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A man launches his boat from point A on a bank of a straight river, 4 km wide, and wants to reach point B, 4 km downstream on the opposite bank, as quickly as possible (see the figure below). He could row his boat directly across the river to point C and then run to B, or he could row directly to B, or he could row to some point D between C and B and then run to B. If he can row 6 km/h and run 8 km/h, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared to the speed at which the man rows.)

Respuesta :

MissPhiladelphia
Let's consider the man rows to point D, at x km from C...so we have DB= (4-x) km 

AC = 4 km 
CD = x km 
AD = √(4^2 + x^2) ..... Distance travelled through river 

Travel on land = (4 - x) 

f(t) = distance / time 
= [(x^2 + 16)^1/2]/6 + (4-x)/8 

f'(t) = 1/6 * (x^2 + 16)^(-1/2) * (2x) + 1/8 * (-1) 
= x / [3*√(x^2 + 16)] -1/8 
= [8x - 3*√(x^2 + 16)] / [24*√(x^2 + 16)] 

At f'(t) = 0 

0 = 8x - 3*√(x^2 + 16) 
8x = 3*√(x^2 + 16) 
64x^2 = 9(x^2 + 16) 
x = 12/√55 x>0 
= 1.618 Km .... distance travelled throgh water 

4 - x = 2.382 Km ......distance on foot

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