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Optimisation Question

A rectangular box, with no top, is made from thin card.
The volume of the box is 500 cm3.
The base of the box is a square with sides of length x cm

(a)Show that the area, A cm2, of card used to make
such an open box is given by

A= x^2 + (2000/x)

(b) find the minimum amount of card needed to make this box

I have no idea how to do this.

Reply 1

TenOfThem

Original post by 50ShadesOfRay

Write down a formula using x and h (height) for volume
Set this = to the volume
Re-arrange to get h

Use this to write down an expression for the surface area

Reply 2

krisshP

Original post by 50ShadesOfRay

What part a or b or both or what? What did you do?

Reply 3

krisshP

For the second part you want the lowest A. Think about differentiation and a minimum point :smile:

Reply 4

50ShadesOfRay

Original post by TenOfThem

Write down a formula using x and h (height) for volume
Set this = to the volume
Re-arrange to get h

Use this to write down an expression for the surface area

What about B, i can't work it out?

Reply 5

50ShadesOfRay

Original post by krisshP

For the second part you want the lowest A. Think about differentiation and a minimum point :smile:

Sorry i am confused, can you walk me threw it

Reply 6

krisshP

You have an equation for A. Differentiate this equation to get dA/dx and set it equal to 0 and solve as the lowest A is at a turning point :smile:

. Does that ring any bells for you?

Edit: think of a graph of A against x if it helps where the graph is a U shaped parabola, so it has a minimum point :smile:

(edited 10 years ago)

Reply 7

50ShadesOfRay

Original post by krisshP

You have an equation for A. Differentiate this equation to get dA/dx and set it equal to 0 and solve as the lowest A is at a turning point :smile:

. Does that ring any bells for you?

Edit: think of a graph of A against x if it helps where the graph is a U shaped parabola, so it has a minimum point :smile:

Okay i understand that, however i am unsure how to set

-200x^-2 + 2x = 0

the answer is 600 by the way

(edited 10 years ago)

Reply 8

krisshP

Original post by 50ShadesOfRay

Okay i understand that, however i am unsure how to set

-200x^-2 + 2x = 0

A= x^2 + (2000/x)
A=x^2 +2000x^-1
dA/dx=2x-2000x^-2
dA/dx=2x-2000/(x^2)
At minimum point dA/dx=0
2x-2000/(x^2)=0

Solve this, btw you messed up your differentiation a bit or did a typo error.

If you still struggle

Spoiler

Reply 9

50ShadesOfRay

Original post by krisshP

Spoiler

that doesn't give 300 though

Reply 10

krisshP

Original post by 50ShadesOfRay

that doesn't give 300 though

It gives x=10, right? Look at the question again. So you know worked out that the squared base of the box has sides of length 10cm. They wanted minimum amount of card needed, so you need A instead of x :smile:

Reply 11

50ShadesOfRay

Original post by krisshP

You are the biggest legend! :biggrin:

Reply 12

josefwakeling

Original post by TenOfThem

Write down a formula using x and h (height) for volume
Set this = to the volume
Re-arrange to get h

Use this to write down an expression for the surface area