$$ \left.\begin{array}{|l}\begin{array}{r} ext { (a) Find the following side lengths for the net. } \ A=3 \ B=8 \mathrm{~cm} \ C\end{array} \ D=4 \mathrm{~cm} \ ext { (b) Use the net to find the surface area of the prism. } \ \square \mathrm{cm}^{2}\end{array} ight] $$.

Question

$$ \left.\begin{array}{|l}\begin{array}{r}	ext { (a) Find the following side lengths for the net. } \ A=3 \ B=8 \mathrm{~cm} \ C\end{array} \ D=4 \mathrm{~cm} \ 	ext { (b) Use the net to find the surface area of the prism. } \ \square \mathrm{cm}^{2}\end{array}ight] $$.

UpStudy AI · Accepted Answer

To solve the problem, we need to find the missing side length $$ C $$ and then use the net to calculate the surface area of the prism.

### Step 1: Find the missing side length $$ C $$

From the information provided, we have:
- $$ A = 3 \, \text{cm} $$
- $$ B = 8 \, \text{cm} $$
- $$ D = 4 \, \text{cm} $$

Assuming that the net represents a rectangular prism, we can deduce that the lengths of the sides are related. In a rectangular prism, opposite sides are equal. Therefore, we can assume that:
- $$ C $$ is the height of the prism, which is equal to $$ D $$.

Thus, we have:
$$
C = D = 4 \, \text{cm}
$$

### Step 2: Calculate the surface area of the prism

The surface area $$ S $$ of a rectangular prism can be calculated using the formula:
$$
S = 2(AB + AC + BC)
$$
where:
- $$ A $$ is the length,
- $$ B $$ is the width,
- $$ C $$ is the height.

Substituting the known values:
- $$ A = 3 \, \text{cm} $$
- $$ B = 8 \, \text{cm} $$
- $$ C = 4 \, \text{cm} $$

We can calculate the surface area:
$$
S = 2(3 \cdot 8 + 3 \cdot 4 + 8 \cdot 4)
$$

Calculating each term:
1. $$ 3 \cdot 8 = 24 $$
2. $$ 3 \cdot 4 = 12 $$
3. $$ 8 \cdot 4 = 32 $$

Now, substituting these values back into the surface area formula:
$$
S = 2(24 + 12 + 32) = 2(68) = 136 \, \text{cm}^2
$$

### Final Answer

The missing side length $$ C $$ is $$ 4 \, \text{cm} $$, and the surface area of the prism is $$ 136 \, \text{cm}^2 $$.
$$ C = 4 \, \text{cm} $$ and the surface area of the prism is $$ 136 \, \text{cm}^2 $$.

\( \left.\begin{array}{|l}\begin{array}{r}\text { (a) Find the following side lengths for the net. } \\ A=3 \\ B=8 \mathrm{~cm} \\ C\end{array} \\ D=4 \mathrm{~cm} \\ \text { (b) Use the net to find the surface area of the prism. } \\ \square \mathrm{cm}^{2}\end{array}\right] \).

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