A university is doing a study to record the growth of mosquitos. There is originally 25 mosquitoes. The growth of the mosquitos is 0.03% every 7 days. How many days until there are 20 435 mosquitos?
What I did was use the growth formula A=25(1+0.03)^t to figure out how many mosquitoes were there after a week, ( I got 25.75) then used the formula A=25.75(1.03)^t/7 and got 1580. Did I do this right?
20435=25(1+.03)^t/7
1.03^t/7=817.4
take the log of each side.
t/7 * log 1.03 = log 817.4
t= 7* log 817.4 /log1.03
I get about 1600 days for the longer period. I agree with you.
okay, good, I think i did it right then (it was actually a question on my exam that I just completed, it was the only question I was unsure about)
Yes
good job, you must have done some rounding
what you want is
20435 = 25(1.03)^t , where t is the number of weeks
8174 = 1.03^t
take log of both sides
log 817.4 = log (1.03^t)
t = log 817.4 / log 1.03 = 226.87 weeks
or
1588 days
Yeah probably, I also don't remember if it was 20435 or 20425 exactly, but it was approximately that.
Your approach is almost correct, but there is a small mistake in your calculation. Here's how you can solve it correctly:
Let's break down the problem step by step:
1. The original number of mosquitoes is given as 25.
2. The growth rate is 0.03% every 7 days. This means that the number of mosquitoes will increase by 0.03% every week.
3. To calculate how many days it will take to reach a certain number of mosquitoes, we need to use the exponential growth formula: A = P(1 + r/n)^(nt), where:
- A is the final amount (in this case, 20,435 mosquitoes),
- P is the initial amount (25 mosquitoes),
- r is the growth rate (0.03% or 0.0003 as a decimal),
- n is the number of times the growth is compounded per time period (in this case, once per week),
- t is the number of time periods (in this case, the number of days).
Now let's calculate the number of days it will take:
1. Start by finding the weekly growth factor: (1 + 0.0003)^1 = 1.0003. This means the number of mosquitoes will increase by 0.03% or 0.0003 times the previous amount every week.
2. To find the number of weeks it takes to reach 20,435 mosquitoes, we need to solve for t in the equation: 25 * (1.0003)^t = 20,435.
3. Divide both sides of the equation by 25 to isolate the exponential term: (1.0003)^t = 817.4.
4. Take the logarithm of both sides (with the base of your choice, let's use natural logarithm): ln((1.0003)^t) = ln(817.4).
5. Apply the power rule of logarithms to bring down the exponent: t * ln(1.0003) = ln(817.4).
6. Solve for t by dividing both sides by ln(1.0003): t = ln(817.4) / ln(1.0003).
7. Use a calculator to calculate t, which turns out to be approximately 3303.3 weeks.
8. Since there are 7 days in a week, multiply the number of weeks by 7 to find the number of days: 3303.3 weeks * 7 days/week ≈ 23123.3 days.
9. Therefore, it will take approximately 23,123 days for the number of mosquitoes to reach 20,435.
In summary, you made a small calculation error in your approach. Instead of 1580, the correct answer is approximately 23123 days.