Question

A. Complete the chart based on the initial conditions:

An ant population starts at 1200 for a certain colony. As each month goes by in the winter, 6.5% of the population dies off. How many of the ants should be alive after 4 months? (Round to the whole ant.)

Month Ant Population
(start) 1200
1
2
3
4



B. Consider the following table that describes the mass of a kitten as weeks go by. If you know the kitten grows by 10% each week for the first 5 weeks, fill in the table for the values each week.

Week Mass (grams)
(start) 80
1
2
3
4

Answer 1

a.

`Month\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Pop(whole \ ant)\\month1 => x_1=80*0.94^1=1128\\month2=>x_2=1200*0.94^2=1060\\month3=>x_3=1200*0.94^3=996\\month4=>x_4=1200*0.94^4=936`

b.

`week Number\ \ \ \ \ \ \ \ \ \ \ \ \ \ Mass(g)\\week1 => x_1=80*1.1^1=88g\\week2=>x_2=80*1.1^2=96.8g\\week3=>x_3=80*1.1^3=106.48g\\week4=>x_4=80*1.1^4=117.128g`

Explanation:

a. From the information provided, we can deduce that the population death's follows a Geometric sequence in the form
`(a,ar,ar^2,ar^3...)` where
`a-first \ term` and
`r-common \ ratio`

#Since the population is reducing,
`r` can is obtained as
`r=1-r=0.94`

#The
`n^t^h` term is obtained using the formula
`x_n=ar^(^n^-^1^)`, given a=1200

The number of ants alive after every month (in first 4 months)

`Month\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Pop(whole \ ant)\\month1 => x_1=80*0.94^1=1128\\month2=>x_2=1200*0.94^2=1060\\month3=>x_3=1200*0.94^3=996\\month4=>x_4=1200*0.94^4=936`

The ant's alive after 4 months is obtained as the value of
`x_5`

`x_n=ar^(^n^-^1^)\\1-x_5=1-1200* 0.94^4=936.89\\\approx 936`

Hence, 936 ants are alive after 4 months.

b. As with the above question, the kitten population follows a geometric sequence:
`(a,ar,ar^2,ar^3...)`.

#Since it's a growing population , the common ration is the sum of 100% + the growth rate,

`r=1.1` and
`a=80` and
`x_n=ar^(^n^-^1^)`

The population after 4weeks will be:

`week Number\ \ \ \ \ \ \ \ \ \ \ \ \ \ Mass(g)\\week1 => x_1=80*1.1^1=88g\\week2=>x_2=80*1.1^2=96.8g\\week3=>x_3=80*1.1^3=106.48g\\week4=>x_4=80*1.1^4=117.128g`