# Rate of production of eggs

Eggs are produced at a rate of R(t)eggs per hour,where t=0 represents 12:00 midnight and R(t)(in thousands of eggs) is :-

R(t)= -10cospi/12t+10

a)how many eggs are produced in one day.

b)When are the eggs produced at the fastest rate

c)A machine can produce eggs at a constant rate. At the end of 1 week the same number of eggs are produced by the machine . What rate is the machine producing eggs in terms of R(t).

d)Each chicken produces the same amount of eggs on one 24hr period.how many chickens are there if each chicken can produce 20 eggs per day

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Eggs are produced at a rate of R(t)eggs per hour,where t=0 represents 12:00 midnight and R(t)(in thousands of eggs) is :-

R(t)= -10 cos pi/12 t+10

a)how many eggs are produced in one day.

No of eggs produced can be found by integrating R(t) from time t=0 to t=24

0 ∫24 R(t) dt= 0 ∫24 (-10 cos (pi /12) t+ 10 ) dt

=[ -(10 sin (pi /12) t)/ (pi /12) + 10 t ]0 24

The first term is a sine function between o and 2 pi . Hence its value will be zero. The second term will give a value of 240.

Thus 240,000 eggs will be produced in one day. ( The value of R(t) is in thousands of eggs)

b)When are the eggs produced at the fastest rate .

The maxima and minima can be found by differentiating the rate with respect to t and equating it to zero.

d/ dt (-10 cos (pi /12) t+ 10 )=0

or 10 sin (pi /12) t =0

This is satisfied at t=0 and at t=12

At t=0

R(t)= (-10 cos (pi /12) X 0+ 10 )=0

This is the minimum value.

At t=12

R(t)= (-10 cos (pi /12) X 12+ 10 )= -10 cos pi + 10= -10(-1) +10=20

This is the maximum value.

At noon (t=12) the eggs are produced at the fastest rate. (20, 000 eggs per hour)

c)A machine can produce eggs at a constant rate. At the end of 1 week the same number of eggs are produced by the machine . What rate is the machine producing eggs in terms of R(t).

The machine is producing 240,000 eggs per day= 240,000/24= 10,000 eggs per hour

R(t)= 10 t

Where R(t) is in thousands of eggs per hour.

d)Each chicken produces the same amount of eggs on one 24hr period .how many chickens are there if each chicken can produce 20 eggs per day.

Total no of eggs produced in a day=240,000

Each chicken produces 20 eggs per day

No of chicken= 240,000/20=12,000

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