Question

dr. rivera recorded 200 bees present in a population. assuming the population of bees grows at a rate of 6% per year, find the growth factor and write an exponential model for this situation in terms of t, the number of years passed. round to two decimal places, when necessary. show your work here hint: to add an exponent x^t, type \exponent\ or press \^\ \\( y = 200(7.42)^t \\) \\( y = 201(7.42)^t \\) \\( y = 201.06(213.1236)^t \\) \\( y = 200(1.06)^t \\) in a census, green haven had a population of 360. assuming the population of green haven decays at a rate of 6% per year, find the decay factor and write exponential model for this situation in terms of t, the number of years passed. round to two decimal places, when necessary. show your work here hint: to add an exponent x^t, type \exponent\ or press \^\ \\( a(t) = -360(6.58)^t \\) \\( a(t) = 360(0.94)^t \\) \\( a(t) = -360(0.94)^t \\) \\( a(t) = -2520(0.94)^t \\)

Explanation:

First Problem (Bee Population Growth)

Step1: Recall exponential growth formula

The general form of an exponential growth model is \( y = a(1 + r)^t \), where \( a \) is the initial amount, \( r \) is the growth rate (as a decimal), and \( t \) is time.
Here, \( a = 200 \) (initial number of bees), and \( r = 6\% = 0.06 \).

Step2: Calculate the growth factor

The growth factor is \( 1 + r \). Substituting \( r = 0.06 \), we get \( 1 + 0.06 = 1.06 \).

Step3: Write the exponential model

Substituting \( a = 200 \) and the growth factor \( 1.06 \) into the formula, we get \( y = 200(1.06)^t \).

Step1: Recall exponential decay formula

The general form of an exponential decay model is \( A(t)=a(1 - r)^t \), where \( a \) is the initial amount, \( r \) is the decay rate (as a decimal), and \( t \) is time.
Here, \( a = 360 \) (initial population), and \( r = 6\% = 0.06 \).

Step2: Calculate the decay factor

The decay factor is \( 1 - r \). Substituting \( r = 0.06 \), we get \( 1 - 0.06 = 0.94 \).

Step3: Write the exponential model

Substituting \( a = 360 \) and the decay factor \( 0.94 \) into the formula, we get \( A(t)=360(0.94)^t \). Also, population can't be negative, so we discard options with negative coefficients.

Answer:

\( y = 200(1.06)^t \) (corresponding to the option \( y = 200(1.06)^t \))

Second Problem (Green Haven Population Decay)