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A carton is such that the sum of its height h, length x, and width y equals 108 inches. The...

Question:

A carton is such that the sum of its height {eq}h {/eq}, length {eq}x {/eq}, and width {eq}y {/eq} equals {eq}108 {/eq} inches. The carton has a square base.

We wish to maximize the volume of the box. To find the dimensions of the carton with maximum volume, what function do we wish to maximize?

A) {eq}\displaystyle\; V(x) = 108x^{2} {/eq}

B) {eq}\displaystyle\; V(x) = 108x^{2} - 2x^{3} {/eq}

C) {eq}\displaystyle\; V(x) = 432x^{3} {/eq}

D) {eq}\displaystyle\; V(x) = x^{2} + \frac{108}{2x} {/eq}

E) {eq}\displaystyle\; V(x) = 108 - 4x {/eq}

Maximize the Volume of a Cartoon:

A three-dimensional cartoon box with its three dimensions length, width and height are given. In this problem, we have to uncover the volume function that we have to maximize. Before maximizing the function, Calculus tells us that we have to transform the function into a single variable. For this transformation, we have need some initial conditions. In this problem, the initial conditions are the sum of the three dimensions of the cartoon equals {eq}108 {/eq} and the base of the cartoon is square which are used to transform the volume function into a single variable function.

Answer and Explanation:

We are given:

The length, width and height of a cartoon are {eq}x \, , \, y \, \text{and} \, h {/eq} respectively.

And:

{eq}x + y + h = 108 \hspace{1 cm} \text{(Equation 1)} {/eq}

Since the base of the cartoon is a square so we can say its length is equal to its width.

Hence:

{eq}x = y {/eq}


From (Equation 1), we can also write:

{eq}x + x + h = 108 \hspace{1 cm} \left[ \because x = y \right] \\ \Rightarrow 2x + h = 108 \\ \Rightarrow h = 108 - 2x {/eq}


Now:

Volume of the cartoon is:

{eq}\begin{align*} V &= xyh \\ &= x \times x \times \left( 108 - 2x \right) \hspace{1 cm} \left[ \because x = y \,\, \text{ and } \,\, h = 108 - 2x \right] \\ &= x^2 \left( 108 - 2x \right) \\ &= 108x^2 - 2x^3 \end{align*} {/eq}


Hence:

To find the dimensions of the cartoon with maximum volume, we have to maximize the following volume function:

{eq}V(x) =108x^2 - 2x^3 {/eq}

So:

Option (B) is correct.


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