Maximum/minimum

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Frank Doheny

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Frank Doheny 8 years, 5 months ago

Published this.

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Frank Doheny 8 years, 7 months ago

Created this.

Name	Status	Author	Last Modified
Maximum/minimum	draft	Frank Doheny	18/11/2016 11:37
cormac's copy of Maximum/minimum	draft	cormac breen	04/03/2017 17:43
cormac's copy of cormac's copy of Maximum/minimum	draft	cormac breen	07/03/2017 14:22
Simon's copy of Maximum/minimum	draft	Simon Thomas	14/03/2019 11:18
Maria's copy of cormac's copy of Maximum/minimum	draft	Maria Aneiros	27/05/2019 05:47
Maria's copy of Maximum/minimum	draft	Maria Aneiros	27/05/2019 05:47

Question statement

A rectangular sheet of metal of length = $\var{l}cm$ and width = $\var{w}cm$ has a square of side $x\,cm$ cut from each corner. The ends and sides will be folded upwards to form an open box.

Determine the value of $x$ that will maximise the volume of this box.

Give any introductory information the student needs.

			Name	Type	Generated Value

			l	integer	60
			w	integer	16

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A random number between and (inclusive) with step size

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Generated value: `integer`

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Input the value for $x$ correct to one decimal place.

$x =$ Unnamed gap

Advice

The length of the box is $\var{l}-2x$ , the width is $\var{w}-2x$ and the height is $x$ .

The volume is then given by

$V=(\var{l}-2x).(\var{w}-2x).x$

$V=\simplify{4x^3-2*({w}+{l})x^2+{l}*{w}x}$

$\frac{dV}{dx}=\simplify{12x^2-4*({w}+{l})x+{l}*{w}}$

This is a quadratic equation.

$x=\frac{\simplify{4*({w}+{l})}\pm\sqrt(\simplify{16*({w}+{l})^2-48*{w}*{l}})}{24}$

$x=\frac{\simplify{{w}+{l}}\pm\sqrt(\simplify{{w}^2-{w}*{l}+{l}^2})}{6}$

$\frac{d^2V}{dx^2}=\simplify{24x-4*({w}+{l})}$

when $x=\simplify{({w}+{l}-sqrt({w}^2-{w}*{l}+{l}^2))/6}$ $\frac{d^2V}{dx^2}<0$ and therefore is the value that gives a maximum.

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Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Frank Doheny 8 years, 5 months ago

Frank Doheny 8 years, 7 months ago

Error in variable testing condition

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Generated value: `integer`

Parts

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Maximum/minimum

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Frank Doheny 8 years, 5 months ago

Frank Doheny 8 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`