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Math Help - The Derivative Function

  1. #1
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    The Derivative Function

    Suppose you are in a car (C) that that has run out of petrol 2 km away from a T-intersection (T). You know that there is a Petrol Station (G) 4 km away from the intersection as shown in the diagram. You can walk along the road at a speed of 5 km. per hour. You could also cut across an undulating field and intersect the road closer to the petrol station (S), but your cross country walking speed will be reduced to 3 km. per hour. You must decide which option is quickest, since the petrol station closes in 1 and a half hours.

    a)
    From the diagram, derive the following formula, which gives the total time required to walk to the garage using any straight path across the field T(x)= [square root(x^2 +4)]/3 + (4-x/5)


    b)
    Explain the significance of the situation when x = 0, and why T(x) cannot be used to calculate the total time taken when x = 0?
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  2. #2
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    Quote Originally Posted by Snowboarder View Post
    Suppose you are in a car (C) that that has run out of petrol 2 km away from a T-intersection (T). You know that there is a Petrol Station (G) 4 km away from the intersection – as shown in the diagram. You can walk along the road at a speed of 5 km. per hour. You could also cut across an undulating field and intersect the road closer to the petrol station (S), but your cross country walking speed will be reduced to 3 km. per hour. You must decide which option is quickest, since the petrol station closes in 1 and a half hours.

    a) From the diagram, derive the following formula, which gives the total time required to walk to the garage using any straight path across the field T(x)= [square root(x^2 +4)]/3 + (4-x/5)


    b) Explain the significance of the situation when x = 0, and why T(x) cannot be used to calculate the total time taken when x = 0?
    You might use the discussion in this thread: http://www.mathhelpforum.com/math-he...d-problem.html

    or this one: http://www.mathhelpforum.com/math-he...imization.html

    as a guide.
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  3. #3
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    i do not know how to do this cause pitagoras doesn't work here.
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  4. #4
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    Quote Originally Posted by Snowboarder View Post
    i do not know how to do this cause pitagoras doesn't work here.
    T = \frac{CS}{3} + \frac{SG}{5}, measured in hours.

    You're right ...... pitagoras doesn't work here. But Pythagoras works just fine. So I'd suggest using his Theorem to get CS.
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  5. #5
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    oki thx i've done it point a) but do not know how can i explain the significance of the situation when x=0 and why T(x) can not be used to calculate the total time taken when x=0.
    Also i have to determine the quickest route to take to the petrol station.
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  6. #6
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    Quote Originally Posted by Snowboarder View Post
    oki thx i've done it point a) but do not know how can i explain the significance of the situation when x=0 and why T(x) can not be used to calculate the total time taken when x=0.
    [snip]
    That's something you need to think about, which means you have to go back and look at the diagram. What is the route to the petrol station when x = 0 .....?

    Quote Originally Posted by Snowboarder View Post
    [snip]
    Also i have to determine the quickest route to take to the petrol station.
    Use calculus to find the x-coordinate of the minimum turning point of the given time function T.
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