To find the volume of the statue, we’ll use the fact that the statue and the model are mathematically similar. Here’s the process:

Step 1: Find the Ratio of the Surface Areas

The surface area of the statue is $3600 \, \text{cm}^2$, and the surface area of the model is $625 \, \text{cm}^2$.

Since they are similar, the ratio of the areas (surface areas) is given by: $\text{Area Ratio} = \frac{\text{Surface Area of Statue}}{\text{Surface Area of Model}} = \frac{3600}{625}$

Simplifying this, we get: $\text{Area Ratio} = \frac{3600 \div 25}{625 \div 25} = \frac{144}{25}$

So, the area ratio is $\frac{144}{25}$.

Step 2: Determine the Ratio of the Lengths

The area ratio for similar figures is the square of the length ratio. So, if the length ratio is $k$, then: $k^2 = \frac{144}{25}$

To find $k$, take the square root of both sides: $k = \sqrt{\frac{144}{25}} = \frac{\sqrt{144}}{\sqrt{25}} = \frac{12}{5}$

Thus, the length ratio is $\frac{12}{5}$.

Step 3: Determine the Ratio of the Volumes

The volume ratio for similar figures is the cube of the length ratio. So: $\text{Volume Ratio} = k^3 = \left(\frac{12}{5}\right)^3$

Calculating this: $\text{Volume Ratio} = \frac{12^3}{5^3} = \frac{1728}{125}$

Step 4: Find the Volume of the Statue

Now, we know the volume of the model is $750 \, \text{cm}^3$, and we can use the volume ratio to find the volume of the statue: $\text{Volume of Statue} = \text{Volume of Model} \times \frac{1728}{125}$

Calculating this: $\text{Volume of Statue} = 750 \times \frac{1728}{125} = 10368 \, \text{cm}^3$

Final Answer

The volume of the statue is: $\boxed{10368 \, \text{cm}^3}$

Would you like more details or have any questions?

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Related Questions

1. If the length of the model is 20 cm, what is the length of the statue?
2. How would the volume change if the model’s surface area was 500 cm² instead of 625 cm²?
3. What would the volume of the statue be if the model’s volume were doubled?
4. What is the volume ratio if the surface area of the model is 900 cm² and the statue remains the same?
5. How does the area ratio relate to the volume ratio in similar geometric objects?

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Tip: For similar objects, remember that area scales by the square of the length ratio, and volume scales by the cube of the length ratio.