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A box without a top is to be made from a rectangular piece of cardboard, with dimensions 8 in. by 10 in., by cutting out square corners with side length x and folding up the sides.



(a)   Write an equation for the volume V of the box in terms of x.

(b)   Use technology to estimate the value of x, to the nearest tenth, that gives the greatest volume. Explain your process.


My Answers?: 

(a) Equation for volume (v):





Thank You For Helping Me!!!

 May 10, 2018


volume  =  (length)(width)(height)


length  =   10 - x - x   =   10 - 2x
width=   8 - x - x   =   8 - 2x
height=   x


volume  =  (10 - 2x)(8 - 2x)( x )

volume  =  (80 - 36x + 4x2)( x )

volume  =  80x - 36x2 + 4x3

volume  =  4x3 - 36x2 + 80x      Your answer is exactly right! smiley👍



We want to know what value of  x  produces the greatest volume.


Let's look at a graph of the equation   y  =  4x3 - 36x2 + 80x   .



Now we want to know what value of  x  produces the greatest value of  y .


Notice that there is not an absolute maximum value of  y  on the graph....as  x  gets bigger than  5,  y  keeps getting bigger.


However, since there is no way to cut the corners of the box such that  x  is larger than 4 , we only care about what  x  value produces the greatest value of  y  for  x values in the domain  [0, 4] .


The largest value of  y  that can be made from  x  values in the domain  [0, 4]  is about 52.5, and it occurs when  x  is about 1.5 .   smiley

 May 10, 2018

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