If a box with a square cross section is to be sent by a delivery service, there are restrictions on its size such that its volume is given by 
V = x2(81 − 3x),
 where x is the length of each side of the cross section (in inches).
(a) Is V a function of x?
Yes, V is a function of x.
No, V is not a function of x.    
Correct: Your answer is correct.
Exactly!
(b) If 
V = V(x),
 find V(5) and V(24). (If V is not a function of x, enter DNE.)
V(5)	= 
1650
 
Correct: Your answer is correct.
 in3
V(24)	= 
5184
 
Correct: Your answer is correct.
 in3

(c) What restrictions must be placed on x (the domain) so that the problem makes physical sense? (Enter your answer using interval notation. If V is not a function of x, enter DNE.)

To find the domain restrictions for \( x \) so that the problem makes physical sense, we need to consider the constraints imposed by the equation for the volume \( V(x) = x^2(81 - 3x) \).

### Step 1: Non-Negative Volume
The volume \( V(x) \) must be non-negative, meaning:
\[
V(x) \geq 0
\]
This implies that both \( x^2 \) and \( 81 - 3x \) must be non-negative. Since \( x^2 \) is always non-negative for all real numbers, the condition \( 81 - 3x \geq 0 \) must be satisfied.

### Step 2: Solve for \( x \)
\[
81 - 3x \geq 0
\]
\[
3x \leq 81
\]
\[
x \leq 27
\]
Thus, \( x \) must be less than or equal to 27.

### Step 3: Physical Meaning of \( x \)
Since \( x \) represents the length of the side of a square cross-section, \( x \) must also be positive. Hence:
\[
x > 0
\]

### Step 4: Combine the Restrictions
The domain of \( x \) that makes physical sense is therefore:
\[
0 < x \leq 27
\]

### Final Answer in Interval Notation
The restrictions on \( x \) are given by the interval:
\[
(0, 27]
\]

This ensures that the side length \( x \) is positive and that the volume \( V(x) \) is non-negative.

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. What happens to the volume \( V(x) \) as \( x \) approaches 27?
2. How would the domain change if the equation for \( V(x) \) was \( V(x) = x^2(100 - 2x) \)?
3. Can the function \( V(x) = x^2(81 - 3x) \) have any maximum or minimum points within the interval \( (0, 27] \)?
4. How would the restrictions change if \( x \) could be any non-negative real number, not just a positive integer?
5. What is the physical interpretation of the term \( 81 - 3x \) in the context of this problem?

**Tip:** Always consider both the mathematical and physical constraints when determining the domain of a function in applied problems.

Explore how to calculate the volume of a box with a square cross section using the formula V(x) = x^2(81 - 3x). Learn about domain restrictions for x to ensure the volume makes physical sense. Suitable for students in Grades 10-12.

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