Perimeter, Area, and Volume Using a net to find the surface area of a triangular prism (a) Find the following side lengths for the net. $$ \begin{array}{l} A=15 \mathrm{yd} \ B=2 \mathrm{yd} \ C=8 \mathrm{yd} \ D=17 \mathrm{yd} \end{array} $$ (b) Use the net to find the surface area of the prism. $$ \square $$ $$ \mathrm{yd}^{2} $$

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**Step 1. Identify the components of the prism**

A triangular prism has two triangular bases and three rectangular lateral faces. The net shows:
- Two congruent triangles that form the bases.
- Three rectangles that “wrap” around the sides of the triangle.

From the given measures, we identify:
- The triangular bases have side lengths  
  $$
  15\text{ yd},\quad 8\text{ yd},\quad 17\text{ yd}
  $$
- The height (or length) of the prism is given as  
  $$
  2\text{ yd}
  $$
  (denoted by $$B=2\text{ yd}$$).  
  This height is the common side-length for the rectangular lateral faces.

Thus, the three lateral (rectangular) faces have dimensions:
- One rectangle: $$15\text{ yd} \times 2\text{ yd}$$
- One rectangle: $$8\text{ yd} \times 2\text{ yd}$$
- One rectangle: $$17\text{ yd} \times 2\text{ yd}$$

**Step 2. Compute the area of the triangular bases**

The triangle has side lengths $$15\text{ yd}$$, $$8\text{ yd}$$, and $$17\text{ yd}$$. First, find the semiperimeter $$s$$:
$$
s = \frac{15+8+17}{2} = \frac{40}{2} = 20\text{ yd}
$$

Using Heron’s formula, the area $$T$$ of the triangle is:
$$
T = \sqrt{s(s-15)(s-8)(s-17)}
$$
Substitute the values:
$$
T = \sqrt{20(20-15)(20-8)(20-17)} = \sqrt{20 \times 5 \times 12 \times 3}
$$
Calculate the product inside the square root:
$$
20 \times 5 = 100,\quad 12 \times 3 = 36,\quad \text{so}\quad 100 \times 36 = 3600
$$
Thus,
$$
T = \sqrt{3600} = 60\text{ yd}^2
$$

Because there are two triangular bases, their total area is:
$$
2T = 2 \times 60 = 120\text{ yd}^2
$$

**Step 3. Compute the lateral surface area**

Each rectangle’s area is found by multiplying the corresponding side of the triangle by the height of the prism (2 yd). Therefore:
- Area of rectangle with side $$15\text{ yd}$$: $$15 \times 2 = 30\text{ yd}^2$$
- Area of rectangle with side $$8\text{ yd}$$: $$8 \times 2 = 16\text{ yd}^2$$
- Area of rectangle with side $$17\text{ yd}$$: $$17 \times 2 = 34\text{ yd}^2$$

Thus, the lateral surface area is:
$$
30 + 16 + 34 = 80\text{ yd}^2
$$

**Step 4. Calculate the total surface area**

The total surface area (SA) of the prism is the sum of the areas of the two triangles and the three rectangles:
$$
\text{SA} = 120\text{ yd}^2 + 80\text{ yd}^2 = 200\text{ yd}^2
$$

**Answer**

(a) The net has two triangular bases with sides $$15\text{ yd}$$, $$8\text{ yd}$$, and $$17\text{ yd}$$ and three rectangles with dimensions $$15 \times 2\text{ yd}$$, $$8 \times 2\text{ yd}$$, and $$17 \times 2\text{ yd}$$.

(b) The surface area of the prism is  
$$
200\, \mathrm{yd}^2.
$$
(a) The net has two triangular bases with sides 15 yd, 8 yd, and 17 yd, and three rectangles with dimensions 15×2 yd, 8×2 yd, and 17×2 yd. (b) The surface area of the prism is 200 yd².

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