This is a classic optimization question and it is useful to consider the concepts at play here before generating equations and arriving at an answer: Consider the two, opposite, limiting cases.
Case A: Kyle swims directly toward the child. On the plus side, this minimizes the distance he travels, so should save time. However, the down side is that he will swim the whole distance, and since he runs faster than he swims, this minimizes his average speed and increases his time.
Case B: He runs all the way 30 m along the shore then swims directly perpendicular to the shoreline. This maximizes the running leg of his trip on the plus side, but maximizes his total distance.
Is there a route in between those two cases that will minimize the time? (In this case yes, in other cases, it may be best to swim the whole way. It is never ideal to run the full 30 m.)
Let x be the distance along shore that he does not run. We will be able to write an equation for his time as function of x only, then we can take the derivative, set it = 0, and solve for the ideal x. In order to do this, we should recall that time = distance/speed.
Distance running = (30 - x) so time running = (30 - x) / 4
Distance swimming (by pythag thm) = √(x2 + 502) so time swimming = √(x2 + 502) / 1.1
T(x) = (30 - x) / 4 + √(x2 + 502) / 1.1
T'(x) = - 1/4 + x / 1.1√(x2 + 502) = 0
4x = 1.1√(x2 + 502)
16x2 = 1.21(x2 + 2500)
14.79x2 = 3025
x ~ 14.301 m so Kyle should run about 15.699 m along the shore before diving in.
It is an interesting result that the ratio of his speeds, 1.1 : 4 , is exactly the ratio of the ideal length of shoreline he does not run to the length of the ideal swimming distance.
