Prism P and prism Q are similar.
The ratio of the surface area of prism P to the surface area of prism Q is 1 : 3.The volume of prism Q is 86cm cubed. Calculate the volume of prism P.

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

Prism P and prism Q are similar. The ratio of the surface area of prism P to the surface area of prism Q is 1:3. The volume of prism Q is 86 cm³. Calculate the volume of prism P.

Learn how to calculate the volume of prism P given the surface area ratio of 1:3 between similar prisms P and Q and the volume of prism Q as 86 cm³. This problem involves concepts of similar solids, surface area, and volume ratios. Suitable for students in Grades 9-11.

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