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)

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)

Transcript text: 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 volum section (in inches). (a) Is $V$ a function of $x$ ? $Y e s, V$ 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.) \[ \begin{array}{l} V(11)=3872 \\ V(23)=4232 \end{array} \] \[ 7 l \begin{array}{ll} & \mathrm{in}^{3} \\ \mathrm{in} 3 \end{array} \] (c) What restrictions must be placed on $x$ (the domain) so that the problem makes physical sense? (Enter your ans

Solution

Solution Steps

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

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 $.
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.
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.

Step 1: 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 $ (the side length of the square base), $ V $ is indeed a function of $ x $.

Step 2: Calculate $ V(11) $ and $ V(23) $

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 $: \[ V(23) = 23^2 \cdot 10 = 5290 \, \text{in}^3 \]

Step 3: Determine the restrictions on $ x $

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: \[ x > 0 \quad \text{and} \quad x \text{ must be within the delivery service's size restrictions} \]