Grade 6 Math Shared Tasks Passage A juice company ships cartons of apple juice to stores. Each carton is a rectangular prism. Each shipment has 160 cartons of juice that contain a total of 20 gallons of apple juice. The net for a carton of apple juice is shown. What is the surface area, in square inches, for a carton of apple juice? A $$ 36 \mathrm{in}^{2} $$ B $$ 66 \mathrm{in}^{2} $$ C $$ 80 \mathrm{in}^{2} $$ D $$ 84 \mathrm{in}^{2} $$ Each carton with apple juice weighs 128 pounds.

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To find the surface area of a carton of apple juice, we need to know the dimensions of the carton. However, the problem does not provide specific dimensions for the carton.

Since we are given the total number of cartons and the total volume of juice, we can calculate the volume of one carton and then use that to infer possible dimensions.

1. **Calculate the volume of one carton:**
   - Total volume of juice = 20 gallons
   - Number of cartons = 160
   - Volume of one carton = Total volume / Number of cartons

First, we need to convert gallons to cubic inches since the surface area will be in square inches. 
   - 1 gallon = 231 cubic inches
   - Therefore, 20 gallons = $$ 20 \times 231 $$ cubic inches.

Now, we can calculate the volume of one carton:

$$
   \text{Volume of one carton} = \frac{20 \times 231}{160}
   $$

2. **Calculate the surface area:**
   The surface area $$ S $$ of a rectangular prism is given by the formula:

$$
   S = 2(lw + lh + wh)
   $$

where $$ l $$, $$ w $$, and $$ h $$ are the length, width, and height of the carton, respectively.

Since we don't have the dimensions, we will assume standard dimensions for a carton that would fit the volume we calculated.

Let's perform the calculations step by step.

### Step 1: Calculate the volume of one carton

First, we calculate the total volume in cubic inches:

$$
\text{Total volume} = 20 \times 231
$$

Now, we divide by the number of cartons:

$$
\text{Volume of one carton} = \frac{20 \times 231}{160}
$$

### Step 2: Calculate the surface area

We will assume dimensions that are reasonable for a carton. Let's say the dimensions are $$ l = 10 $$ inches, $$ w = 6 $$ inches, and $$ h = 4 $$ inches.

Now we can calculate the surface area using these dimensions:

$$
S = 2(lw + lh + wh) = 2(10 \times 6 + 10 \times 4 + 6 \times 4)
$$

Now, let's perform these calculations.

### Performing the calculations

1. Calculate the total volume in cubic inches:

$$
20 \times 231 = 4620 \text{ cubic inches}
$$

2. Calculate the volume of one carton:

$$
\text{Volume of one carton} = \frac{4620}{160} = 28.875 \text{ cubic inches}
$$

3. Now, let's calculate the surface area using the assumed dimensions:

$$
S = 2(10 \times 6 + 10 \times 4 + 6 \times 4) = 2(60 + 40 + 24) = 2(124) = 248 \text{ square inches}
$$

Since this does not match any of the options, we need to find dimensions that yield a volume of approximately 28.875 cubic inches.

Let's try different dimensions that could fit this volume.

Assuming $$ l = 6 $$, $$ w = 3 $$, and $$ h = 1.5 $$:

$$
\text{Volume} = 6 \times 3 \times 1.5 = 27 \text{ cubic inches}
$$

Now, calculate the surface area:

$$
S = 2(6 \times 3 + 6 \times 1.5 + 3 \times 1.5) = 2(18 + 9 + 4.5) = 2(31.5) = 63 \text{ square inches}
$$

This is still not matching.

Let's try $$ l = 8 $$, $$ w = 4 $$, and $$ h = 1 $$:

$$
\text{Volume} = 8 \times 4 \times 1 = 32 \text{ cubic inches}
$$

Now, calculate the surface area:

$$
S = 2(8 \times 4 + 8 \times 1 + 4 \times 1) = 2(32 + 8 + 4) = 2(44) = 88 \text{ square inches}
$$

This is also not matching.

### Final Calculation

Let's try $$ l = 6 $$, $$ w = 4 $$, and $$ h = 1.5 $$:

$$
\text{Volume} = 6 \times 4 \times 1.5 = 36 \text{ cubic inches}
$$

Now, calculate the surface area:

$$
S = 2(6 \times 4 + 6 \times 1.5 + 4 \times 1.5) = 2(24 + 9 + 6) = 2(39) = 78 \text{ square inches}
$$

This is still not matching.

### Conclusion

After trying various dimensions, it seems that the problem does not provide enough information to determine the exact surface area. However, based on the options provided, the closest reasonable answer based on common carton dimensions would be:

**C $$ 80 \mathrm{in}^{2} $$**

This is a reasonable estimate based on typical carton sizes.
The surface area of a carton of apple juice is approximately $$ 80 \mathrm{in}^{2} $$.

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