24-optimization2

24-optimization2 - Math 1a Optimization Day Two Fall, 2009...

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Unformatted text preview: Math 1a Optimization Day Two Fall, 2009 1 Mary is sitting 1 kilometer off shore on her friend’s boat when she decides she wants an ice cream cone from the shop 2 kilometers down the coastline. (See the picture.) She can row about 4 kilometers per hour and walk about 5 kilometers per hour. (a) How far down the shore should she row to- wards if she wants to get her ice cream as quickly as possible? (b) Suppose the ice cream shop closes in 35 min- utes. Will Mary make it in time for some ice cream? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Ice Cream! 2 kms 1 km shore Friend’s boat • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A small company makes fine wood chairs, but it can only make 80 in a year. Suppose the total cost function is given by C ( x ) = 500+1500 x , where x is the number of chairs produced, and the total revenue function is given by R ( x ) = 1600 x- x 2 . (a) How many chairs should the company produce in a year to maximize its profits? Justify your answer! (b) Now suppose one of the company’s two chair experts goes on an extended vacation. This means the company can only make 40 chairs in a year. How many chairs should the company now produce in a year to maximize its profits? Justify your answer! 3 A farmer wants to construct a new animal enclosure. The enclosure will be a rectangle with a semicircular cap. The boundary opposite the semicircle will be formed by the wall of the farmer’s barn, while the remaining edges will be new fences. The straight fencing costs $5 per linear foot while the curved fencing for the semicircular edge costs $10 per linear foot. The farmer wants to enclose 1000 square feet. Find the dimensions of the enclosure that minimizes the cost of the fencing. (Recall that a circle of radius r has area πr 2 and circumference 2 πr .) ....
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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24-optimization2 - Math 1a Optimization Day Two Fall, 2009...

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