Suppose that flying over land requires `a ` units of energy per kilometer, then flying over water requires `1.1 a ` units of energy per kilometer. Denote also the distance in kilometers between B and C as `x in [ 0 , 13 ] . ` Then the distance between...

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Suppose that flying over land requires `a ` units of energy per kilometer, then flying over water requires `1.1 a ` units of energy per kilometer. Denote also the distance in kilometers between B and C as `x in [ 0 , 13 ] . ` Then the distance between A (the island) and C is `sqrt ( x^2 + 25 ) ` and the distance between C and D is `13 - x . `

This way, the amount of energy E depends on `x ` and is equal to

`E ( x ) = a ( 1.1 sqrt ( x^2 + 25 ) + 13 - x ) .`

This function is continuous on `[ 0 , 13 ] ` and is differentiable on `( 0 , 13 ) . ` The values at the endpoints are `E ( 0 ) = a ( 5.5 + 13 ) = 18.5 a , ` `E ( 13 ) = a*1.1sqrt ( 194 ) approx 15.32 a .`

The derivative of the function is `E' ( x ) = a ( 1.1 * x / sqrt ( x^2 + 25 ) - 1 ) , ` which is zero where `1.1 x = sqrt ( x^2 + 25 ) , ` or `1.21 x^2 = x^2 + 25 , ` or `x^2 = 25 / 0.21 . ` This way the only critical point is `x_1 = 5 / sqrt ( 0.21 ) approx 10.91 . ` The value at this point is approximately `15.29a, ` which is only slightly less that `E(13).`

This means that formally speaking, a bird should fly to the point on the shore **10.91** kilometers from from the point B. But the amount of saved energy would be negligibly small.