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?

Answer 1

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 for both prisms.

The formula for the surface area \( S \) of a rectangular prism is given by:

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

**Calculating the surface area of prism A:**
- Dimensions of prism A:
- Length \( l = 2 \)
- Width \( w = 4 \)
- Height \( h = 6 \)

Using the formula:

\[
S_A = 2(lw + lh + wh) = 2(2 \times 4 + 2 \times 6 + 4 \times 6)
\]

Calculating each term:

- \( lw = 2 \times 4 = 8 \)
- \( lh = 2 \times 6 = 12 \)
- \( wh = 4 \times 6 = 24 \)

Now substituting back into the formula:

\[
S_A = 2(8 + 12 + 24) = 2(44) = 88
\]

**Calculating the surface area of prism B:**
- Dimensions of prism B:
- Length \( l = 6 \)
- Width \( w = 12 \)
- Height \( h = 18 \)

Using the same formula:

\[
S_B = 2(lw + lh + wh) = 2(6 \times 12 + 6 \times 18 + 12 \times 18)
\]

Calculating each term:

- \( lw = 6 \times 12 = 72 \)
- \( lh = 6 \times 18 = 108 \)
- \( wh = 12 \times 18 = 216 \)

Now substituting back into the formula:

\[
S_B = 2(72 + 108 + 216) = 2(396) = 792
\]

**Finding how many times larger the surface area of prism B is than prism A:**

Now we can determine how many times larger \( S_B \) is compared to \( S_A \):

\[
\text{Ratio} = \frac{S_B}{S_A} = \frac{792}{88} = 9
\]

Thus, the surface area of prism B is **9 times larger** than the surface area of prism A.