Question

The energy expended by a bird per day, E, depends on the time spent foraging for food per day, F hours. Foraging for a shorter time requires better territory, which then requires more energy for its defense.1 Find the foraging time that minimizes energy expenditure if E=0.25F+1.7/f^2

Answer 1

Therefore F=2.387 hours gives a minimum value of energy expenditure E.

Explanation:

Given that,

The energy expended by a bird per day

`E=0.25 F+(1.7)/(F^2)`

Differentiating with respect to F

`E'=0.25 -(3.4)/(F^3)`

Again differentiating with respect to F

`E''=(10.2)/(F^4)`

Now set E'=0

`0.25 -(3.4)/(F^3)=0`

`\Rightarrow (3.4)/(F^3)=0.25`

`\Rightarrow F^3=(3.4)/(0.25)`

`\Rightarrow F=2.387`

Now
`E''|_(F=2.387)=(10.2)/(2.387^4)>0`

Since, E''>0 at F=2.387, So at F=2.387 , E has minimum value.

Therefore F=2.387 hours that minimizes energy expenditure.