# Math Help - The Derivative Function

1. ## 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?

2. Originally Posted by Snowboarder
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.

3. i do not know how to do this cause pitagoras doesn't work here.

4. Originally Posted by Snowboarder
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.

5. 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.

6. Originally Posted by Snowboarder
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 .....?

Originally Posted by Snowboarder
[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.