Let's calculate each part step by step.
1. Surface area of a right rectangular prism (w=12 in., l=18 in., h=8 in.)
The formula for the surface area \( S \) of a rectangular prism is given by:
\[ S = 2(lw + lh + wh) \]
Plugging in the given dimensions:
- \( w = 12 \) in.
- \( l = 18 \) in.
- \( h = 8 \) in.
\[ S = 2(18 \cdot 12 + 18 \cdot 8 + 12 \cdot 8) \] Calculating each term:
- \( 18 \cdot 12 = 216 \)
- \( 18 \cdot 8 = 144 \)
- \( 12 \cdot 8 = 96 \)
Now adding them together:
\[ 216 + 144 + 96 = 456 \]
Multiply by 2:
\[ S = 2 \cdot 456 = 912 \text{ in}^2 \]
2. Surface area of a cube with 12-inch edges
The formula for the surface area \( S \) of a cube is given by:
\[ S = 6s^2 \]
where \( s \) is the length of an edge. Given that \( s = 12 \) in., we calculate:
\[ S = 6 \cdot 12^2 = 6 \cdot 144 = 864 \text{ in}^2 \]
3. Surface area of a right rectangular prism with each edge length of 15 ft
Since it's a cube (all edges are the same), we use the same cube formula:
\[ S = 6s^2 \]
with \( s = 15 \) ft:
\[ S = 6 \cdot 15^2 = 6 \cdot 225 = 1350 \text{ ft}^2 \]
4. Surface area of a cereal box (rectangular prism) with dimensions:
- Length = 8 in.
- Height = 12 in.
- Width = 2 in.
Using the same formula for surface area:
\[ S = 2(lw + lh + wh) \]
Plugging in the dimensions:
\[ S = 2(8 \cdot 2 + 8 \cdot 12 + 2 \cdot 12) \]
Calculating each term:
- \( 8 \cdot 2 = 16 \)
- \( 8 \cdot 12 = 96 \)
- \( 2 \cdot 12 = 24 \)
Adding these together:
\[ 16 + 96 + 24 = 136 \]
Multiply by 2:
\[ S = 2 \cdot 136 = 272 \text{ in}^2 \]
5. How many complete boxes can be wrapped with one roll of wrapping paper?
First, we need to find the surface area of one box with dimensions:
- Length = 16 in.
- Width = 14 in.
- Height = 6 in.
Using the surface area formula for a rectangular prism:
\[ S = 2(lw + lh + wh) \]
Plugging in the dimensions:
\[ S = 2(16 \cdot 14 + 16 \cdot 6 + 14 \cdot 6) \]
Calculating each term:
- \( 16 \cdot 14 = 224 \)
- \( 16 \cdot 6 = 96 \)
- \( 14 \cdot 6 = 84 \)
Adding them together:
\[ 224 + 96 + 84 = 404 \]
Multiply by 2:
\[ S = 2 \cdot 404 = 808 \text{ in}^2 \]
Next, calculate the area of the roll of wrapping paper:
\[ \text{Area} = 25 \cdot 405 = 10125 \text{ in}^2 \]
Now, divide the total area of the wrapping paper by the surface area of one box:
\[ \text{Number of boxes} = \frac{10125}{808} \approx 12.5 \]
Since only complete boxes can be wrapped, we take the floor of this number:
\[ \text{Complete boxes} = 12 \]
Summary of Answers:
- Surface area of rectangular prism: 912 in²
- Surface area of cube: 864 in²
- Surface area of rectangular prism (15 ft edges): 1350 ft²
- Surface area of cereal box: 272 in²
- Complete boxes that can be wrapped: 12
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