+327 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!!!
+9466 (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! 
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(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 .
