A triathlete on the swimming leg of a triathlon is 150.0 m from the shore ( a ).

Question

A triathlete on the swimming leg of a triathlon is 150.0 m from the shore (a). The triathlete's bike is 100.0 m from the shore on the land (b). The component of her distance from the bicycle along the shore line, ( x + y in the diagram), is 195.5 m.

A) If the triathlete's running speed is 7.85 m/s and swimming speed is 1.334 m/s, calculate the value of x so that the triathlete reaches her bike in the least amount of time

B) Calculate the minimum time required to reach the bicycle.

PLEASE PROVIDE STEP BY STEP EXPLANATIONS =)

Explanation / Answer

so time = time water + time land

time water = sqrt(150^2 + x^2)/1.334

x + y = 195.5 so y = (195.5-x)

time land = sqrt( 100^2 + y^2)/7.85

time land = sqrt( 100^2 + (195.5-x)^2)/7.85

so

t = sqrt(150^2 + x^2)/1.334+sqrt( 100^2 + (195.5-x)^2)/7.85

dt/dx = 0.5*(150^2 + x^2)^(-1/2)*(2*x)/1.334 + 0.5*(100^2+(195.5-x)^2)^(-1/2)*(2*(195.5-x)*-1)/7.85 = 0

solve for x
x=22.32 m

b)

plug in to find t

t=139.2 s

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