The surface area of a prism is 54 square inches. What is the surface area of a similar prism that is smaller by a scale of 1/3?

since each dimension is 1/3 as big, the volume (product of all three dimensions) is 1/27 as much.

To find the surface area of a similar prism that is smaller by a scale of 1/3, we need to use the concept of similarity. If the dimensions of the original prism are scaled down by a factor of 1/3, then each side of the smaller prism will be 1/3 of the length of the corresponding side of the original prism.

Let's assume the original prism has length (L), width (W), and height (H). The surface area of the original prism can be calculated using the formula:

Surface Area = 2(LW + LH + WH)

Given that the surface area of the original prism is 54 square inches, we can substitute this value into the formula:

54 = 2(LW + LH + WH)

Now, let's consider the smaller prism that is scaled down by a factor of 1/3. The dimensions of the smaller prism will be L/3, W/3, and H/3. We want to find the surface area of this smaller prism.

Using the formula for surface area, the surface area of the smaller prism is:

Surface Area of Smaller Prism = 2((L/3)(W/3) + (L/3)(H/3) + (W/3)(H/3))
= 2(LW/9 + LH/9 + WH/9)
= 2(LW + LH + WH)/9

So, the surface area of the smaller prism is 2(LW + LH + WH)/9. Now, we need to find this value using the given information.

Since the surface area of the original prism is 54 square inches, we can substitute this value into the equation:

54 = 2(LW + LH + WH)

Now, solving for (LW + LH + WH):

(LW + LH + WH) = 54/2
(LW + LH + WH) = 27

Substituting this value into the equation for the surface area of the smaller prism:

Surface Area of Smaller Prism = 2(27)/9
= 54/9
= 6 square inches

Therefore, the surface area of the smaller prism, which is scaled down by 1/3, is 6 square inches.