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/