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.
(a) Equation for volume (v):
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volume = (length)(width)(height)
|length||= 10 - x - x = 10 - 2x|
|width||= 8 - x - x = 8 - 2x|
volume = (10 - 2x)(8 - 2x)( x )
volume = (80 - 36x + 4x2)( x )
volume = 80x - 36x2 + 4x3
volume = 4x3 - 36x2 + 80x Your answer is exactly right! 👍
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 .