MHB Gradients of curves to find average growth rate between two points

[srg263]

Hello Maths Help Board Users,

I was hoping for some guidance on the following problem. Thanks in advance for your contributions and time.

"A fish population grew according to the following quadratic model - the number of fish on day t is given by
P (t) = 800t - t²"

Q) Find the average growth rate between t=400 and t=600.

The population size at t=600 is 120,000 and at t=400 is 160,000

Am i correct in understanding to find the average growth rate between t=400 and t=600 i need to work out the slope between the two points?

Change in population / change in time
= 120,000 - 160,000 / 600 - 400
= -40,000 / 200
= -200 (this is the gradient of the line?)

Many thanks.

 

[MarkFL]

Yes, I think you have the right idea here, that the average growth rate $R$ is the total change in population divided by the total change in time:

$$\overline{R}=\frac{\Delta P}{\Delta t}=\frac{P\left(t_2\right)-P\left(t_1\right)}{t_2-t_1}$$

 

[HallsofIvy]

To be completely correct (if you had a really hard nosed teacher, like me) you should give the answer as

"-200 fish per day"