c_w = cost per km in water
c_L = cost per km on land
where: c_L < c_w (why?)

thus:

function (in y) to be minimized:

taking the first derivative with respect to y

text approved solutions

First, lets assume some parameters and solve the problem just to become familiar with it:

graphing domain:

the cost function:

the cost function derivative:

initial guess:

Mathcad finds the root of the derivative at y = 129.3 miles

c_w = cost per km in water
c_L = cost per km on land
where: c_L < c_w

Lets solve the general problem analytically:

or

the cost function

taking the first derivative of C(y) with respect to y

setting this derivative = zero in order to find the minimum

rearranging

squaring both sides (and remembering that c_L was negative before squaring!)

rearranging and some algebra

rearranging and some algebra

rearranging and some algebra

rearranging and some algebra

squaring both sides

remembering c_L was negative:

which obtains the numerical solution and is the text approved solution

to obtain the solution as a function of x, just substitute y into:

you can do this as an exercise!