To find out how many times larger the surface area of prism B is compared to prism A, we first need to calculate the surface area of each prism using the formula for the surface area of a rectangular prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Step 1: Calculate the surface area of prism A
For prism A with dimensions \( l = 2, w = 4, h = 6 \):
\[ \text{Surface Area}_A = 2(lw + lh + wh) \]
Calculating each term:
- \( lw = 2 \times 4 = 8 \)
- \( lh = 2 \times 6 = 12 \)
- \( wh = 4 \times 6 = 24 \)
Now, summing these values:
\[ lw + lh + wh = 8 + 12 + 24 = 44 \]
Calculating the total surface area:
\[ \text{Surface Area}_A = 2 \times 44 = 88 \]
Step 2: Calculate the surface area of prism B
For prism B with dimensions \( l = 6, w = 12, h = 18 \):
\[ \text{Surface Area}_B = 2(lw + lh + wh) \]
Calculating each term:
- \( lw = 6 \times 12 = 72 \)
- \( lh = 6 \times 18 = 108 \)
- \( wh = 12 \times 18 = 216 \)
Now, summing these values:
\[ lw + lh + wh = 72 + 108 + 216 = 396 \]
Calculating the total surface area:
\[ \text{Surface Area}_B = 2 \times 396 = 792 \]
Step 3: Find how many times larger the surface area of prism B is than prism A
Now we can compare the surface areas:
\[ \text{Times larger} = \frac{\text{Surface Area}_B}{\text{Surface Area}_A} = \frac{792}{88} = 9 \]
Therefore, the surface area of prism B is 9 times larger than the surface area of prism A.
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