Question

the relative growth rate of a certain bacteria colony is 21% per hour. suppose there are 7 bacteria initially. use the exponential growth model ( p(t) = p_0 e^{kt} ) to answer the following
a find a function that describes the population of bacteria after ( t ) hours.
b how many bacteria should be expected after 1 day?
a ( p(t) = square )
(type an exact answer in terms of ( e ). use integers or decimals for any numbers in the expression.)

Explanation:

Step1: Identify initial values and rate

Initial population \( P_0 = 7 \), growth rate \( r = 21\% = 0.21 \) per hour. For exponential growth, \( k = r = 0.21 \).

Step2: Write the population function

The model is \( P(t) = P_0 e^{kt} \), so substitute values: \( P(t) = 7e^{0.21t} \).

Step3: Calculate population after 1 day

1 day = 24 hours. Substitute \( t = 24 \): \( P(24) = 7e^{0.21 \times 24} = 7e^{5.04} \).

Answer:

a. \( 7e^{0.21t} \)
b. \( 7e^{5.04} \) (or approximately 7×154.47 ≈ 1081.29 if decimal is needed)