To solve this problem, we will use the properties of similar prisms.

Given:

• The ratio of the surface areas of prism P to prism Q is 1:3.
• The volume of prism Q is 86 cm³.

Step 1: Relating the surface area and volume

For similar prisms, the ratio of their surface areas is the square of the ratio of their corresponding linear dimensions, and the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions.

Let the linear scale factor between the prisms be $k$. Then:

• The ratio of their surface areas is $k^2$, which is 1:3.
• The ratio of their volumes is $k^3$.

Step 2: Finding the linear scale factor

Since the ratio of the surface areas is 1:3, we know: $k^2 = \frac{1}{3}$

Taking the square root of both sides: $k = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Step 3: Finding the volume ratio

The ratio of the volumes is $k^3$. Therefore: [ k^3 = \left(\frac{\sqrt{3}}{3}\right)^3 = \frac{1}{3\sqrt{3}} \times \frac{\sqrt{3}}{3} = \frac{1}{3\sqrt{3}} and