Cost of Phone CallDate: 02/27/97 at 16:05:31 From: CAI Lab User Subject: problem Hi Dr. Math! A direct-dial long distance call between two cities costs $1.04 for the first 2 minutes and $0.36 for each additional minutes or fraction thereof. Use the greatest integer function to write the cost (C) of a call in terms of the time (t) (in minutes). Can you help me? Thanks! Date: 03/02/97 at 23:21:08 From: Doctor Luis Subject: Re: problem The graph of the function you describe looks like this (notice the position of the holes - @): cost($) ^ ... | . ... | . ... | . 2.48 | @** 2.12 | @** 1.76 | @** 1.40 | @** 1.04 ****** | | --|--------------------------> | 1 2 3 4 5 time (min) This function is really composed of two parts, one in the interval from 0 to 2 (I'm assuming you'll be charged for the first two minutes even if the duration of your call is less), and another for the interval containing values greater than 2. If you want to write this function as a single equation, you can think of it as the sum of a constant term (the 1.04 part) and a step function MULTIPLIED by a function (say u(t)) which is 0 when t is less than or equal to 2 and is 1 when t is greater than 2. In symbols: c(t) = (1.04) + s(t)*u(t) The step function s(t) for this problem can be represented in terms of the greatest integer function [x] = max{z|z <= x} by the expression: s(t) = -(0.36)*[2-t] The function u(t) is, by definition: 0 , if t <= 2 u(t) = 1 , if t > 2 Now, u(t) can be represented by a single expression if we introduce the "signum" function sgn x, and the absolute value function abs x, which are defined by: -1 , if x<0 x , if x >= 0 sgn(x) = 0 , if x=0 abs(x) = 1 , if x>0 -x , if x < 0 (note that sgn x gives us the sign of x, hence the name) With these functions defined, one can easily prove that u(t) can be written as: u(t) = (sgn(abs(t-2))+sgn(t-2))/2 Thus, the single formula expressing the cost of the phone call in terms of the time can be represented by: c(t) = (1.04)-(0.36)*[2-t]*(sgn(abs(t-2))+sgn(t-2))/2 -Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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