i need help solving this

Question

This exercise uses the exponential growth model.

Some bullfrogs were introduced into a small pond. The following graph shows the bullfrog population for the next few years. Assume that the population grows exponentially.

The t n-coordinate plane is given. The n-axis is labeled Frog Population. The curve enters the window at n = 100 on the positive n-axis, goes up and right becoming more steep, passes through the approximate point (2, 225), passes through the approximate point (4, 506), and exits the window in the first quadrant.

(a)

What was the initial bullfrog population?

bullfrogs

(b)

Find a function that models the bullfrog population t years since the bullfrogs were put into the pond. (Round your r value to four decimal places.)

n(t) =

(c)

What is the projected bullfrog population after 14 years?

bullfrogs

(d)

Estimate how long (in years) it takes the population to reach 142,000. (Round your answer to the nearest whole number.)

years after introduction

Answer 1

a) n(t)=N₀e^rt

When t = 0, 100=N₀(1)

N₀= 100

b) n(t) = 100e^rt

225 = 100e^2r (1)

506 = 100e^4r (2)

Divide (2) by (1)

2.25 = e^4r-2r

2r = ln(2.25)

r = 0.4052

n(t) = 100 e^0.4052t

c) n(14) = 100 e^0.4052(14)

d)14200 = 100e^0.4052t

t = ln(142)/0.4052 ≈ 12 years

Answer 2