Question

a box with a hinged lid is to be made out of a rectangular piece of cardboard that measures 4 inches by 8 inches. six squares will be cut from the cardboard: one square will be cut from each of the corners, and one square will be cut from the middle of each of the 8 - inch sides. (see figure 1.) the remaining cardboard will be folded to form the box and its lid. (see figure 2.) letting x represent the side - lengths (in inches) of the squares, use the aleks graphing calculator to find the value of x that maximizes the volume enclosed by this box. then give the maximum volume. round your responses to two decimal places.
value of x that maximizes volume:
maximum volume:

Explanation:

Step1: Determine the dimensions of the box

The length of the box \(L=(8 - 3x)\) inches, the width \(W=(4 - 2x)\) inches and the height \(H = x\) inches. The volume \(V\) of the box is given by the formula \(V(x)=x(8 - 3x)(4 - 2x)\).

Step2: Expand the volume function

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Step3: Find the derivative of the volume function

Using the power - rule \((x^n)'=nx^{n - 1}\), we have \(V'(x)=18x^{2}-56x + 32\).

Step4: Set the derivative equal to zero

$$18x^{2}-56x + 32 = 0$$

Divide through by \(2\): \(9x^{2}-28x + 16=0\).

Step5: Solve the quadratic equation

The quadratic formula for \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here, \(a = 9\), \(b=-28\), \(c = 16\).

$$x=\frac{28\pm\sqrt{(-28)^{2}-4\times9\times16}}{2\times9}=\frac{28\pm\sqrt{784 - 576}}{18}=\frac{28\pm\sqrt{208}}{18}=\frac{28\pm4\sqrt{13}}{18}=\frac{14\pm2\sqrt{13}}{9}$$

\(x_1=\frac{14 + 2\sqrt{13}}{9}\approx2.49\) and \(x_2=\frac{14 - 2\sqrt{13}}{9}\approx0.73\). But \(x = 2.49\) makes \(4-2x=4 - 2\times2.49=-0.98<0\), so we discard it.

Step6: Find the maximum volume

Substitute \(x = 0.73\) into the volume function \(V(x)=6x^{3}-28x^{2}+32x\).

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Answer:

Value of \(x\) that maximizes volume: \(0.73\) in
Maximum volume: \(9.71\) in³