Thread: Applications of Maxima/Minima

An island is at point A, 4km offshore from the nearest point B on a straight beach. A woman on the island wishes to go to a point C, 6km down the beach from B. She can go by rowboat at 5km/hr to a point P Between B and C. And then walk at 8km/hr along a straight path from P to C.
a.) Estimate on your graphics calculator the route from A to C that she can travel in the least time.
b.) Confirm Your estimates in part a analytically.

Originally Posted by ^_^Engineer_Adam^_^

An island is at point A, 4km offshore from the nearest point B on a straight beach. A woman on the island wishes to go to a point C, 6km down the beach from B. She can go by rowboat at 5km/hr to a point P Between B and C. And then walk at 8km/hr along a straight path from P to C.
a.) Estimate on your graphics calculator the route from A to C that she can travel in the least time.
b.) Confirm Your estimates in part a analytically.

Me, analytically only.

The question asks for route for the least time, so we find the distance for the least time.

distance = rate*time,
So, time = distance / rate -----------**

Rates are given, so we find distances.

Let x = BP, in km.
So, 6-x = PC, in km.

In right triangle ABP, by Pythagorean theorem,
AB = sqrt(4^2 +x^2) = sqrt(16 +x^2)

Then, the times,
Let T = total time spent from A to P to C,
T = [sqrt(16 +x^2) / 5] +(6-x)/8 -------------------------(i)
Differentiate both sides with respect to x,
dT/dx = (1/5)[(1/2)(2x /sqrt(16 +x^2))] +(1/8)[-1]
dT/dx = (1/5)[x /sqrt(16 +x^2)] -1/8
Equate dT/dx to zero,
0 = (1/5)[x/sqrt(16 +x^2)] -1/8
Clear the fractions, multiply both sides by 5*8*sqrt(16 +x^2).
0 = 8x -5sqrt(16 +x^2)
sqrt(16 +x^2) = 8x/5
sqrt(16 +x^2) = 1.6x
Square both sides,
16 +x^2 = (2.56)x^2
16 = (1.56)x^2
x^2 = 16/1.56 = 10.2564
x = sqrt(10.2564) = 3.20256 km -----------for least T.

Hence, for the route, or distance of the route, when x = 3.20256 km,
D = sqrt(16 +x^2) +(6-x)
D = sqrt(16 +10.2564) +(6 -3.20256)
D = 7.92 km. ---------------------------answer.

Originally Posted by ^_^Engineer_Adam^_^

An island is at point A, 4km offshore from the nearest point B on a straight beach. A woman on the island wishes to go to a point C, 6km down the beach from B. She can go by rowboat at 5km/hr to a point P Between B and C. And then walk at 8km/hr along a straight path from P to C.
a.) Estimate on your graphics calculator the route from A to C that she can travel in the least time.
...

Hello,

to a)
To estimate the time needed take the most "simplest" ways she can go:

i) directly from A to C. The distance AC = √(16+36) ≈ 7.211 km. For this distance she needs 1 h; 26 min; 32 s

ii) from A to B and from B to C. The time needed is 4/5 h + 6/8 h = 31/20 h = 1 h; 33 min;

That means the minimum of time must be smaller than 1 h; 26 min; 32 s

With ticbol 's result she needs 1 h; 22 min; 28 s

EB