Final answer:
To find the expression for the number of bacteria after a given time, we use the formula A = P * e^(kx), where A is the final number of bacteria, P is the initial number of bacteria, k is the growth rate constant, and x is the time in hours. We can find the number of bacteria after 8 hours, the rate of growth after 8 hours, and when the population will reach 7000 using this formula.
Step-by-step explanation:
To find an expression for the number of bacteria after x hours, we can use the formula for exponential growth: A = P * e^(kx), where A is the final number of bacteria, P is the initial number of bacteria, k is the growth rate constant, and x is the time in hours. Since we know that after 4 hours the population has increased to 320, we can plug in these values to solve for k. Once we have k, we can use it to find the number of bacteria after any other given time by plugging it into the equation for A.
To find the number of bacteria after 8 hours, we substitute x = 8 into the equation and solve for A.
To find the rate of growth after 8 hours, we take the derivative of the equation with respect to x and evaluate it at x = 8.
To find when the population will reach 7000, we set A = 7000 in the equation and solve for x.