The size of a certain insect population is given by p(t)=200e.02t, where t is measured in days. at what time will the population equal 800?

Functions in physics is a mathematical concept used to represent different types of physical quantities in nature. Some types of functions are- linear, quadratic, cubic, exponential and reciprocal functions. A functions relates a set of inputs with another set of desired outputs. Functions are essential for formulating physical relationships in theoretical and practical sciences. For example, functions can be widely used in the representation of electric and magnetic fields in space and time, they are used in quantum mechanics, it is very widely used in determining the position and motion of an object in space. We perform a variety of mathematical operations on functions to make some conclusions and to answer some typical questions related to a physical system.

Here in this question, we have a function which represents a certain population of insect, the function is given by,

[tex]P(t)=200e^0^.^0^2^t[/tex]

where [tex]t[/tex] represents the number of days.

We have to find the total time in days for the population to reach 800 insects, that means,

[tex]P(t)=800[/tex]

Hence,

[tex]200e^0^.^0^2^t=800[/tex]

[tex]e^0^.^0^2^t=\frac{800}{400}[/tex]

[tex]e^0^.^0^2^t=4[/tex]

Taking natural log both sides

[tex]ln e^0^.^0^2^t=ln 4\\[/tex]

[tex]0.02t=ln4[/tex]

[tex]t=\frac{ln4}{0.02}[/tex]

[tex]t=69.3\\[/tex]

[tex]t=69[/tex] days (appproximately)

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