To calculate the surface area of the triangular prism, we need to determine the areas of all its faces. The prism consists of:

1. Two triangular faces (front and back): These triangles have a base of $15 \, \text{m}$ and a height of $9 \, \text{m}$.

2. Three rectangular faces:

   • A rectangular face along the base of the triangles ($15 \, \text{m} \times 12 \, \text{m}$).
   • Two rectangular faces along the sides of the triangles:
     • $12 \, \text{m} \times 16 \, \text{m}$.
     • Another rectangular face connecting the slanted side of the triangle (diagonal) to $12 \, \text{m}$.

Step-by-Step Calculation:

1. Area of each triangular face:

The area of one triangle is given by: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area of triangle} = \frac{1}{2} \times 15 \, \text{m} \times 9 \, \text{m} = 67.5 \, \text{m}^2$ Since there are two triangles: $\text{Total area of triangles} = 2 \times 67.5 = 135 \, \text{m}^2$

2. Area of the rectangular faces:

• Base rectangle: $15 \, \text{m} \times 12 \, \text{m} = 180 \, \text{m}^2$
• Side rectangle 1: $12 \, \text{m} \times 16 \, \text{m} = 192 \, \text{m}^2$
• Side rectangle 2 (slanted side): We first need to calculate the length of the slanted side using the Pythagorean theorem. The slanted side forms a right triangle with the base ($15 \, \text{m}$) and height ($9 \, \text{m}$): $\text{Slant length} = \sqrt{15^2 + 9^2} = \sqrt{225 + 81} = \sqrt{306} \approx 17.49 \, \text{m}$ So, the area of this rectangle is: $17.49 \, \text{m} \times 12 \, \text{m} \approx 209.88 \, \text{m}^2$

3. Total surface area:

Summing all the areas: $\text{Total surface area} = \text{Triangles} + \text{Base rectangle} + \text{Side rectangle 1} + \text{Side rectangle 2}$ $\text{Total surface area} = 135 + 180 + 192 + 209.88 = 716.88 \, \text{m}^2$

Final Answer:

The total surface area of the triangular prism is approximately: $716.88 \, \text{m}^2$

Would you like additional clarification or assistance? Here are five related questions to explore:

1. How is the Pythagorean theorem applied in finding the slanted length?
2. Can you explain surface area for other prism shapes like cylinders or rectangular prisms?
3. What are the units and why are they squared in surface area?
4. How would the calculations change if the dimensions of the prism changed?
5. Could we use alternative methods to find surface area more efficiently?

Tip: Always double-check your dimensions and ensure you account for all the faces of a 3D shape!