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.