If the dimensions of a rectangular prism are doubled, how does that affect the surface area and volume?

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When the dimensions of a **rectangular prism** are doubled, both its **surface area** and **volume** increase, but they do so by different factors. Here's a detailed breakdown:

### Original Dimensions:
- **Length (l)**
- **Width (w)**
- **Height (h)**

### Original Surface Area and Volume:
- **Surface Area (SA):**
  $$
  SA = 2(lw + lh + wh)
  $$
- **Volume (V):**
  $$
  V = l \times w \times h
  $$

### After Doubling the Dimensions:
- **New Length (2l)**
- **New Width (2w)**
- **New Height (2h)**

#### New Surface Area:
$$
\begin{align*}
SA_{\text{new}} &= 2[(2l)(2w) + (2l)(2h) + (2w)(2h)] \
&= 2[4lw + 4lh + 4wh] \
&= 2 \times 4(lw + lh + wh) \
&= 8(lw + lh + wh) \
&= 4 \times [2(lw + lh + wh)] \
&= 4 \times SA_{\text{original}}
\end{align*}
$$
- **Result:** The **surface area** is **multiplied by 4**.

#### New Volume:
$$
\begin{align*}
V_{\text{new}} &= (2l) \times (2w) \times (2h) \
&= 8 \times (l \times w \times h) \
&= 8 \times V_{\text{original}}
\end{align*}
$$
- **Result:** The **volume** is **multiplied by 8**.

### Summary:
- **Surface Area:** Increases by a factor of **4**.
- **Volume:** Increases by a factor of **8**.

### Example:
Suppose you have a rectangular prism with dimensions:
- **Length:** 2 units
- **Width:** 3 units
- **Height:** 4 units

**Original Surface Area:**
$$
SA = 2(2 \times 3 + 2 \times 4 + 3 \times 4) = 2(6 + 8 + 12) = 2 \times 26 = 52 \text{ square units}
$$

**Original Volume:**
$$
V = 2 \times 3 \times 4 = 24 \text{ cubic units}
$$

**After Doubling Dimensions (4, 6, 8):**

**New Surface Area:**
$$
SA_{\text{new}} = 2(4 \times 6 + 4 \times 8 + 6 \times 8) = 2(24 + 32 + 48) = 2 \times 104 = 208 \text{ square units} \quad (\text{which is } 4 \times 52)
$$

**New Volume:**
$$
V_{\text{new}} = 4 \times 6 \times 8 = 192 \text{ cubic units} \quad (\text{which is } 8 \times 24)
$$

### Conclusion:
Doubling each dimension of a rectangular prism **quadruples its surface area** and **increases its volume eightfold**.
Doubling each dimension of a rectangular prism increases its surface area by 4 times and its volume by 8 times.

If the dimensions of a rectangular prism are doubled, how does that affect the surface area and volume?

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