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Loudon County Phone Company charges a $100 activation fee plus a monthly fee of $0.05 per minute for every minute over 200 minutes each month. (So, there is no monthly charge for 0 – 200 minutes of use per month). (a) Write an equation that can be used to determine a new subscriber’s first month’s phone bill, B(x), given the amount of minutes called, x. (Assume that the subscriber will make over 200 minutes of calls)
(b) Determine the subscriber’s first month’s bill if the total minutes used was 650. Show work.
(c) Determine the number of minutes used if the new subscriber’s first month bill was $114. Show work.
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1 Answer

Hi Theresa!  From a finance perspective, you're looking at the difference between fixed costs (activation fee) and variable costs (usage fees).  To solve these questions, you'll need to figure out which numbers are added/subtracted, versus those that are multiplied/divided.
 
(a)  For your formula, you have these variables:
 
         r    Rate for usage, per minute           You know this is 0.05
         c    Cap for free usage                       You know this is 200
         m   Total number of minutes of use    You don't know what this is
         a    Activation fee                              You know this is 100
 
     A monthly bill will always need to multiply r by m - c, which is the usage rate times the number of minutes above 200.  So, that's (m-c)*r   or, if you use the numbers, (m-200)*0.05.  Next, the first month's bill will need to also add 100 to that result.  I would recommend that you begin your formula with the usage piece, and then add the activation fee.  Be sure to enclose the usage algebra in parentheses before adding the activation fee.
 
(b)  This is easy, once you've developed the formula for question (a).  Simply use 650 for the variable m, and do the math.  In other words, subtract 200 from 650, then multiply that difference by the usage rate of $0.05.  Remember to add the extra $100 for the activation fee, after you've computed the usage overage charge.
 
(c)  For this solution, you'll need to back into the number of minutes by reconstructing the algebra.  In question (b), you knew the minutes but not the total bill.  Now, you know the total bill but not the minutes.  Just switch your variables around.  Since this is the first month bill, you'll need to subtract the $100 activation fee first.  That'll leave you with $14 for the usage overage.  What number of minutes yields $14 when you subtract 200 minutes and then multiply the resulting minutes by $0.05?
 
      Here's the algebra:
      14 = ( m - 200 ) * 0.05
      14 ÷ 0.05 = m - 200
      ( 14 ÷ 0.05 ) + 200 = m
 
Hope this helps!
 
Bill
 

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