Respuesta :
Answer:
762 days
Step-by-step explanation:
Given
[tex]Rate = 200 + 10t + 13t^2[/tex]
Let the rate be R.
So the rate change with time is represented as:
[tex]\frac{dR}{dt} = 200 +10t + 13t^2[/tex]
So:
[tex]dR = (200 + 10t + 13t^2)\ dt[/tex]
To get the number of insects between day 0 and day 3, we need to integrate dR and set the bounds to 0 and 3
i.e.
[tex]dR = (200 + 10t + 13t^2)\ dt[/tex] becomes
[tex]\int\limits^3_0 {dR} \, =\int\limits^3_0 (200 + 10t + 13t^2)\ dt[/tex]
[tex]R =\int\limits^3_0 (200 + 10t + 13t^2)\ dt[/tex]
Integrate
[tex]R = 200t + \frac{10t^2}{2} + \frac{13t^3}{3} [3,0][/tex]
Solve for R by substituting 0 and 3 for t
[tex]R = (200*3 + \frac{10*3^2}{2} + \frac{13*3^3}{3}) - ( 200*0 + \frac{10*0^2}{2} + \frac{13*0^3}{3})[/tex]
[tex]R = (200*3 + \frac{10*9}{2} + \frac{13*27}{3}) - (0 + \frac{10}{2} + \frac{0}{3})[/tex]
[tex]R = (200*3 + 5*9 + 13*9) - (0)[/tex]
[tex]R = 600 + 45 + 117 - 0[/tex]
[tex]R =762[/tex]
The population of insect between the required interval is 762
The required change in the population of insects between day 0 and day 3 is 762.
Given that,
A population of insects increases at the rate of 200++ 10t + 13t2.
We have to determine,
What is the change in the population of insects between day 0 and day 3.
According to the question,
The rate change with time is represented as:
[tex]\frac{dr}{dt} = 200+10t+13t^{2}[/tex]
To determine the change in the population of insects between day 0 and day 3,
Integrating the given function with respect to time with limit 0 days to 3 days.
[tex]\int\limits^3_0 dr = \int\limits^3_0 (200+10t+13t^{2} ). dt\\\\\int\limits^3_0dr =\int\limits^3_0 \, 200dt + \int\limits^3_0 10tdt +\int\limits^3_0{13t^{2} dt\\\\\\\\R = [200t]^3_0 + [5t^{2} ]^3_0 + [\frac{13t^{3} }{3} ]^3_0\\\\R = 200[(3) - (0)] + 5 (3^{2} - 0^{2} ) + \frac{13}{3} [ 3^{3} - 0^{3} ]\\\\R = 200(3) + 5 (9) + \frac{13}{3}(27)\\\\R = 600+45+117\\\\R = 762[/tex]
Hence, The required change in the population of insects between day 0 and day 3 is 762.
To know more about Rate change click the link given below.
https://brainly.com/question/24540629
