SOLUTION: Hello I'm having trouble setting this problem up. Any help would be appreciated. Thanks. A traveler having 18 miles to go, calculates that his usual rate would make him one-half

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Question 1201248: Hello I'm having trouble setting this problem up. Any help would be appreciated. Thanks.
A traveler having 18 miles to go, calculates that his usual rate would make
him one-half hour late for an appointment; he finds that in order to arrive on
time he must travel at a rate one-half mile an hour faster. What is his usual
rate?
Found 3 solutions by ikleyn, josgarithmetic, MathTherapy:
Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
.
Hello I'm having trouble setting this problem up. Any help would be appreciated. Thanks.
A traveler having 18 miles to go, calculates that his usual rate would make
him one-half hour late for an appointment; he finds that in order to arrive on
time he must travel at a rate one-half mile an hour faster. What is his usual rate?
~~~~~~~~~~~~ 

Let x be his usual rate, in miles per hour.


The time to travel 18 miles at his usual rate is  18%2Fx  hours.


The time to travel with the faster rate, (x+0.5) miles per hour, is  18%2F%28x%2B0.5%29 hours.


The longer time is  1%2F2 of an hour longer:

    18%2Fx - 18%2F%28x%2B0.5%29 = 1%2F2  of an hour.    (1)


    +----------------------------------------------+
    |    At this point, the setup is complete:     |
    |    you just have an equation to solve.       |
    +----------------------------------------------+


To solve it, multiply both sides by 2x*(x+0.5) = 2x^2 + x.  You will get

    36(x+0.5) - 36x = x^2 + 0.5x

    36x + 18 - 36x = x^2 + 0.5x

    x^2 + 0.5x - 18 = 0

    x%5B1%2C2%5D = %28-0.5+%2B-+sqrt%280.5%5E2+%2B+4%2A1%2A18%29%29%2F2 = %28-0.5+%2B-+sqrt%2872.25%29%29%2F2 = %28-0.5+%2B-+8.5%29%2F2.


Obviously, only positive root fits  x = %28-0.5+%2B+8.5%29%2F2 = 8%2F2 = 4  miles per hour.


Check it, by substituting into equation (1)

    18%2F4+-+18%2F4.5 = 4.5 - 4 = 0.5 = 1%2F2  of an hour.   ! correct !


ANSWER.  The usual rate is 4 miles per hour.

Solved.



Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
Using the description

            SPEED        TIME          DIST.

USUAL       r            18/r            18

FASTER      r+1/2       18/(r+1/2)       18 

DIFFERENCE                1/2

18%2Fr-18%2F%28r%2B1%2F2%29=1%2F2---------simplify and solve.

36%2Fr-36%2F%28r%2B1%2F2%29=1
36%28r%2B1%2F2%29-36r=r%28r%2B1%2F2%29
36r%2B18-36r=r%28r%2B1%2F2%29
18=r%28r%2B1%2F2%29
r%5E2%2Br%2F2-18=0
r%5E2%2B%281%2F2%29r-18=0
discrim,1%2F4-4%28-18%29=72%261%2F4
r=%28-1%2F2%2B+sqrt%2872%261%2F4%29%29%2F2
highlight%28r=4%29

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
Hello I'm having trouble setting this problem up. Any help would be appreciated. Thanks.
 A traveler having 18 miles to go, calculates that his usual rate would make
him one-half hour late for an appointment; he finds that in order to arrive on
time he must travel at a rate one-half mile an hour faster. What is his usual
rate?

Let usual/regular speed be S
Then time taken to get to appointment = 18%2FS
Increasing speed by matrix%281%2C2%2C+1%2F2%2C+mph%29, or .5 mph means that his new speed to get there on-time is: (S + .5) mph.
So, time he'd take to get to the appointment, based on his new speed = 18%2F%28S+%2B+.5%29
At the new speed , he'll get there on-time by "shaving" 1%2F2-hour, or .5 hour off of the previous late-time, or 18%2FS+-+.5
We then get the following TIME equation: matrix%281%2C3%2C+18%2F%28S+%2B+.5%29%2C+%22=%22%2C+18%2FS+-+.5%29
                                          matrix%281%2C3%2C+18%28S%29%2C+%22=%22%2C+18%28S+%2B+.5%29+-+.5S%28S+%2B+.5%29%29 ------ Multiplying by LCD, S%28S+%2B+.5%29
                                    
                                     matrix%281%2C3%2C+2S%5E2+%2B+S+-+36%2C+%22=%22%2C+0%29 ----- Multiplying by 4
                                 (S - 4)(2S + 9) = 0 ----- Factoring trinomial
                                  S - 4 = 0            or      2S + 9 = 0
           Normal/Usual speed, or S = 4 mph       or      2S = - 9 (ignore)