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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):

(8-2x)(10-2x)x=

(80-36x+4x^2)x=

4x^3-36x^2+80x

 

Thank You For Helping Me!!!

KennedyPape  May 10, 2018
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(a)

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👍

 

(b)

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   .

https://www.desmos.com/calculator/ptpji7kvzf

 

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

hectictar  May 10, 2018

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