maximisation minimisation help :( (1 Viewer)

[AnandDNA]

the steel frame of a rectangular prism is three times as long as it wide and h is the height. The prism has a volume of 4374m^3. find the dimensions of the frame so that the minimum amount of steel is used.

 

[AnandDNA]

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[Chinmoku03]

Are you sure that the length and width are not related to h, the height value?

 

[ianc]

hi there

let's call the width x, and so this means the length is 3x

V = length*width*height
= 3hx^2 = 4374

rearrange to make h the subject:
h = 1458 / (x^2)

so now we've got our h depending on x

the amount of steel needed is simply done by drawing a diagram and adding up all the lengths (remember not surface area, but the frame)

S = 4(length+width+height)
= 4(3x+x+h)
= 16x + 4h

(substitute the expression for h)

= 16x + 5832 / (x^2)

so we need to differentiate S with respect to x, and solve dS/dx=0 to find the value for x such that S is at a minimum

16 + (-2)5832 / (x^3) = 0
16x^3 = 11664
x = 9

h=18

so the width is 9, length is 27, height is 18

hope this helps

it's a tricky question

 

[AnandDNA]

oh yeh thanks heaps. I didnt realise we could have found an expression by adding up all the lengths. Thanks heaps