7) A population of ants is growing at a rate of 8% a year. If there are 160 ants in the initial population, find
the number of ants after 6 years.
Why plusing this 100 % and rate of 8%
Why did the ant population go to school? To learn how to multiply!
Now, let's get down to business. To find the number of ants after 6 years, we'll use the formula:
N = P(1 + r)^t
Where:
N = Number of ants after t years
P = Initial population
r = Rate of growth (in decimal form)
t = Number of years
Plugging in the values:
P = 160 ants
r = 8% = 0.08
t = 6 years
N = 160(1 + 0.08)^6
N = 160(1.08)^6
N ≈ 282.43
So, after 6 years, we can approximate the number of ants to be around 282.43 (though, unfortunately, ants don't come in decimal numbers). Keep in mind that this is just an approximation, as the actual number of ants may vary due to other factors.
To find the number of ants after 6 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final amount (number of ants)
P = Initial amount (number of ants)
r = Annual growth rate (8% or 0.08 in decimal form)
n = Number of times the growth is compounded per year (assuming it is compounded annually, n = 1)
t = Number of years
Given that the initial population is 160 and the growth rate is 8%, we can substitute these values into the formula and solve for A.
A = 160(1 + 0.08/1)^(1*6)
A = 160(1.08)^6
A ≈ 302.6526
Therefore, the number of ants after 6 years would be approximately 303 ants.
To find the number of ants after 6 years, we need to calculate the growth of the population over those 6 years.
1) Start with the initial population of ants: 160 ants.
2) Calculate the growth rate per year: 8%.
3) Convert the growth rate to decimal form: 8% = 0.08.
4) To calculate the number of ants after 1 year, multiply the initial population by the growth rate: 160 * (1 + 0.08) = 160 * 1.08 = 172.8.
The population after 1 year is approximately 173 ants.
5) Repeat the above step for each subsequent year. To find the population after 2 years, multiply the population after 1 year (173) by the growth rate: 173 * 1.08 = 186.84.
The population after 2 years is approximately 187 ants.
6) Continue this process for 6 years: multiplying the population each year by the growth rate.
After 3 years: 187 * 1.08 = 201.96, approximately 202 ants.
After 4 years: 202 * 1.08 = 218.16, approximately 218 ants.
After 5 years: 218 * 1.08 = 235.44, approximately 235 ants.
After 6 years: 235 * 1.08 = 253.80, approximately 254 ants.
Therefore, the number of ants after 6 years is approximately 254 ants.
8% + 100% = 108% = 108 / 100 = 1.08
The n-th term of a geometric sequence with initial value a1 and common ratio r is given by:
an = a1 ∙ r ⁿ⁻¹
In tis case :
a1 = 160
r = 1.08
Need to notice that term two is after one year, so term 7 will be after 6 years.
n = 7
so:
a6 = a1 ∙ r⁷⁻¹
a6 = 160 ∙ r⁶
a6 = 160 ∙ 1.08⁶
a6 = 160 ∙ 1.586874322944
a6 = 253.89989167104
The number of ants after 6 years will be 253.