M408K Hm. 12 Solutions

M408K Hm. 12 Solutions - Mahon Kevin Homework 12 Due 3:00...

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Mahon, Kevin – Homework 12 – Due: Nov 17 2005, 3:00 am – Inst: Edward Odell 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points A trailer rental agency rents 14 trailers per day at a rate of $8 per day. It discovers that for each $2 increase in rate, one fewer trailer is rented. Determine the rate, r max , which maximizes the rental income. 1. r max = $18 correct 2. r max = $22 3. r max = $20 4. none of these 5. r max = $16 Explanation: Let r be the daily rate of renting a trailer. Then the number of trailers rented daily is 14 - ( r - 8) 2 . Hence the daily income, I ( r ), of the agency is given by I ( r ) = r 14 - ( r - 8) 2 . The income will be maximized, therefore, at the critical points of I ( r ). Now after differen- tiation, I 0 ( r ) = 14 - ( r - 8) 2 - r 2 . Thus the critical points of I ( r ) occur at 14 - ( r - 8) 2 - r 2 = 0 , i.e., r = 18 . Consequently, r max = $18 is the rate that maximizes income. keywords: Stewart5e, 002 (part 1 of 1) 10 points Circuit City has been selling 100 television sets a week at $520 each. A market survey indicates that for each $20 rebate offered to a buyer, the number of sets sold will increase by 5 per week. How large a rebate should Circuit City offer a buyer in order to maximize its revenue? 1. rebate = $75 2. none of these 3. rebate = $80 4. rebate = $70 5. rebate = $60 correct 6. rebate = $65 Explanation: Let $20 x be the rebate offered to a buyer. Then the price of a TV will be $(520 - 20 x ) and the number of sets sold at this price will be 100 + 5 x . The revenue with this rebate is thus R ( x ) = (520 - 20 x )(100 + 5 x ) = 100(26 - x )(20 + x ) = 100(520 + 6 x - x 2 ) . But then R 0 ( x ) = 100(6 - 2 x ) , while R 00 ( x ) = - 100 × 2 < 0 .
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Mahon, Kevin – Homework 12 – Due: Nov 17 2005, 3:00 am – Inst: Edward Odell 2 Consequently, the Revenue is maximized at x 0 = 3, in which case the rebate = $60 . keywords: Stewart5e, 003 (part 1 of 1) 10 points A high speed office copier has an initial price of $5000. A service contract cost $240 for the first year and increases by $100 per year thereafter. It can be shown that over x years the total cost of the copier is given by C ( x ) = 5000 + 190 x + 50 x 2 . When is the average cost per year smallest? (This is often referred to as the replacement time for a piece of equipment.) 1. replacement time = 13 years 2. replacement time = 11 years 3. replacement time = 12 years 4. replacement time = 9 years 5. replacement time = 10 years correct Explanation: The average cost c ( x ) = C ( x ) x = 5000 + 190 x + 50 x 2 x = 5000 x + 190 + 50 x . In practical terms it represents the total cost (initial cost plus maintenance costs) of the computer averaged out over an x year period. The marginal average cost, therefore, is given by c 0 ( t ) = - 5000 x 2 + 50 , and so the only critical point of c ( x ) for x > 0 is the positive solution of c 0 ( x ) = 0, i.e. , replacement time = r 5000 50 = 10 years . keywords: Stewart5e, 004 (part 1 of 2) 10 points The marketing department of a company has determined that the cost (in dollars) of producing x units of a new product is given by C ( x ) = 400 + 320 x, while the demand equation for that product will be given by p = 1600 - 1 25 x.
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