<h2>
Answer with explanation:</h2>
We know that the exponential function is given by:
![f(x)=ab^x](https://tex.z-dn.net/?f=f%28x%29%3Dab%5Ex)
where a is the initial amount.
and b is the change in the amount and is given by:
if the function is increasing by a rate of r
and
if the function is decreasing by a rate of r.
a)
The initial amount of fish in the trout are: 7
i.e. a=7
Also, the population doubles every year.
This means that that b=2
Hence, the population after t years is given by the function P(t) as:
![P(t)=7(2)^t](https://tex.z-dn.net/?f=P%28t%29%3D7%282%29%5Et)
b)
The original amount of the machine is: $ 3,000
i.e. a=3,000
Also, the value of machine decreases by a rate of 7%
i.e.
![r=7\%\\\\i.e.\\\\r=0.07](https://tex.z-dn.net/?f=r%3D7%5C%25%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cr%3D0.07)
Hence, we have:
![b=1-r\\\\i.e\\\\b=1-0.07\\\\i.e.\\\\b=0.93](https://tex.z-dn.net/?f=b%3D1-r%5C%5C%5C%5Ci.e%5C%5C%5C%5Cb%3D1-0.07%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cb%3D0.93)
Hence, the function which represent the price of the machine after t years i.e. P(t) is given by:
![P(t)=3000(0.93)^t](https://tex.z-dn.net/?f=P%28t%29%3D3000%280.93%29%5Et)
c)
The initial population of colony of ants i.e. a=300.
The number of ants increases at a rate of 1.5% every month.
i.e. ![r=1.5%\\\\i.e.\\\\r=0.015](https://tex.z-dn.net/?f=r%3D1.5%25%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cr%3D0.015)
i.e.
![b=1+r\\\\i.e.\\\\b=1+0.015\\\\i.e.\\\\b=1.015](https://tex.z-dn.net/?f=b%3D1%2Br%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cb%3D1%2B0.015%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cb%3D1.015)
Hence, the function P(t) which represents the population of ants after t months is given by:
![P(t)=300(1.015)^t](https://tex.z-dn.net/?f=P%28t%29%3D300%281.015%29%5Et)
d)
The initial infected cells i.e. a=300
The infected cells are decaying at a rate of 1.5% per minute.
i.e.
![r=1.5%\\\\i.e.\\\\r=0.015](https://tex.z-dn.net/?f=r%3D1.5%25%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cr%3D0.015)
Since, there is a decay hence,
![b=1-r\\\\i.e.\\\\b=1-0.015\\\\i.e.\\\\b=0.985](https://tex.z-dn.net/?f=b%3D1-r%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cb%3D1-0.015%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cb%3D0.985)
Hence, the function P(t) which represents the number of infected cells after t minutes is given by:
![P(t)=300(0.985)^t](https://tex.z-dn.net/?f=P%28t%29%3D300%280.985%29%5Et)