Can someone explain this to me please the answer is 2023 but idk how u get to that btw the equation for the population of the rabbits was r = 50 {e}^(0.5t) for the year 2016 ​

Question

Can someone explain this to me please

the answer is 2023 but idk how u get to that

btw the equation for the population of the rabbits was

`r = 50 {e}^(0.5t)`
for the year 2016
​

Answer 1

8th year

Explanation:

r > C

50(e^0.5t) > 1000(e^0.1t)

(e^0.5t)/(e^0.1t) > 20

e^(0.5t-0.1t) > 20

e^0.4t > 20

ln(e^0.4t) > ln20

0.4t × lne > ln20

t > ln(20)/0.4

t > 7.489330685

Population of rabbits first exceeds the population of crickets during the 8th year

Answer 2

Explanation:

Start by finding when the populations become equal.

C = R

1000e^(0.1t) = 50e^(0.5t)

Divide both sides by 50.

20e^(0.1t) = e^(0.5t)

Divide both sides by e^(0.1t).

20 = e^(0.4t)

Take natural log of both sides.

ln 20 = 0.4t

Multiply both sides by 2.5

t = 2.5 ln 20

t ≈ 7.5

The population of rabbits first exceeds the population of crickets in the middle of the 7th year after 2016, or 2023.