Surface Area Calculation: Effect of Tripling Width on Rectangular Prism

To solve this problem, let's follow these steps:

• Step 1: Calculate the original surface area of the prism.
• Step 2: Calculate the surface area after tripling the width.
• Step 3: Determine the change in surface area.

First, the original surface area of the rectangular prism is given by the formula:
$SA = 2(lw + lh + wh)$.

Step 1: Substitute the original dimensions:
$SA_{\text{original}} = 2(lw + lh + wh)$.

Step 2: Now, when we triple the width, the new width is $3w$. Substitute $3w$ into the surface area formula:
$SA_{\text{new}} = 2(l(3w) + lh + (3w)h)$.
This simplifies to:
$SA_{\text{new}} = 2(3lw + lh + 3wh) = 6lw + 2lh + 6wh$.

Step 3: Subtract the original surface area from the new one to find the change:
$\Delta SA = SA_{\text{new}} - SA_{\text{original}} = (6lw + 2lh + 6wh) - (2lw + 2lh + 2wh)$.
Thus, $\Delta SA = 4lw + 4wh$.

This change can be factorized further as:
$\Delta SA = 4(l(w + h))$.

Therefore, the surface area will increase by $4(w + h)l$.

Thus, the correct answer is: It will increase by $(w + h) \cdot l \cdot 4$. This is choice 3 and 4.