Mixing Alcohol, Cable TV Pricing
Date: 8/2/96 at 1:58:36 From: Alfred T Chu Subject: Mixing Alcohol, Cable Price Word Problems Hello Dr. Math, Here are two word problems that I don't quite get... 1. A petroleum distributor has two gasohol storage tanks, the first containing 9 percent alcohol and the second containing 12 percent alcohol. They receive an order for 300,000 gallons of 10 percent alcohol. How can they mix alcohol from the two tanks to fill this order? 2. A cable television company plans to begin operations in a small town. The company knows that about 600 people will subscribe to the service if the price per subscriber is $5 per month, but that for each 5-cent increase in the monthly subscription price, 4 of the original 600 people will decide not to subscribe. The office begins operations, and its total revenue for the first month is $1500. How many people have subscribed to the cable television service? Sincerely, Alfred
Date: 8/2/96 at 13:21:33 From: Doctor Anthony Subject: Re: Mixing Alcohol, Cable Price Word Problems Word Problem 1: Mixing Alcohol Let x = volume of type(1) 100-x = volume of type(2) in 100 gallons of mixture. Then 0.09x + 0.12(100-x) = 10 gallons -0.03x + 12 = 10 2 = 0.03x and so x = 66.6667 gallons So we use 66+(2/3) gallons type (1) and 33+(1/3) gallons type (2) per 100 gallons of mixture. For 300,000 gallons, we multiply up by 3000 to get 200,000 and 100,000 of type (1) and type (2) respectively. Question 2: Cable Rates Let x = number of increments of 5-cents added to the price. So the number of subscribers = 600-4x, and subscription = 5+.05x So we have: (600-4x)(5+.05x) = 1500 3000 - 20x + 30x - 0.2x^2 = 1500 -0.2x^2 + 10x + 1500 = 0 0.2x^2 - 10x - 1500 = 0 so x = -65.1387 or x = 115.13 The negative value we ignore, as the question does not say that they would gain subscribers by reducing the price. So they have increased the price by 115 increments of 5 cents = $5.75 Total cost of subscribing = $5 + $5.75 = $10.75 Total number of subscribers = 600-460 = 140 Total income = 140*10.75 = $1505 - which checks approximately. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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