Question

Given a rectangular prism labeled A with the dimensions l = 2, w = 4, and h = 6, and a second prism labeled B with the dimensions l = 6, w = 12, and h = 18, how many times larger will the surface area of prism B be than prism A?

Prism B's surface area is __ times larger than the surface area of prism A.

Answer 1

To find how many times larger the surface area of prism B is compared to prism A, we first need to calculate the surface area of both prisms.

The formula for the surface area \( S \) of a rectangular prism with length \( l \), width \( w \), and height \( h \) is given by:

\[
S = 2(lw + lh + wh)
\]

### Calculating the Surface Area of Prism A:
For prism A:
- \( l = 2 \)
- \( w = 4 \)
- \( h = 6 \)

Calculating each part:
1. \( lw = 2 \times 4 = 8 \)
2. \( lh = 2 \times 6 = 12 \)
3. \( wh = 4 \times 6 = 24 \)

Now add these areas together:
\[
lw + lh + wh = 8 + 12 + 24 = 44
\]

Now calculate the surface area:
\[
S_A = 2 \times 44 = 88
\]

### Calculating the Surface Area of Prism B:
For prism B:
- \( l = 6 \)
- \( w = 12 \)
- \( h = 18 \)

Calculating each part:
1. \( lw = 6 \times 12 = 72 \)
2. \( lh = 6 \times 18 = 108 \)
3. \( wh = 12 \times 18 = 216 \)

Now add these areas together:
\[
lw + lh + wh = 72 + 108 + 216 = 396
\]

Now calculate the surface area:
\[
S_B = 2 \times 396 = 792
\]

### Comparing the Surface Areas:
Now, we can find how many times larger the surface area of prism B is compared to prism A:
\[
\text{Ratio} = \frac{S_B}{S_A} = \frac{792}{88}
\]

Calculating this gives:
\[
\frac{792}{88} = 9
\]

### Conclusion:
Prism B's surface area is **9 times larger** than the surface area of prism A. Thus, the final answer is:

Prism B's surface area is **9 times larger** than the surface area of prism A.