a family of 100 ants invades your home. if its population increases at a rate of 20% per week, in how many weeks will there be 200 ants? 400 ants? and 800 ants?
n = 100 (1 + .20)^w
plug in n (number of ants), solve for w
log(n/100) = w log(1.2)
thank you
To find out in how many weeks there will be 200 ants, 400 ants, and 800 ants, we need to calculate how many weeks it takes for the population to reach these numbers based on the given growth rate of 20% per week.
Let's start by setting up an equation to represent the population growth:
New Population = Initial Population × (1 + Growth Rate)^Number of Weeks
We know that the initial population is 100 ants and the growth rate is 20% per week. So, the equation becomes:
New Population = 100 × (1 + 0.20)^Number of Weeks
Now, we can solve for the number of weeks required to reach each of the target populations.
1. 200 ants:
200 = 100 × (1 + 0.20)^Number of Weeks
To find the number of weeks for 200 ants, we need to isolate the exponent. Divide both sides of the equation by 100:
200/100 = (1 + 0.20)^Number of Weeks
2 = (1.20)^Number of Weeks
To eliminate the exponent, we can take the logarithm of both sides. Let's choose the natural logarithm (ln):
ln(2) = ln[(1.20)^Number of Weeks]
By applying the logarithm property, we can bring the exponent down:
ln(2) = Number of Weeks × ln(1.20)
Now, divide both sides by ln(1.20):
Number of Weeks = ln(2) / ln(1.20)
Using a calculator, we find that this takes approximately 3.81 weeks.
2. 400 ants:
Similar to the previous calculation, we can set up the equation:
400 = 100 × (1 + 0.20)^Number of Weeks
Divide both sides by 100:
400/100 = (1 + 0.20)^Number of Weeks
4 = (1.20)^Number of Weeks
Again, take the natural logarithm of both sides:
ln(4) = Number of Weeks × ln(1.20)
Divide both sides by ln(1.20):
Number of Weeks = ln(4) / ln(1.20)
Using a calculator, we find that this takes approximately 7.09 weeks.
3. 800 ants:
Using the same process as above:
800 = 100 × (1 + 0.20)^Number of Weeks
Divide both sides by 100:
800/100 = (1 + 0.20)^Number of Weeks
8 = (1.20)^Number of Weeks
Take the natural logarithm:
ln(8) = Number of Weeks × ln(1.20)
Divide both sides by ln(1.20):
Number of Weeks = ln(8) / ln(1.20)
Using a calculator, we find that this takes approximately 11.86 weeks.
Therefore, in approximately 3.81 weeks, there will be 200 ants; in approximately 7.09 weeks, there will be 400 ants; and in approximately 11.86 weeks, there will be 800 ants.