Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point B on the shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart. If it takes 1.1 times as much energy to fly over water as land, to what point C should the bird fly to minimize the total energy expended in returning to its nesting area?

The bird should fly to the point C approximately 10.91 kilometers from the point B.

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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...

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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.

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