a candy box is made from a piece of cardboard that measures 39 by 21 inches. squares of equal size will be…

a candy box is made from a piece of cardboard that measures 39 by 21 inches. squares of equal size will be cut out of each corner. the sides will then be folded up to form a rectangular box. what size square should be cut from each corner to obtain maximum volume?\na square with a side of length □ inches should be cut away from each corner to obtain the maximum volume.\n(round to the nearest hundredth as needed.)

a candy box is made from a piece of cardboard that measures 39 by 21 inches. squares of equal size will be cut out of each corner. the sides will then be folded up to form a rectangular box. what size square should be cut from each corner to obtain maximum volume?\na square with a side of length □ inches should be cut away from each corner to obtain the maximum volume.\n(round to the nearest hundredth as needed.)

Answer

Explanation:

Step1: Set up the volume function

Let the side - length of the square cut from each corner be (x) inches. The length of the box is (L=(39 - 2x)) inches, the width of the box is (W=(21 - 2x)) inches, and the height of the box is (H = x) inches. The volume (V) of the box is given by the formula (V=(39 - 2x)(21 - 2x)x). Expand the function: [ \begin{align*} V&=(39\times21-39\times2x-2x\times21 + 4x^{2})x\ &=(819-78x - 42x+4x^{2})x\ &=819x-120x^{2}+4x^{3} \end{align*} ]

Step2: Find the derivative of the volume function

Differentiate (V(x)=4x^{3}-120x^{2}+819x) with respect to (x). Using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}), we have (V^\prime(x)=12x^{2}-240x + 819).

Step3: Set the derivative equal to zero

Set (V^\prime(x)=0), so (12x^{2}-240x + 819 = 0). Divide through by (3): (4x^{2}-80x + 273=0). Use the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}), where (a = 4), (b=-80), and (c = 273). [ \begin{align*} x&=\frac{80\pm\sqrt{(-80)^{2}-4\times4\times273}}{2\times4}\ &=\frac{80\pm\sqrt{6400-4368}}{8}\ &=\frac{80\pm\sqrt{2032}}{8}\ &=\frac{80\pm45.08}{8} \end{align*} ] We get (x_1=\frac{80 + 45.08}{8}=\frac{125.08}{8}=15.635) and (x_2=\frac{80 - 45.08}{8}=\frac{34.92}{8}=4.365).

Step4: Check the domain and the second - derivative

The domain of (x) is (0\lt x\lt10.5) (since (21-2x>0)). So (x = 15.635) is not in the domain. Differentiate (V^\prime(x)) to get the second - derivative (V^{\prime\prime}(x)=24x-240). When (x = 4.365), (V^{\prime\prime}(4.365)=24\times4.365-240=104.76-240=-135.24<0).

Answer:

(4.37) inches