Question

The number of bats in a colony is growing exponentially. After 2 years, there were 180 bats. After 5 years, there were 1440 bats. If the colony continues to grow at the same rate, how many bats are expected to be in the colony after 9 years

Answer 1

Answer:

23040 bats

Step-by-step explanation:

Let N(t) be the number of bats at time t

We know that exponential function

$y=ab^t$

According to question

$N(t)=ab^t$

Where t (in years)

Substitute t=2 and N(2)=180

$180=ab^2$...(1)

Substitute t=5 and N(5)=1440

$1440=ab^5$...(2)

Equation (1) divided by equation (2)

$\frac{180}{1440}=\frac{ab^2}{ab^5}=\frac{1}{b^{5-2}}$

By using the property $a^x\div a^y=a^{x-y}$

$\frac{1}{8}=\frac{1}{b^3}$

$b^3=8=2\times 2\times 2=2^3$

$b=2$

Substitute the values of b in equation (1)

$180=a(2)^2=4a$

$a=\frac{180}{4}=45$

Substitute t=9

$N(9)=45(2)^9=23040 bats$

Hence, after 9 years the expected bats in the colony=23040 bats