A fenced enclosure consists of a rectangle of length L and width 2R, and a semic

Question

A fenced enclosure consists of a rectangle of length L and width 2R, and a semicircle of radius R, as shown in the figure below. The enclosure is to be built to have an area A of 1600ft^2. The cost of the fence is $40/ft for the curved portion and $30/ft for the straight sides. Use the min function to determine with a resolution of .01 ft the values of R and L required to minimize the total cost of the fence.

Figure is at P13 here:

http://books.google.com/books?id=iFmg5rlmOSkC&pg=PA179&lpg=PA179&dq=a+fenced+enclosure+consists+of+a+rectangle&source=bl&ots=tkp_8E6LdE&sig=G6k2CW0Yrc94sA8x8-8EH_wgem0&hl=en&ei=k4CGTZyTNcbLgQf4gpG9CA&sa=X&oi=book_result&ct=result&resnum=4&sqi=2&ved=0CDMQ6AEwAw#v=onepage&q=a%20fenced%20enclosure%20consists%20of%20a%20rectangle&f=false

I attempted to find area and cost below:

Area=2RL + (piR^2)/2 =1600

Cost=(2RL*30)+((piR^2)/2*40)
help please

Explanation / Answer

The equation for area is correct that you wrote but the equation for cost is wrong. See the fence is always on the boundary and at the area so the fence will be at the semi-circular path of the curved area and at the rectangular area of the land so cost=30*(2R+2L)+40*(pi*R) now solving for L from first equation and subtituting it in 2nd equation now we have an equation for cost and R so now differentiating the equation and solving for R we get R=11.77 so now substituting this value in the equation got by solving the first equation.... L=58.72

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