Math Assignment

MATH ASSIGNMENT
1. A beekeeper estimates that the number of bees in a particular hive at the start of 2019 is 500,
and that at the start of any subsequent day the number of bees in the hive has increased by
1% compared to what it was exactly one day earlier
a) The function of b(t) can be modelled as:
The growth is exponentially growing thus:
b ( t )= A ektWhere b : R →(0 , ∞ )
at t=0 :b ( 0 )=500 ∴ A=500 At t=1:b ( 1 ) =500 ( 1+ 0.01 ) =505∴505=500 ek (1 )
k =ln ( 505
500 )=0.009950330853b ( t )=500 e0.0099503 t
b) The inverse of b−1 exist, since the time taken for bees to reach to a certain population can
be calculated. The date on when census was taken can be determined. In other words, we
are inputting the population of bees in the hive and outputting the time since the start of
this year 2019.
c) The inverse of b−1let b ( t )= y=500 e0.0099503 t y
500 =e0.0099503t
t=
ln ( y
500 )
0.0099503
∴ b−1 ( t ) =
ln ( t
500 )
0.0099503
d) Inverse function estimates the time taken for bees to have that population, or the age of
the bees in that beehive. Estimates the date on when the population of the bees was
counted.
The function b’(t) can be described as taking the population of bees in the hive as input
and outputting the time since the start of this year.

e) The derivative of b(t)
b' ( t )= d
dt ( 500 e0.0099503 t )¿ 4.975165427 e0.099503 t
The derivative indicates the change in the number of bees in given time.
f) Date in which population of bees in the hive reach 10000
t=
ln ( 10000
500 )
0.0099503 =301 days
From 1st January, 2019, 301st day will be on: 28th October, 2019.
g) The function c :[0 , 365]→(0 , ∞)which gives the population of bees in the hive
throughout 2019. The population in day 305 is 1000, and 306 is 1010
1000= A ek 3051010= A e306 k 1010
1000 = A e306 k
A ek 305 →1.01=ek ∴ k=ln (1.01 )=0.00995030853
A= 1000
e305 ( 0.0099503308 ) =48.0818183085
Thus, the function that models the population of bees in the hive throughout 2019 is
piecewise function given as:
c ( t ) = { 500 e0.0099503t ,∧0 ≤ t ≤304
48.0818183085 e0.0099503 t ,∧304<t ≤365
h) The population of bees in the hive at the end of the year, at t=365, thus;
c ( t ) =48.0818183085 e0.009950330853 ( 365 )=1817
2. For each customer X other than Wendy, they first calculate
w ( X )= { n
10 if n<10
1if n ≥10