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abozer, Research Assistant
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Experience:  B.S. Mechanical Engineering
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# 1) A carpenter is building a rectangular room with a fixed

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1) A carpenter is building a rectangular room with a fixed perimeter of 360 ft. What dimensions would yield the maximum area? What is the maximum area?
2)Aki's Bicycle Designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by C(x)=0.5x^2-0.4x+3.216, where C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?
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abozer :

Hi,

abozer :

1)

abozer :

A = xy

abozer :

P = 2(x + y)

abozer :

Perimeter is 360,

abozer :

360 = 2(x + y)

abozer :

180 = x + y

abozer :

y = (180 - x)

abozer :

Substitute this in the first equation,

abozer :

A = x(180 - x)

abozer :

A = 180x - x^2

abozer :

Take the derivative to find the point where A is maximum,

abozer :

A' = 180 - 2x

abozer :

A' = 0 at maximum,

abozer :

0 = 180 - 2x

abozer :

x = 90

abozer :

y = 180 - 90

abozer :

y = 90

abozer :

So the dimensions are,

abozer :

length = 90

abozer :

width = 90

abozer :

2)

abozer :

Cost is minimum at the point where the derivative of the average cost function is zero,

abozer :

C'(x) = x - 0.4

abozer :

x - 0.4 = 0

abozer :

x = 0.4

abozer :

So the average cost is minimum when 40 bicycles are built.

abozer :

Let me know if you have any questions.

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