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Let's break down the problem in the image step by step.

We are given the dimensions of an open rectangular box:
- Length: \( 2x + 3 \)
- Width: \( x + 1 \)
- Height: \( x \)

### Part (a): Expression for Volume \( V(x) \)
The volume \( V(x) \) of a rectangular box is given by the formula:
\[
V(x) = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the given dimensions:
\[
V(x) = (2x + 3)(x + 1)(x)
\]

### Part (b): Expression for Surface Area \( S(x) \)
Since the box is open, it has no top. The surface area of the box is the sum of the areas of the five remaining faces:
- Two side faces: \( 2 \times \text{Height} \times \text{Length} = 2(x)(2x + 3) \)
- Two other side faces: \( 2 \times \text{Height} \times \text{Width} = 2(x)(x + 1) \)
- Bottom face: \( \text{Length} \times \text{Width} = (2x + 3)(x + 1) \)

Thus, the surface area \( S(x) \) is:
\[
S(x) = 2(x)(2x + 3) + 2(x)(x + 1) + (2x + 3)(x + 1)
\]

### Part (c): Values of \( V(x) \) and \( S(x) \) when \( x = 5 \)
Now we can substitute \( x = 5 \) into the expressions for \( V(x) \) and \( S(x) \).

Let me calculate these values.### Results:

1. **Expression for Volume \( V(x) \):**
   \[
   V(x) = 2x^3 + 5x^2 + 3x
   \]

2. **Expression for Surface Area \( S(x) \):**
   \[
   S(x) = 8x^2 + 13x + 3
   \]

3. **When \( x = 5 \):**
   - \( V(5) = 390 \, \text{cm}^3 \)
   - \( S(5) = 268 \, \text{cm}^2 \)

Let me know if you'd like more details or have further questions.

Here are some related questions you can consider:
1. How would the expressions change if the box had a top?
2. What is the volume when \( x = 3 \)?
3. How do we find the critical points for the volume and surface area?
4. Can you find the rate of change of volume with respect to \( x \)?
5. What is the effect of increasing \( x \) on the surface area?

**Tip:** Always double-check units when calculating volume and surface area to ensure consistency.

The dimensions (in cm) of an open rectangular box are given in the diagram. Find (a) an expression for the volume, V(x), of the box, (b) an expression for the external surface area, S(x), of the box, (c) the value of (i) V(x) and (ii) S(x) when x = 5.

Learn how to find the expression for volume and surface area of an open rectangular box with dimensions based on x. This problem covers key algebraic and geometric concepts. Example values are calculated for x = 5.

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