Question

pre calc question

Answer 1

You want to minimize the TOTAL TIME. Split that into two parts:

Tw = walking time.
Tr = rowing time.

Now make a diagram. He is at M, and wants to reach Q. So he aims the boat at P:


|
+ Q(3,1)
|
|
| P(x,0)
-+-------+--*----+-------+----- Coast
| 1 2 3
|
|
|
M(0,-2)


MP = sqrt(x^2 + 4)

PQ = sqrt((x-3)^2 + 1)

Row MP in Tr = MP/2 = sqrt(x^2 + 4)/2

Walk PQ in Tw = PQ/4 = sqrt((x-3)^2 + 1)/4

Total time is the sum of those;

T = sqrt(x^2 + 4)/2 + sqrt((x-3)^2 + 1)/4

Now do the usual stuff:

2x 2(x - 3)
dT/dx = ---------------- + --------------------
4 sqrt(x^2 + 4) 8 sqrt((x-3)^2 + 1)

x (x - 3)
dT/dx = ---------------- + --------------------
2 sqrt(x^2 + 4) 4 sqrt((x-3)^2 + 1)


Set that equal to zero:


x (x - 3)
0 = ---------------- + --------------------
2 sqrt(x^2 + 4) 4 sqrt((x-3)^2 + 1)


2x sqrt((x-3)^2 + 1) + (x - 3)sqrt(x^2 + 4)
0 = ----------------------------------------...
4 sqrt(x^2 + 4) sqrt((x-3)^2 + 1)

Set the top to zero:

2x sqrt((x-3)^2 + 1) + (x - 3)sqrt(x^2 + 4) = 0

2x sqrt((x-3)^2 + 1) = (3 - x)sqrt(x^2 + 4)

Square both sides:

4x^2((x - 3)^2 + 1) = (9 - 6x + x^2)(x^2 + 4)

4x^2(x^2 - 6x + 9 + 1) = (9 - 6x + x^2)(x^2 + 4)

4x^4 - 24x^3 + 40x^2 = 9x^2 - 6x^3 + x^4 + 36 - 24x + 4x^2

3x^4 - 18x^3 + 27x^2 = + 36 - 24x

3x^4 - 18x^3 + 27x^2 + 24x - 36 = 0

x^4 - 6x^3 + 9x^2 + 8x - 12 = 0

x = 1 is an obvious solution:


1 - 6 9 8 - 12 (1)

1 -5 4 12
------------------------
1 - 5 4 12 zero


x^3 - 5x^2 + 4x + 12 = 0 is the reduced equation.

A quick-and-dirty graph of x^3 - 5x^2 + 4x + 12 shows only a root for a negative number.

So x = 1 is the solution

Answer 2

https://answers.yahoo.com/question/index?qid=20130403135357AAHZFha