Can anyone help me with this question

Answer(I would recommend going through explanation thoroughly, ask in comments if still unsure)

Part A) y = 8.1 • e^ ln(2)/12t

Part B) 10.8 mm^2

Explanation

Part a)

You are told the initial area is 8.1 mm^2 this is when t (time) = 0.

Y = a • e^bt where b and a are constant to be found for your questions

Now regardless of the value of b, if t = 0 then the expression
e^bt will always equal 1 as bt will always equal 0 if time is 0 (note: anything to the power of 0 is 1)

Therefore when t=0 (initial area is 8.1mm^2 and y = initial area)is substituted

You obtain

8.1mm^2 = a • e^b0

8.1mm^2 = a • e^0

8.1mm^2 = a

Therefore you have obtained the value of a.

You are also told that the area doubles every 12 minutes, so in 12 minutes time, the area will be 16.2 (8.1mm^2 multiplied by 2)

Hence substitute t = 12 minutes
y = 16.2 and a = 8.1 in the expression to get

16.2 = 8.1 • e^12b

Now rearrange to obtain b (the only unknown value)

16.2/8.1 = e^12b

e^12b = 2

Take ln of both sides (note lne^12b becomes 12b)

12b = ln (2)

b = ln(2) / 12

Therefore your final equation is

y = 8.1 • e^ln2/12 t

Part b)
You know the equation now and you have t= 5 minutes so simply substitute to obtain occupied area (denoted by y)

y = 8.1 • e^ln2/12 • 5

= 10.81220

= 10.8mm^2 (nearest tenth as your question asks for)