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

#1**+1 **

**(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 + 4x^{2})( x )

volume = 80x - 36x^{2} + 4x^{3}

volume = 4x^{3} - 36x^{2} + 80x Your answer is exactly right! 👍

**(b)**

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

Let's look at a graph of the equation y = 4x^{3} - 36x^{2} + 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 .

hectictar
May 10, 2018