8. A cylinder is similar to the one above with a scale factor of $$ 1: 4 $$. Find the volume $$ \& $$ surface area of the new cylinder.

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To determine the **volume** and **surface area** of a new cylinder that is similar to an original cylinder with a **scale factor of $$1:4$$**, we can use the principles of similarity in geometry.

### Understanding Scale Factors in Similar Figures

When two figures are similar, their corresponding linear dimensions are proportional by a scale factor. For three-dimensional shapes like cylinders:

- **Linear dimensions (e.g., radius, height):** Scale by the factor $$ k $$.
- **Surface area:** Scales by $$ k^2 $$.
- **Volume:** Scales by $$ k^3 $$.

Given a **scale factor of $$1:4$$**:
- $$ k = 4 $$ (since the new cylinder is larger).

### Applying the Scale Factor

1. **Volume:**

The volume of similar three-dimensional shapes scales with the cube of the scale factor.

$$
   \text{Scale Factor for Volume} = k^3 = 4^3 = 64
   $$

$$
   \text{Volume of New Cylinder} = 64 \times \text{Volume of Original Cylinder}
   $$

2. **Surface Area:**

The surface area scales with the square of the scale factor.

$$
   \text{Scale Factor for Surface Area} = k^2 = 4^2 = 16
   $$

$$
   \text{Surface Area of New Cylinder} = 16 \times \text{Surface Area of Original Cylinder}
   $$

### Summary

- **Volume of the New Cylinder:**  
  $$ \text{Volume}_{\text{new}} = 64 \times \text{Volume}_{\text{original}} $$

- **Surface Area of the New Cylinder:**  
  $$ \text{Surface Area}_{\text{new}} = 16 \times \text{Surface Area}_{\text{original}} $$

**Note:** To find numerical values, you would multiply the original cylinder's volume and surface area by 64 and 16, respectively.

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If you have the specific dimensions of the original cylinder, you can plug them into the formulas for the volume and surface area to get exact numerical answers for the new cylinder.
To find the volume and surface area of a new cylinder similar to the original with a scale factor of 1:4, multiply the original cylinder's volume by 64 and its surface area by 16.

A cylinder is similar to the one above with a scale factor of .
Find the volume surface area of the new cylinder.

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A cylinder is similar to the one above with a scale factor of . Find the volume surface area of the new cylinder.