Answer with explanation:
We know that the exponential function is given by:
[tex]f(x)=ab^x[/tex]
where a is the initial amount.
and b is the change in the amount and is given by:
[tex]b=1+r[/tex] if the function is increasing by a rate of r
and [tex]b=1-r[/tex] 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:
[tex]P(t)=7(2)^t[/tex]
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.
[tex]r=7\%\\\\i.e.\\\\r=0.07[/tex]
Hence, we have:
[tex]b=1-r\\\\i.e\\\\b=1-0.07\\\\i.e.\\\\b=0.93[/tex]
Hence, the function which represent the price of the machine after t years i.e. P(t) is given by:
[tex]P(t)=3000(0.93)^t[/tex]
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. [tex]r=1.5%\\\\i.e.\\\\r=0.015[/tex]
i.e.
[tex]b=1+r\\\\i.e.\\\\b=1+0.015\\\\i.e.\\\\b=1.015[/tex]
Hence, the function P(t) which represents the population of ants after t months is given by:
[tex]P(t)=300(1.015)^t[/tex]
d)
The initial infected cells i.e. a=300
The infected cells are decaying at a rate of 1.5% per minute.
i.e.
[tex]r=1.5%\\\\i.e.\\\\r=0.015[/tex]
Since, there is a decay hence,
[tex]b=1-r\\\\i.e.\\\\b=1-0.015\\\\i.e.\\\\b=0.985[/tex]
Hence, the function P(t) which represents the number of infected cells after t minutes is given by:
[tex]P(t)=300(0.985)^t[/tex]
