Find an exponential growth model

n(t) = Cat
for the population t years since the bullfrogs were put into the pond with coordinates (0,148) (2,333)

do you mean n = c e^(at) ? I will assume that.

when t = 0 , e^at = e^0 = 1
so
c = 148

whet t = 2
333 = 148 e^(2a)
2.25 = e^(2a)
so
ln 2.25 = 2 a
.8109 = 2 a
a = .405
so
n = 148 e^(.405 t)

To find an exponential growth model, we need to use the given information about the population of bullfrogs at different time points. Let's start by plugging in the given coordinate values into the general exponential growth model equation, n(t) = Cat.

Using the coordinate (0, 148), we have:
148 = Ca(0)

Since anything multiplied by 0 is 0, we can conclude that Ca(0) = 0.

Now, let's use the second coordinate (2, 333):
333 = Ca(2)

Now, we have a system of equations:
Ca(0) = 0
Ca(2) = 333

To solve this system, we can use the second equation to find the value of Ca:
Ca = 333/2
Ca = 166.5

Now that we have found the value of Ca, we can substitute it back into the exponential growth model equation.

Therefore, the exponential growth model for the population of bullfrogs is:
n(t) = 166.5at

To find an exponential growth model for the population of bullfrogs over time, we need to apply the given information to the general form of an exponential growth equation: n(t) = Ca^t.

We are given two coordinates: (0, 148) and (2, 333). These coordinates represent the initial population at time t = 0 and the population after 2 years. Let's use this information to derive the specific equation for exponential growth.

Step 1: Plug in the first set of coordinates (t = 0, n = 148).

148 = Ca^0
148 = Ca^0 = C(1) = C

So, we have found the value of C, which is 148.

The exponential growth model is now n(t) = 148a^t.

Step 2: Plug in the second set of coordinates (t = 2, n = 333).

333 = 148a^2

Step 3: Solve the equation for 'a'.

Divide both sides by 148:

333/148 = a^2

2.25 ≈ a^2

Take the square root of both sides:

√(2.25) = √(a^2)

1.5 ≈ a

The value of 'a' is approximately 1.5.

The exponential growth model for the population of bullfrogs is:

n(t) = 148(1.5)^t