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.