After a shipwreck, 120 rats manage to swim from the wreckage to a deserted island. The rat population on the island grows exponentially, and after 15 months, there are 280 rats on the island. A. Find a function that models the population t months after the arrival of the rats.
B. What will the population be 3 years after the shipwreck?
C. When will the population reach 2000 rats?

a.  

b.   Population after 3 years is 142

c.   50 years

Step-by-step explanation:

Given

Type of growth: Exponential

Initial number of rats = 120

Number of rats (15months) = 280

Solving (a)

Since the growth type is exponential, we make use of the following exponential progression

Where Xo is the initial population;

Xo = 125

is the current population at T month

So;

; when

Substitute these values in the above formula

Divide both sides by 120

Take 15th root of both sides

Subtract 1 from both sides

(Approximated)

Plug in values of R and Xo in

Write as a function

Hence, the function is

Solving (b):

Population after 3 years

In this case, T = 3

So:

(Approximated)

Solving (c): When population will reach 2000

Here: X(T) = 2000

So:

So:

Divide both sides by 120

Take Log of both sides

Apply law of logarithm

Divide both sides by Log(1.058)

Approximate

check the picture below.

well, the angle at the vertex d is clearly not a right-angle, let's check for the one at the vertex f which does look like,   however if it's indeed, that means ef is perpendicular to df, in which case their slopes will be negative reciprocal of each other.

so the negative reciprocal of the slope of ef is 1, however the slope of df is not 1, is 1/2, so nope, they're not perpendicular lines.

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step-by-step explanation: