Question

A study is done on the population of a certain fish species in a lake. Suppose that the population size P(t) after t years is given by the following exponential function.

P(t)=420(0.78)^t
does the function represent growth or decay?
By what percentage does the population change each year?

Answer 1

Function represents decay.

Each year the population decreases 22%.

Explanation:

An exponential function can be written as,

`f(x)=ab^x`

Where, a represents the initial value of the function,

And, b represents the change factor ( growth or decay )

(i) If b > 1, then, the function is exponential growth function,

(ii) if 0 < b < 1, then, the function is exponential decay function,

Here, the given exponential function that represents the fish population after t years,

`P(t)=420(0.78)^t`

By comparing,

The change factor of the function P(t), b = 0.78

Since, 0 < 0.78 < 1,

Thus, the given function represents decay.

Also, decay factor = 1 - decay rate( in decimals ),

Let r be the change rate ( decay rate ) per year,

⇒ 1 - r = 0.78

⇒ -r = 0.78 - 1 ⇒ r = 0.22 = 22 %

Hence, each year the population decreases 22%.

Answer 2

Answer
part 1) A decay
part 2) 22%

Explanation
part 1
This is an exponential function.
The number been raised to the power of time is a fraction which is less than 1. This means that as the time increases the fraction is decreasing hence the population.

part 2
The formula for getting the population is P(t)=420×〖0.78〗^t
This means that the current population is 420 and 0.78 is the percentage.
This will be equal to 0.78=78/100
As a percentage =78/100×100=78%
This means that the population decrease by (100-78) = 22%