 \text{ cm} \), and \( h = x \text{ cm} \).

Substituting these values:
\[
\text{Surface Area} = 2(3 \cdot 3 + 3 \cdot x + 3 \cdot x) = 2(9 + 6x) = 18 + 12x
\]
The surface area must be less than or equal to \( 210 \text{ cm}^2 \):
\[
18 + 12x \leq 210
\]
Solving for \( x \):
\[
12x \leq 192
\]
\[
x \leq 16
\]

#### Step 2: Volume Constraint
The volume of a cuboid is given by:
\[
\text{Volume} = l \cdot w \cdot h = 3 \cdot 3 \cdot x = 9x
\]
The volume must be at least \( 90 \text{ cm}^3 \):
\[
9x \geq 90
\]
Solving for \( x \):
\[
x \geq 10
\]

#### Conclusion for Part (a)
Combining the two constraints, the range of values for \( x \) is:
\[
10 \leq x \leq 16
\]

### Part (b)(i): Is it possible for \( \angle CBY \) to be equal to \( 50^\circ \)?

To determine if \( \angle CBY \) can be \( 50^\circ \), we would need to find the shortest path for the ant from \( B \) to \( E \) along the surface. The path typically involves unfolding the cuboid into a plane and finding the straight-line distance on this flattened representation. Calculating the exact angle would involve trigonometric analysis based on the path's geometry, which would depend on the position of \( Y \).

### Part (b)(ii): Is the path length less than or equal to 18 cm?

To find the shortest path from \( B \) to \( E \), consider unfolding the cuboid into a rectangle. The rectangle formed by unfolding \( BCDEHG \) would have dimensions \( 3 \text{ cm} \times (3 + x) \). The shortest path can be found using the Pythagorean theorem:
\[
\text{Shortest Path} = \sqrt{(3 + x)^2 + 3^2}
\]
Substitute \( 10 \leq x \leq 16 \) and calculate if the path length satisfies the condition.

Would you like more details on any of these parts?

### Additional Questions:
1. What happens to the range of \( x \) if the surface area constraint changes?
2. How would the shortest path change if the base dimensions were different?
3. What trigonometric functions would be needed to solve for \( \angle CBY \)?
4. How does increasing the cuboid's height affect the shortest path?
5. Can you find another method to calculate the shortest path besides unfolding?

### Tip:
For complex geometric problems, visualizing the 3D object as a 2D net can simplify the calculations significantly.

A cuboid has a square base of length 3 cm. The total surface area of the cuboid is not greater than 210 cm² and the volume of the cuboid is not less than 90 cm³. Given that DE is x cm, find the range of values for x. An ant crawls from point B to point E on the surface of the cuboid via any point Y on CH, taking the shortest path. (i) Is it possible for ∠CBY to be equal to 50°? Explain your answer. (ii) Chloe claims that the length of the path will not be greater than 18 cm. Do you agree? Explain your answer.

Learn how to find the range of x for a cuboid with a square base of 3 cm, given constraints on its surface area and volume. The problem involves applying surface area and volume formulas, finding the shortest path using the Pythagorean Theorem, and determining if an angle of 50° is possible. Suitable for Grades 9-11.

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