Reasoning strategy Making models

label Mathematics
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You cut square corners off a peice of cardboard with dimensions 16 inch by 20 inch. You then fold the cardboard to create a box with no lid. To the nearest inch what dimensions will give you the greatest volume?

May 26th, 2015

Draw a rectangle about 16 by 20 cm. Label it 10 by 15 inches. Draw a small square at each corner. Label it's side as x

From the diagram you can see the box dimensions will be:

The length = (20-2x)
The width = (16-2x)
The height = x

V = (20-2x)*(16-2x) * x
V = (320 - 40x - 32x + 4x^2) * x
Which is:
V = x(4x^2 - 72x + 320)
V = 4x^3 - 72x^2 + 320x; is the function of the volume

Draw a graph the function and find the value of x that produces the maximum
volume.

Looking at the graph we can see max vol occurs when x = 3

Substitute 3 for x in the original dimensions

The length = 14 in.
The width = 10 in.
The height = 3
in.



May 26th, 2015

I do not understand. Can you explain?

May 26th, 2015

Can you explain further?


May 26th, 2015

you make a quadratic equation using the values.

x will be the sides of the squares cut off

May 26th, 2015

formula for volume is length x width x height

which is given by:

The length = (20-2x)
The width = (16-2x)
The height = x

multiplying the 3 will give you V = 4x^3 - 72x^2 + 320x which is a quadratic equation

May 26th, 2015

once you graph this equation, look at the highest value of the graph at the x axis which will be 3.

substitute this in the original value of:

The length = (20-2x) = 14
The width = (16-2x) = 10
The height = x = 3


this is as detailed as it gets.

i hope you understand now

all the best



May 26th, 2015

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