Fit a bounded growth model

A scientist is studying the growth of a bacteria colony in a petri dish. He places bacteria at one spot of the dish and he measures the area $A$ the bacteria cover on day $t$. His results are recorded below:

Time, $t$ (days)	0	1	2	3	4	5	6	7
Area, $A$ ($\mathrm{cm}^2$)	1.00	5.00	27.10	41.39	46.95	49.95	51.07	52.05

(i) Draw a scatter diagram to illustrate the data.1 mark

(ii) The scientist would like to predict the future growth of the bacteria. Explain why, in this context, a linear model is not likely to be appropriate.1 mark

It is proposed to fit a model of the form $\ln(D-A)=a+bt$, where $D$ is a suitable constant. The product moment correlation coefficient between $t$ and $\ln(D-A)$ is denoted by $r$. The following table gives values of $r$ for some possible values of $D$.

$D$	53	54	55
$r$		−0.99132	−0.98767

(iii) Calculate the value of $r$ for $D=53$, giving your answer correct to 5 decimal places. Hence, explain which value of $D$ is the most appropriate.2 marks

(iv) With the value of $D$ in (iii), calculate the values of $a$ and $b$ correct to 5 decimal places, and use them to predict the area covered by the bacteria after 20 days.3 marks

(v) Give an interpretation, in context, of the value of $D$.1 mark