Let's analyze the problem step-by-step.

Given Information:

• The prism has a length of 30 cm.
• The cross-section WXYZ is a square with an area of 144 cm².

(i) Calculating the Volume of the Prism

1. Find the side length of the square cross-section:

   Since the cross-section is a square, its area $A$ can be calculated by: $\text{Area} = \text{side}^2$

   Given that the area is 144 cm², we solve for the side length: $\text{side} = \sqrt{144} = 12 \, \text{cm}$

2. Calculate the Volume of the Prism:

   The volume $V$ of a prism is given by: $V = \text{Base Area} \times \text{Height (Length of the prism)}$

   The base area is the area of the square cross-section (144 cm²), and the height (length of the prism) is 30 cm:

   $V = 144 \, \text{cm}^2 \times 30 \, \text{cm} = 4320 \, \text{cm}^3$

(ii) Calculating the Total Surface Area of the Prism

1. Find the Surface Area of the Two Square Ends:

   Each end is a square with an area of 144 cm². There are 2 such ends: $\text{Area of two ends} = 2 \times 144 = 288 \, \text{cm}^2$

2. Find the Surface Area of the Four Rectangular Faces:

   Each of the four faces is a rectangle with one side equal to the side of the square (12 cm) and the other side equal to the length of the prism (30 cm).

   The area of one rectangular face is: $\text{Area of one rectangular face} = 12 \, \text{cm} \times 30 \, \text{cm} = 360 \, \text{cm}^2$

   Since there are 4 such faces: $\text{Area of four faces} = 4 \times 360 = 1440 \, \text{cm}^2$

3. Calculate the Total Surface Area:

   The total surface area $SA$ of the prism is the sum of the areas of the two ends and the four faces: $SA = 288 \, \text{cm}^2 + 1440 \, \text{cm}^2 = 1728 \, \text{cm}^2$

Final Answers:

1. Volume of the prism: $\mathbf{4320 \, \text{cm}^3}$
2. Total surface area of the prism: $\mathbf{1728 \, \text{cm}^2}$

Would you like further details on any of these steps, or do you have any other questions?

Related Questions:

1. What is the formula for the volume of a rectangular prism?
2. How do you calculate the surface area of a triangular prism?
3. What is the significance of understanding the surface area in real-world contexts?
4. Can you determine the volume of a prism if only given the perimeter of its base and its height?
5. How does the formula for the volume of a cylinder differ from that of a prism?

Tip: To find the surface area of a prism, always add the areas of the two bases to the lateral surface area.