Questions: Se tiene una caja con las siguientes medidas: Alto: x Ancho: 2-x Largo: 4-2x Determina: a) El modelo matemático V(x) para representar el volumen de la caja. b) Gráfica V(x) en los valores indicados y determina el volumen máximo que esta caja puede contener. Completa la tabla: x V(x)= (x, y) 0.6 0.9 1 1.3 1.6 2

Se tiene una caja con las siguientes medidas: Alto: x Ancho: 2-x Largo: 4-2x Determina: a) El modelo matemático V(x) para representar el volumen de la caja. b) Gráfica V(x) en los valores indicados y determina el volumen máximo que esta caja puede contener. Completa la tabla: x V(x)= (x, y) 0.6 0.9 1 1.3 1.6 2
Transcript text: Se tiene una caja con las siguientes medidas: \begin{tabular}{|l|} \hline Alto: $x$ \\ \hline Ancho: $2-x$ \\ \hline Largo: $4-2 x$ \\ \hline \end{tabular} Determina: a) El modelo matemático $V(x)$ para representar el volumen de la caja. $\square$ b) Gráfica $V(x)$ en los valores indicados y determina el volumen máximo que esta caja puede contener. Completa la tabla: \begin{tabular}{|c|l|l|} \hline$x$ & $V(x)=$ & $(x, y)$ \\ \hline 0.6 & & \\ \hline 0.9 & & \\ \hline 1 & & \\ \hline 1.3 & & \\ \hline 1.6 & & \\ \hline 2 & & \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Determinar el modelo matemático \( V(x) \)

El volumen de la caja se puede calcular como el producto de sus dimensiones: alto, ancho y largo.

\[ V(x) = x \cdot (2 - x) \cdot (4 - 2x) \]

Step 2: Simplificar la expresión de \( V(x) \)

Simplificamos la expresión para \( V(x) \):

\[ V(x) = x (2 - x) (4 - 2x) \]

Primero multiplicamos \( (2 - x) \) y \( (4 - 2x) \):

\[ (2 - x)(4 - 2x) = 8 - 4x - 2x + 2x^2 = 8 - 6x + 2x^2 \]

Luego multiplicamos por \( x \):

\[ V(x) = x (8 - 6x + 2x^2) = 8x - 6x^2 + 2x^3 \]

Step 3: Completar la tabla

Calculamos \( V(x) \) para los valores dados de \( x \):

\[ \begin{array}{|c|c|c|} \hline x & V(x) & (x, y) \\ \hline 0.6 & V(0.6) = 8(0.6) - 6(0.6)^2 + 2(0.6)^3 = 3.168 & (0.6, 3.168) \\ \hline 0.9 & V(0.9) = 8(0.9) - 6(0.9)^2 + 2(0.9)^3 = 4.374 & (0.9, 4.374) \\ \hline 1 & V(1) = 8(1) - 6(1)^2 + 2(1)^3 = 4 & (1, 4) \\ \hline 1.3 & V(1.3) = 8(1.3) - 6(1.3)^2 + 2(1.3)^3 = 3.042 & (1.3, 3.042) \\ \hline 1.6 & V(1.6) = 8(1.6) - 6(1.6)^2 + 2(1.6)^3 = 0.512 & (1.6, 0.512) \\ \hline 2 & V(2) = 8(2) - 6(2)^2 + 2(2)^3 = -8 & (2, -8) \\ \hline \end{array} \]

Final Answer

\[ V(x) = 8x - 6x^2 + 2x^3 \]

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