Question

An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What should be the length of the side x of the square cut out of each corner to get a box with the maximum volume

Answer 1

Answer:

x  = 7.45 inch for the maximum volume.

The area of the square = x² = 7.45² = 55.5025  inch²

Step-by-step explanation:

From the given information:

An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side.

The objective is to determine the length of the side x of the square cut out of each corner to get a box with the maximum volume

The volume of the box = l×b×h

The volume of the box = $(42 - 2x) \times (48-2x) \times (x)$

The volume of the box = $(2016 - 84x - 96x +4x^2)x$

The volume of the box = $(2016 -180x+4x^2)x$

The volume of the box = $(2016x -180x^2+4x^3)$

The volume of the box = $4x^3 - 180x^2 +2016x$

For the maximum volume V' = 0

V' = $12x^2 - 360x + 2016$

Using the quadratic formula; we have:

$= \dfrac{-b \pm \sqrt{b^2 -4ac}}{2a}$

where;

a = 12 , b = -360 c = 2016

$= \dfrac{-(-360) \pm \sqrt{(-360)^2 -4(12)(2016)}}{2(12)}$

$= \dfrac{360 \pm \sqrt{129600 -96768}}{24}$

$= \dfrac{360 \pm \sqrt{32832}}{24}$

$= \dfrac{360 \pm 181.196}{24}$

$= \dfrac{360 + 181.196}{24} \ \ \ OR \ \ \ \dfrac{360 - 181.196}{24}$

$= \dfrac{541.196}{24} \ \ \ OR \ \ \ \dfrac{178.804}{24}$

$= 22.55 \ \ \ OR \ \ \ 7.45$

For the maximum value , we check the points in the second derivative term

V'' = 24x - 360

V'' ( 22.55) = 24(22.55) - 360

V'' ( 22.55) = 541.2 - 360  

V'' ( 22.55) = 181.2   (minimum)

V'' ( 7.45) = 24(7.45) - 360

V'' ( 7.45) = 178.8 - 360  

V'' ( 7.45) = -181.2  < 0  (maximum)

Therefore, x  = 7.45 inch for the maximum volume.

The area of the square = x² = 7.45² = 55.5025  inch²