If a box with a square cross section is to be sent by the postal service, there are restrictions on its size such that its volume is limited by

*V* = *x*2(182 − 7*x*),

where *x* is the length of each side of the cross section (in inches).

(a) If *V* = *V*(*x*), find *V*(10) and *V*(20).

*V*(10)

=

*V*(20)

=

(b) What restrictions must be placed on *x* (the domain) so that the problem makes physical sense? (Enter your answer using interval notation.)

We are given the volume equation for a box with a square cross-section, \( V(x) = x^2(182 - 7x) \), where \( x \) is the length of each side of the square cross-section in inches.

### Part (a): Finding \( V(10) \) and \( V(20) \)

We need to find the volume of the box when \( x = 10 \) and when \( x = 20 \).

#### Finding \( V(10) \):
\[
V(10) = 10^2(182 - 7 \cdot 10) = 100 \cdot (182 - 70) = 100 \cdot 112 = 11200 \, \text{in}^3
\]

#### Finding \( V(20) \):
\[
V(20) = 20^2(182 - 7 \cdot 20) = 400 \cdot (182 - 140) = 400 \cdot 42 = 16800 \, \text{in}^3
\]

So,
\[
V(10) = 11200 \, \text{in}^3, \quad V(20) = 16800 \, \text{in}^3
\]

### Part (b): Domain Restrictions on \( x \)

For the problem to make physical sense:
1. The side length \( x \) of the cross-section must be non-negative, meaning \( x \geq 0 \).
2. The volume must also be positive, which means \( V(x) > 0 \).

From the equation \( V(x) = x^2(182 - 7x) \), we see that the term \( 182 - 7x \) must also be positive for the volume to be greater than zero.

So, \( 182 - 7x > 0 \), which gives:
\[
x < \frac{182}{7} \approx 26
\]

Thus, \( x \) must be between 0 and 26 (not including 26 because the volume becomes zero at \( x = 26 \)).

Therefore, the domain for \( x \) in interval notation is:
\[
x \in (0, 26)
\]

### Final Answers:
(a)
- \( V(10) = 11200 \, \text{in}^3 \)
- \( V(20) = 16800 \, \text{in}^3 \)

(b) The domain for \( x \) is \( (0, 26) \).

---

Would you like more details or have any questions?

Here are 5 related questions:
1. How can we find the maximum volume the box can have?
2. What happens to the volume if \( x \) approaches 26?
3. How would you find the critical points of \( V(x) \) to maximize the volume?
4. What is the volume when \( x = 0 \)?
5. How does the volume change as \( x \) increases?

**Tip:** To maximize the volume, we can take the derivative of the volume function \( V(x) \) and find where it equals zero!

If a box with a square cross section is to be sent by the postal service, there are restrictions on its size such that its volume is limited by V = x^2(182 − 7x), where x is the length of each side of the cross section (in inches). (a) If V = V(x), find V(10) and V(20). (b) What restrictions must be placed on x (the domain) so that the problem makes physical sense? (Enter your answer using interval notation.)

Explore how to calculate the volume of a box with a square cross section using the function V(x) = x^2(182 - 7x). Learn how to find specific values like V(10) = 11200 in³ and V(20) = 16800 in³, and understand the domain restrictions for x in interval notation (0, 26). Suitable for high school students.

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