Deriving Question Help (1 Viewer)

[D90]

Hi, I did this test just before the holidays started and I am only just looking at it now. I had no idea how to do it during the test and when I look back over the answer I was given I am still confused. I have done my best to list the whole question, i got full marks for part I) but did not get any marks for part II).
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QUESTION C:
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A man is on a boat at point B on a lake and AD is a straight stretch of the lake's edge. B is 3km from point A on the river bank. The man wishes to travel from point B to point D. He intends to row in a straight line to point C and then walk to D. He can row at 4km/hr and walk at 5km/hr.
(Heres my best attempt at the diagram included with the question)

Point B





Point A-----------------Point C--------------------Point D


-Point(s) BAC form a right angle triangle
-Point B to Point A=3km
-Point A to Point C=xkm
-Point B to Point C=(sqaure root)x^2+9
-Point A to Point D=6km

I) Explain why T=(sqaure root)x^2+9 OVER 4 + 6-x OVER 5

(i got this question correct so no help needed with part I)

II) Find the value of x which will enable him to complete the trip in the minimum time.
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This is where I struggle, part II, any help on this would be greatly appreciated because I just can't seem to do it, as soon as I start it I get stuck and I don't fully understand the answer I have been given. Once again any help is greatly appreciated.

 

[Steth0scope]

To find the minimum Time you need to derive T with respect to x.

By letting dt/dx = 0, you can find values of x for which T has a maximum or minimum. Then you have to test each value of x to see if it is a minimum or maximum (in this case there is only one value of x for which there is a minimum or maximum and it turns out to be a minimum - which is what the questions wants).

I'm trying to upload a pdf scan with the solution but it wont let me.

 

[D90]

Ok, i have started but have gotton stuck again. So far I have derived T to get:

1/2(x^2+9)^-1/2 OVER 4 -(Minus)- 1 OVER 5

Is this correct? What is the next step?

Thanks once more for your help.

 

[undalay]

Derive and differentiate are different things...

 

[undalay]

Ok, i have started but have gotton stuck again. So far I have derived T to get:

1/2(x^2+9)^-1/2 OVER 4 -(Minus)- 1 OVER 5

Is this correct? What is the next step?

Thanks once more for your help.

T' = 1/2(x^2+9)^-1/2 x 2x over 4 - 1 over 5
You missed the x2x when u differentiated.

SImplifying:
T' = x over 4root(x^2+9) - 1 over 5
Let T'=0
Cross multiplying
T'=0 = 5x - 4root(x^2+9)
5x = 4root(x^2+9)
25x^2 = 16(x^2+9)
9x^2 = 144
x^2 = 16
x = (+-)4
Obviously x isn't going to be -4
so x= 4
Then you need to prove that x =4 is a minimum turning point. (i.e. that when x = 4, t is the lowest it can be or rather when x=4, time taken is at the minimum).
Once u prove that x=4 is a minimum TP, you're done.

 

[Steth0scope]

Correct.

 

[D90]

Thanks for the help guys it makes alot more sense now. One question though, when you are up to the stage where you get 5x = 4root(x^2+9) how does the 4 in the next line become 16? Is it because when you remove the sqaure root and multiply 5 by the indice of 2 you also multiply 4 by that same indice of 2? Thanks once again for the help.

 

[undalay]

You square both sides.

 

[D90]

Ok great, i thought that was why. Thanks once again for the help.