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Problem Analysis

We are given:

1. The volume of the prism is $480 \, \text{cm}^3$.
2. The prism has a triangular cross-section with dimensions:
   • Base of the triangle: $13 \, \text{cm}$
   • Height of the triangle: $h \, \text{cm}$ (which we need to find)
   • Length of the prism: $24 \, \text{cm}$
   • Width of the prism (the perpendicular side): $8 \, \text{cm}$

The task is to:

1. Find the value of $h$ (the height of the triangle).
2. Find the total surface area of the prism.

Solution

(a) Finding the value of $h$

The volume of a prism with a triangular cross-section is given by: $\text{Volume} = \text{Area of cross-section} \times \text{Length}$

The area of the triangular cross-section is: $\text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 13 \times h$

So, the volume of the prism can be written as: $480 = \left( \frac{1}{2} \times 13 \times h \right) \times 24$

Solving for $h$:

1. Simplify the equation: $480 = \frac{13 \times h \times 24}{2}$ $480 = 13 \times h \times 12$
2. Divide both sides by $13 \times 12$: $h = \frac{480}{13 \times 12} = \frac{480}{156}$
3. Calculate $h$: $h = 3.08 \, \text{cm (approximately)}$

(b) Finding the total surface area of the prism

The surface area of the prism includes:

1. Two triangular faces.
2. Three rectangular faces (one for each side of the triangle).

1. Area of the triangular faces

The area of each triangular face is: $\text{Area of triangle} = \frac{1}{2} \times 13 \times 3.08 = 20.02 \, \text{cm}^2$

Since there are two triangular faces: $\text{Total area of triangles} = 2 \times 20.02 = 40.04 \, \text{cm}^2$

2. Area of the rectangular faces

• Face 1 (base $13 \, \text{cm}$ by length $24 \, \text{cm}$): $13 \times 24 = 312 \, \text{cm}^2$
• Face 2 (height $3.08 \, \text{cm}$ by length $24 \, \text{cm}$): $3.08 \times 24 = 73.92 \, \text{cm}^2$
• Face 3 (width $8 \, \text{cm}$ by length $24 \, \text{cm}$): $8 \times 24 = 192 \, \text{cm}^2$

Adding the areas of the three rectangular faces: $312 + 73.92 + 192 = 577.92 \, \text{cm}^2$

Total Surface Area

Combining the areas of the triangular and rectangular faces: $\text{Total Surface Area} = 40.04 + 577.92 = 617.96 \, \text{cm}^2$

Final Answers

1. $h = 3.08 \, \text{cm}$
2. Total surface area of the prism = $617.96 \, \text{cm}^2$

Would you like a detailed explanation of any part of the solution, or have additional questions? Here are some related questions to expand understanding:

1. What would change in the calculations if the length of the prism increased?
2. How does the surface area formula differ for prisms with different cross-sectional shapes?
3. How would the volume change if the height $h$ was doubled?
4. What other geometric formulas could help solve similar volume problems?
5. How would you find the surface area if the triangular cross-section was an equilateral triangle?

Tip: Always confirm all given dimensions and units in geometry problems before starting the calculations to avoid mistakes.