A population of rabbits starts off at time = 0 with 225 rabbits and increases at a rate of r(t) rabbits per month, where t is in months.
A. What does the expression (24 integral 36)r(t)dt represent?
B. What does the expression 225 + (0 integral 18) r(t)dt represent?
C. Suppose you're told that 1/24(0 integral 24) r(t)dt=4.25. How many rabbits are there in the population after 2 years?
Could someone explain these to me please? Thanks
oobleck what is your problem? I've seen you on other forums here as well acting like the most pretentious snob ever, not everyone gets complex math the first time around.
So what's the answer to C?
A. the population growth during the 3rd year
B. population after 18 months
C. well, ∫[0,24] r(t) dt is the population, so this is the average population during the 1st two years.
So I multiply 24*4.25 and add 225?
A. To understand what the expression (24 integral 36) r(t)dt represents, we need to break it down:
1. "integral" represents integration, which is a mathematical process used to find the area under a curve. In this case, it represents finding the accumulated value of the function r(t) over a specific time interval.
2. The range of integration is from 24 to 36, which means we are integrating the function r(t) over the time period from 24 to 36 months.
3. The term r(t) represents the rate of increase in the rabbit population at time t.
Putting it all together, the expression (24 integral 36) r(t)dt calculates the accumulated growth of the rabbit population from 24 to 36 months based on the rate of increase given by the function r(t).
B. Similarly, let's analyze the expression 225 + (0 integral 18) r(t)dt:
1. The term 225 represents the initial population of rabbits at time t=0, which is given in the question.
2. The integral from 0 to 18 represents the integration of the function r(t) from 0 to 18 months. So, it represents the accumulated growth of the rabbit population during the first 18 months.
When we add the initial population to the accumulated growth, the expression 225 + (0 integral 18) r(t)dt gives us the total population of rabbits after 18 months.
C. In this case, we are given that (1/24) * (0 integral 24) r(t)dt = 4.25:
1. The integral from 0 to 24 represents the total accumulated growth of the rabbit population over 24 months.
2. The term (1/24) is a coefficient used to scale the integral result to an average monthly growth rate.
Given that the expression evaluates to 4.25, we can conclude that the average monthly growth rate is 4.25 rabbits per month.
To find the total number of rabbits after 2 years, we need to multiply this average monthly growth rate by the number of months in 2 years (which is 24). Multiplying 4.25 by 24 gives us the final answer of 102 rabbits.