When exposed to sunlight, the # of bacteria in a culture decreases exponentially at the rate of 10% per hour. What is the best approximation for the number of hours required for the initial # of bacteria to decrease by 50%.
When exposed to sunlight, the # of bacteria in a culture decreases exponentially at the rate of 10% per hour. What is the best approximation for the number of hours required for the initial # of bacteria to decrease by 50%.
Let A(0) denote the initial amount of bacteria and A(t) the amount of bacteria after t hours of exposion to sun light.
After 1 h there remains 90% of the bacteria. After t hours remains $\displaystyle 0.9^t$ of the bacteria.
You know that $\displaystyle \frac{A(t)}{A(0)}=\frac12$
Use the equation
$\displaystyle A(t) = A(0) \cdot (0.9)^t$
plug in the values you know and solve for t.
I've got t is roughly 6h 35 min
Learning to translate is the point, here.
"# of bacteria in a culture"
Define this. B(t) = # of bacteria in a culture - at time t in hours.
"decreases exponentially"
A clue about $\displaystyle B(t) = ae^{bt}$
"decreases exponentially at the rate of 10% per hour."
A clue about the parameter: $\displaystyle B(1) = ae^{b} = 0.90*ae^{0} = B(0)$
"number of hours required for the initial # of bacteria to decrease by 50%."
B(x) = $\displaystyle 0.90^{x} = 0.50$ -- Solve for x.