If it takes a pigeon only 75% as much energy to fly over land as - PowerPoint PPT Presentation

The problem Read the problem Formulæ Cut down independent variables Domain Calculus If it takes a pigeon only 75% as much energy to fly over land as over water, what path will it take when flying from a point 1 mi from shore to its nest which is 2 mi down the shoreline?

The problem Read the problem Formulæ Cut down independent variables Domain Calculus Pigeon wants to minimise energy Best approach: fly straight to a point on the shore, then fly straight along the shoreline

The problem Read the problem Formulæ Cut down independent variables Domain Calculus √ By Pythagorean theorem, distance flown over water is 1 + x 2 Distance flown on land is 2 − x x is horizontal distance flown before reaching shoreline (in miles)

The problem Read the problem Formulæ Cut down independent variables Domain Calculus Total energy spent is � 1 + x 2 + k land · (2 − x ) E tot = E water + E land = k water · Given k land = 0 . 75 · k water

The problem Read the problem Formulæ Cut down independent variables Domain Calculus Only one independent variable x ; nothing to do!

The problem Read the problem Formulæ Cut down independent variables Domain Calculus x = 0: fly straight to shore x < 0: start flying away from nest x = 2: fly straight to nest x > 2: fly past nest Domain is [0 , 2]

The problem Read the problem Formulæ Cut down independent variables Domain Calculus � 1 + x 2 + k land · (2 − x ) E tot = k water · d x d x E tot = k water · √ 1 + x 2 − k land

The problem Read the problem Formulæ Cut down independent variables Domain Calculus d x d x E tot = k water · √ 1 + x 2 − k land Denominator is never 0; always defined

The problem Read the problem Formulæ Cut down independent variables Domain Calculus d x d x E tot = k water · √ 1 + x 2 − k land x 1 + x 2 = k land k water · √ � k water · x = k land · 1 + x 2 water · x 2 = k 2 k 2 land (1 + x 2 ) land ) x 2 = k 2 ( k 2 water − k 2 land � k 2 land x = ± k 2 water − k 2 land

The problem Read the problem Formulæ Cut down independent variables Domain Calculus � k 2 land x = ± k 2 water − k 2 land Negative numbers are not in domain Use k land = 0 . 75 · k water , and cancel k 2 water in numerator and denominator of expression for x : � 0 . 75 2 · k 2 √ water x = 1 . 286 ≈ 1 . 134 . ≈ (1 − 0 . 75 2 ) k 2 water

The problem Read the problem Formulæ Cut down independent variables Domain Calculus x E tot 0 k water + 2 k land 1 . 134 1 . 512 · k water + 0 . 8661 · k land 2 2 . 236 · k water Use k land = 0 . 75 · k water : x E tot 0 3 . 5 · k water 1 . 134 2 . 161 · k water 2 2 . 236 · k water

The problem Read the problem Formulæ Cut down independent variables Domain Calculus Best path is to fly straight to a point 1 . 134 mi along the shoreline, then fly the rest of the way along the shoreline