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 cm^3. calculate the volume of prism P.

To solve for the volume of prism P given that the ratio of the surface areas of prisms P and Q is \(1:3\) and the volume of prism Q is \(86 \, \text{cm}^3\), we can follow these steps:

### Step 1: Determine the ratio of the linear dimensions
As mentioned earlier, the surface area ratio between two similar prisms is the square of the ratio of their corresponding linear dimensions.

Given:
\[
\frac{\text{Surface Area of P}}{\text{Surface Area of Q}} = \frac{1}{3}
\]
Let \( r \) be the ratio of the corresponding linear dimensions of the two prisms.

Then:
\[
r^2 = \frac{1}{3}
\]
Taking the square root of both sides:
\[
r = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]

### Step 2: Determine the ratio of the volumes
The volume ratio between two similar prisms is the cube of the ratio of their corresponding linear dimensions.

So, the volume ratio is:
\[
\left(\frac{\text{Volume of P}}{\text{Volume of Q}}\right) = r^3
\]
Substitute \( r = \frac{\sqrt{3}}{3} \) into the equation:
\[
r^3 = \left(\frac{\sqrt{3}}{3}\right)^3 = \frac{(\sqrt{3})^3}{3^3} = \frac{3\sqrt{3}}{27} = \frac{\sqrt{3}}{9}
\]

### Step 3: Calculate the volume of prism P
Given that the volume of prism Q is \( 86 \, \text{cm}^3 \), we can use the volume ratio to find the volume of prism P.

Let the volume of prism P be \( V_P \):
\[
\frac{V_P}{86} = \frac{\sqrt{3}}{9}
\]

So:
\[
V_P = 86 \times \frac{\sqrt{3}}{9} = \frac{86\sqrt{3}}{9} \approx \frac{86 \times 1.732}{9} \approx \frac{148.952}{9} \approx 16.55 \, \text{cm}^3
\]

### Final Answer:
The volume of prism P is approximately \( 16.55 \, \text{cm}^3 \).

Would you like to explore further or clarify anything about this problem? Here are some related questions:

1. How would the volume change if the surface area ratio was 1:4?
2. What is the relationship between the surface area ratio and the volume ratio for any similar shapes?
3. How does the volume of a prism change if its height is doubled?
4. Can you explain why the volume ratio is the cube of the linear dimension ratio?
5. How would the surface area change if the linear dimensions of a prism were halved?

**Tip:** Remember that when dealing with similar shapes, different properties (such as surface area and volume) scale differently with the ratio of their linear dimensions.

Learn how to calculate the volume of prism P when prism P and prism Q are similar, the surface area ratio is 1:3, and the volume of prism Q is 86 cm^3. Follow step-by-step calculations and understand the relationship between surface area and volume ratios for similar prisms. Suitable for Grades 7-9.

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