Calvin wants to make a opentop box He will cut out the corne

Calvin wants to make a open-top box. He will cut out the corners of a rectangular piece of cardboard ((14 feet by 8 feet) and fold up the sides. What is the domain of the volume function? what is the maximum volume of the box? Please give a explanation on how you find the domain, I already know the answers but I don\'t understand how to find the domain!

Solution

He will cut out the corners of a rectangular piece of cardboard ((14 feet by 8 feet) and fold up the sides

If x ft is cut from corners then length = 14 -x-x = 14 -2x

width = 8 -x -x = 8 -2x

Volume , V(x) = (14 -2x)(8-2x)*x

Its a polynomial of 3rd degree with no denominator terms , so as to give an asymtote.Therfore,

domain would be ( -inf , inf) for function general

However it function is volume so, V(x) >0 whch gives

Domain : ( 0, 4 )

To find maximum : find V\'(x) = 12x^2 - 88x + 112

Critical points : 12x^2 - 88x + 112 =0 , solve for x

x = ( 11 - sqrt37)/3 , ( 11+ sqrt37)/3

Plug x= ( 11 - sqrt37)/3 = , ( 11+ sqrt37)/3 in V(x) to check for maximum

x = ( 11 - sqrt37)/3   = 1.639 ; x = ( 11 + sqrt37)/3   = 5.694

V(1.639) = 82.981 ft^3 ; V( 5.694) = -50.389

So, maximum volume = 82.981 ft^3