The total number Q(t) of bacteria at any time t is given by the integral of the rate of growth function:
Q(t) = ∫ q(t) dt = ∫ 2^t dt = (2^t)/(ln(2)) + C.
After 4 hours, there are approximately 5,000 bacteria in the Petri dish.
Rate of Growth Function: The rate of growth of bacteria is given by q(t)=2^t, where t is in hours and q(t) is in thousands of bacteria per hour.
Initial Bacterial Culture: The initial culture starts with 3,000 bacteria.
Total Number of Bacteria Function: The total number Q(t) of bacteria at any time t is given by the integral of the rate of growth function: Q(t) = ∫ q(t) dt = ∫ 2^t dt = (2^t)/(ln(2)) + C
Initial Condition: Using the initial condition that Q(0) = 3000, we can solve for C: 3000 = (2^0)/(ln(2)) + C, C = 3000 - (1/(ln(2)))
Final Function Q(t): Substituting the value of C back into the total number function: Q(t) = (2^t)/(ln(2)) + 3000 - (1/(ln(2)))
Number of Bacteria After 4 Hours: To find the number of bacteria after 4 hours, substitute t = 4 into Q(t): Q(4) = (2^4)/(ln(2)) + 3000 - (1/(ln(2))), Q(4) ≈ 5000 (rounded to the nearest thousand). Therefore, after 4 hours, there are approximately 5,000 bacteria in the Petri dish.