The sides of a rectangle are to be made from thin bars of two type of metal. Two

Question

The sides of a rectangle are to be made from thin bars of two type of metal. Two opposite sidra are to be made of metal A. and the other two opposite sides are to be made of metal B. Metal A costs $5 per foot and met si B costs $8 per foot. The area of the rectangle must be 20 square feet Find the lengths of the sides that minimize the materials cost of making the sides of the rectangle The dimensions given must be exact and not approximate, and make clear which length is which Use Calculus to to justify your conclusion, and, as always, show your work

Explanation / Answer

Let metal A's side length be a and metal B's side length be b

Area of rectangle = ab and perimeter = 2(a+b)

cost of material required = 2a*5 + 2b*8 = 10a+16b

Given ab=20 => b=20/a

We want to minimize 10a+16b = 10a+320/a

f(a) = 10a+320/a

f(a) is minimum when f'(a)=0 and f''(a) > 0

f'(a) = 10-320/a^2=0 => a=4root(2) or -4root(2)

a cannot be negative, so a=4root(2)

b=20/a = 20/4root(2) = 5/root(2)

Therefore,

a=4root(2) and b=5/root(2)

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