Questions: 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 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. (b) If V=V(x), find V(11) and V(23). (If V is not a function of x, enter DNE.) V(11)=3872 V(23)=4232 in^3 (c) What restrictions must be placed on x (the domain) so that the problem makes physical sense? (Enter your answer)

Solution Steps

To solve the given problem, we need to follow these steps:

1. Part (a): Determine if $V$ is a function of $x$. Since the volume $V$ of a box with a square cross-section can be expressed in terms of $x$, $V$ is indeed a function of $x$.

2. Part (b): Calculate $V(11)$ and $V(23)$. Assuming the volume formula is $V = x^2 \cdot h$ where $h$ is the height, we need to know $h$ to compute these values. If $h$ is not given, we cannot compute these values directly.

3. Part (c): Determine the restrictions on $x$ for the problem to make physical sense. Since $x$ represents a length, it must be a positive value. Additionally, there might be upper limits based on the delivery service's size restrictions.

Since the volume $V$ of a box with a square cross-section can be expressed in terms of $x$ (the side length of the square base), $V$ is indeed a function of $x$.

Given the volume formula $V = x^2 \cdot h$ and assuming the height $h = 10$ inches:

• For $x = 11$: $$V(11) = 11^2 \cdot 10 = 1210 \, \text{in}^3$$
• For $x = 23$: $$V(23) = 23^2 \cdot 10 = 5290 \, \text{in}^3$$

For the problem to make physical sense, $x$ must be a positive value. Additionally, $x$ must be within the delivery service's size restrictions. Therefore, the domain of $x$ is: $$x > 0 \quad \text{and} \quad x \text{ must be within the delivery service's size restrictions}$$

Final Answer