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Math Help - distance and time word problem

  1. #1
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    Question distance and time word problem

    A man walks for 45 minutes at a rate of 3 mph, then jogs for 75 minutes at a rate of 5 mph, then sits and rests for 30 minutes, and finally walks for 1 hour and half.Find the rule of the function that expresses this distance traveled as a function of time [ Caution: Don't mix up the units of time ; use either minutes or hours , not both]

    could someone help me with this problem? I sketched the graph ,and the ranges I got are from 0 to .75 , .75 to 2, 2 to 2.5 and 2.5 to 4 . I know that the first function is 3t and the third one is 6.25.However, I don't know how to write the other two ones.


    Thanks in advance.
    Last edited by vance; February 10th 2009 at 06:41 PM.
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  2. #2
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by vance View Post
    A man walks for 45 minutes at a rate of 3 mph, then jogs for 75 minutes at a rate of 5 mph, then sits and rests for 30 minutes, and finally walks for 1 hour and half.Find the rule of the function that expresses this distance traveled as a function of time [ Caution: Don't mix up the units of time ; use either minutes or hours , not both]

    could someone help me with this problem? I sketched the graph ,and the ranges I got are from 0 to .75 , .75 to 2, 2 to 2.5 and 2.5 to 4 . I know that the first rule is 3t and the third one is 6.25.However, I don't know how to write the other two ones.


    Thanks in advance.
    Hello!

    The function rule for this problem is
    distance = rate x time
    d = rt
    To write the rule as the distance traveled as a function of the time it took to travel that distance you will need to evaluate each set of data using this function. Since you have 3 different rates of speed, and 4 time periods, you will have to calculate each one.

    I changed the times to hours.

    RATE TIME
    0 \rightarrow .5
    3 \rightarrow .75
    5 \rightarrow 1.25
    3 \rightarrow 1.5


    So, distance as a function of time:
    d= rt

    1) d(t) = 0t so d(.5) = (0)(.5) = 0 miles
    2) d(t) = 3t so d(.75) = (3)(.75) = 2.25 miles
    3) d(t) = 5(t) so d(1.25) = (5)(1.25) = 6.25 miles
    4) d(t) = 3t so d(1.5) = (3)(1.5) = 4.5 miles

    Hope this is what you needed! Good luck!
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  3. #3
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    Hello, vance!

    This is a tricky one!
    Luckily, I've done one of these before . . .


    A man walks for 45 minutes at a rate of 3 mph, then jogs for 75 minutes at 5 mph,
    then sits and rests for 30 minutes, and finally walks for 1 hour and half.
    Find the rule of the function that expresses this distance traveled as a function of time.
    This is a piece-wise function.

    The distance function depends on what time period is involved.

    From t=0\text{ to }t = 0.75, the distance function is: . 3t
    . . By the end of the 45 minutes, he has gone 2.25 miles.

    From t = 0.75\text{ to } t = 2, the distance function is: . 2.25 + 5t
    . . By the end of the 75 minutes, he has gone another 6.25 miles, a total of 8.5 miles.

    From t = 2\text{ to }t = 2.5, the distance function is: . 8.5 + 0t \:=\:8.5
    . . By the end of the 30 minutes, he hasn't moved; his total is still 8.5 miles.

    From t = 2.5\text{ to }4, the distance function is: . 8.5+3t
    . . By the end of the 90 minutes, he has gone another 4.5 miles, a total of 13 miles.


    The function would be written like this:

    . . d(t) \;=\;\begin{Bmatrix}3t & & 0 \leq t \leq 0.75 \\ 2.25 + 5t & & 0.75 \leq t < 2 \\ 8.5 & & 2 \leq t \leq 2.5 \\ 8.5+3t & & 2.5 < t < 4 \end{Bmatrix}


    The graph would look like this:
    Code:
          |
          |                               *
          |                            *  :
          |                         *     :
          |                      *        :
          |               * * *           :
          |             * :   :           :
          |           *   :   :           :
          |         *     :   :           :
          |       *       :   :           :
          |     *         :   :           :
          |  *  :         :   :           :
      - - * - - + - - - - + - + - - - - - + - -
          |              2   2          4
          |
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  4. #4
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    Quote Originally Posted by Soroban View Post
    Hello, vance!

    This is a tricky one!
    Luckily, I've done one of these before . . .

    This is a piece-wise function.

    The distance function depends on what time period is involved.

    From t=0\text{ to }t = 0.75, the distance function is: . 3t
    . . By the end of the 45 minutes, he has gone 2.25 miles.

    From t = 0.75\text{ to } t = 2, the distance function is: . 2.25 + 5t
    . . By the end of the 75 minutes, he has gone another 6.25 miles, a total of 8.5 miles.

    From t = 2\text{ to }t = 2.5, the distance function is: . 8.5 + 0t \:=\:8.5
    . . By the end of the 30 minutes, he hasn't moved; his total is still 8.5 miles.

    From t = 2.5\text{ to }4, the distance function is: . 8.5+3t
    . . By the end of the 90 minutes, he has gone another 4.5 miles, a total of 13 miles.


    The function would be written like this:

    . . d(t) \;=\;\begin{Bmatrix}3t & & 0 \leq t \leq 0.75 \\ 2.25 + 5t & & 0.75 \leq t < 2 \\ 8.5 & & 2 \leq t \leq 2.5 \\ 8.5+3t & & 2.5 < t < 4 \end{Bmatrix}


    The graph would look like this:
    Code:
          |
          |                               *
          |                            *  :
          |                         *     :
          |                      *        :
          |               * * *           :
          |             * :   :           :
          |           *   :   :           :
          |         *     :   :           :
          |       *       :   :           :
          |     *         :   :           :
          |  *  :         :   :           :
      - - * - - + - - - - + - + - - - - - + - -
          |              2   2          4
          |

    Thanks soroban. This is what I was looking for!!!
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