Let's solve the questions step by step for the triangular prism shown in the diagram.

Given information:

• The triangular base has a height of $29 \, \text{cm}$ and a base of $34 \, \text{cm}$.
• The length of the prism is $38 \, \text{cm}$.

Step (i) Calculate the volume of the prism:

The volume of a prism is given by:

$V = \text{Base Area} \times \text{Length}$

The base area of the triangular face is:

$\text{Base Area} = \frac{1}{2} \times \text{base of triangle} \times \text{height of triangle}$

Substitute the known values:

$\text{Base Area} = \frac{1}{2} \times 34 \, \text{cm} \times 29 \, \text{cm} = 493 \, \text{cm}^2$

Now, calculate the volume:

$V = 493 \, \text{cm}^2 \times 38 \, \text{cm} = 18,734 \, \text{cm}^3$

So, the volume of the prism is 18,734 cm³.

Step (ii) Calculate the length of the slanted side of the triangle:

To find the slanted side (hypotenuse) of the triangle, we can use the Pythagorean theorem:

$c^2 = a^2 + b^2$

Where:

• $a = 29 \, \text{cm}$ (height),
• $b = 17 \, \text{cm}$ (half of the base, i.e., $\frac{34}{2}$).

Substitute the values:

$c^2 = 29^2 + 17^2 = 841 + 289 = 1130$

$c = \sqrt{1130} \approx 33.63 \, \text{cm}$

So, the length of the slanted side is approximately 33.63 cm.

Step (iii) Calculate the total surface area of the prism:

The surface area consists of:

• The area of the two triangular faces,
• The area of the three rectangular faces.

1. Area of the triangular faces (2 identical triangles): $\text{Area of one triangle} = 493 \, \text{cm}^2$ For two triangles: $2 \times 493 = 986 \, \text{cm}^2$

2. Area of the rectangular faces:

   • Front face: $\text{base of the triangle} \times \text{length} = 34 \times 38 = 1292 \, \text{cm}^2$
   • Bottom face: $\text{slanted side} \times \text{length} = 33.63 \times 38 = 1277.94 \, \text{cm}^2$
   • Back face: $\text{height of the triangle} \times \text{length} = 29 \times 38 = 1102 \, \text{cm}^2$

Now, sum these areas:

$\text{Total Surface Area} = 986 + 1292 + 1277.94 + 1102 = 4657.94 \, \text{cm}^2$

So, the total surface area is approximately 4657.94 cm².

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