An open box of maximum volume is to be made from a square...

An open box of maximum volume is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum volume. \begin{tabular}{|c|c|c|} \hline Height & Length and Width & Volume \\ \hline 1 & $24-2(1)$ & $1[24-2(1)]^{2}=484$ \\ \hline 2 & $24-2(2)$ & $2[24-2(2)]^{2}=800$ \\ \hline \end{tabular} (b) Write the volume $V$ as a function of $x$. (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

An open box of maximum volume is to be made from a square piece of material, 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure).
(a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum volume.
\begin{tabular}{|c|c|c|}
\hline Height & Length and Width & Volume \\
\hline 1 & $24-2(1)$ & $1[24-2(1)]^{2}=484$ \\
\hline 2 & $24-2(2)$ & $2[24-2(2)]^{2}=800$ \\
\hline
\end{tabular}
(b) Write the volume $V$ as a function of $x$.
(c) Use calculus to find the critical number of the function in part (b) and find the maximum value.
(d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

Key Concepts

Optimization

Optimization involves finding the best possible solution, such as maximum or minimum values, for a given function under specified constraints. In this context, it pertains to maximizing the volume of a box by altering design parameters such as the dimensions of the cuts made from the original material.

Function Modeling

Function modeling is the process of translating a real-world scenario into a mathematical function. It involves identifying the relevant variables and constructing a formula—in this case, the volume of the box expressed in terms of the size of the cut squares—that accurately represents the situation.

Geometric Relationships

Geometric relationships focus on understanding how the physical dimensions of an object are interconnected. Here, the dimensions of the box are derived from the original square piece of material, considering how cutting and folding operations alter length, width, and height to form the final box.

Domain Restrictions

Domain restrictions involve determining the set of possible values for the variables based on practical or physical limits of the problem. In the context of forming a box by cutting a square, the size of the cut squares is limited by the dimensions of the original material, ensuring that all resulting lengths remain positive.

Differentiation

Differentiation is a key calculus technique used to analyze how a function changes and to identify rates of change. It is applied here to the volume function to find where the slope equals zero, indicating potential maximum or minimum values.

Critical Point Analysis

Critical point analysis involves finding and testing the points at which the derivative of a function is zero or undefined. These points are candidates for local maxima or minima, and testing these points helps determine the dimensions at which the volume of the box is maximized.

Graphical Analysis

Graphical analysis uses graphing tools to visualize functions, providing an alternative method to confirm analytical findings. By plotting the volume function, one can visually inspect the behavior of the function and verify that the calculated critical point indeed represents the maximum volume.

Transcript

00:04 In this question, we're given a scenario where one square is cut off from each of the corners of a material that is also square in order to form a box by folding up the sides.

00:27 The side length of one of the smaller squares is x, and the side length of the large square that we start with is 24.

00:37 We want to find the maximum possible volume, v, and the x that is used to achieve that maximum volume.

00:50 So first of all, we write a function that we want to maximize in terms of the variable we want to use.

01:00 So we have a volume function in terms of x.

01:06 Since the volume of our box is its length times its width times its height, this will be 24 minus 2x times 24 minus 2x, which are each obtained by taking x away from each side of the 24, each end, i should say, and then that multiplied by the height, which will be x, since these sides are folded up.

01:50 Now we want to maximize this function.

01:53 So recall that when we're dealing with optimization, the possible maximum values either occur at critical points or at endpoints of a domain.

02:05 So first of all, do we have a domain of interest in this question? well, yes, because x cannot be any real number for this to be a realistic situation.

02:16 It must first of all be non -negative, obviously.

02:19 We can't cut off a negative side length, and we can't cut off a side length that will go past half of the original side length, or even up to the half.

02:35 So it can't be equal to half either.

02:37 It must be strictly less.

02:44 So these inequalities are x being greater than or equal to zero, and also x being less than 12.

03:05 So now to find the maximum value of v, let's find its value at each of the possible, each of the candidates for maximum, that is, the endpoints that are included in the domain, which in this case is just zero, and any critical points.

03:28 So first of all, let's find the critical points by differentiating v of x.

03:47 So notice i've expanded v of x to make it easier to differentiate.

03:53 Now i don't have to use chain or product rules, just the usual polynomial differentiation rule.

04:28 Okay, so to determine the critical points, we need to find where this equation is equal to zero.

04:39 It's quite easy to factor this quadratic polynomial.

04:45 We just have to find two numbers that multiply to 48 and add up to negative 16.

04:52 Now 48 has factors of 48 and 1, 24 and 4, neither of whose sums add up to something like 16.

05:02 But then we encounter 12 and 4, which add up to 16...

Please give Ace some feedback