At noon, Professor Simon has a petri dish with 10,000 cells of Bacteria A. In another petri dish, he
has 33,000 cells of Bacteria B. Every hour, Bacteria A grows by 8%. Every hour Bacteria B cells die off,
decreasing the number of cells by 6%. After how many hours will there be more cells of Bacteria A than of
Bacteria B?
PLEASE HELP!! I don't understand this and i really need help.
10000 (1 + .08)^t = 33000 (1 - .06)^t
log(10) + t log(1.08) =
... log(33) + t log(.94)
t [log(1.08) - log(.94)] =
... log(33) - log(10)
t = log(33 / 10) / log(1.08 / .96)
should be .94 in last eqn...not .96
To solve this problem, we need to determine the number of hours it takes for the number of cells of Bacteria A to exceed the number of cells of Bacteria B.
We can start by setting up a formula to represent the growth of Bacteria A and the decrease of Bacteria B over time.
Let's assume 𝑥 represents the number of hours it takes for Bacteria A to surpass Bacteria B.
After 𝑥 hours, the number of cells in Bacteria A will be 10,000 * (1 + 0.08)^𝑥, where (1 + 0.08) represents the growth of Bacteria A by 8% per hour.
After 𝑥 hours, the number of cells in Bacteria B will be 33,000 * (1 - 0.06)^𝑥, where (1 - 0.06) represents the decrease of Bacteria B by 6% per hour.
To find the point where Bacteria A exceeds Bacteria B, we need to solve the following equation:
10,000 * (1 + 0.08)^𝑥 > 33,000 * (1 - 0.06)^𝑥
Now, we need to solve this equation for 𝑥 by using logarithms. Taking the logarithm (base 10) of both sides will help us isolate 𝑥:
log(10,000 * (1 + 0.08)^𝑥) > log(33,000 * (1 - 0.06)^𝑥)
By using logarithmic properties (log(a * b) = log(a) + log(b) and log(a^b) = b * log(a)), the equation becomes:
log(10,000) + 𝑥 * log(1 + 0.08) > log(33,000) + 𝑥 * log(1 - 0.06)
Now we can rearrange this equation to solve for 𝑥:
𝑥 * log(1 + 0.08) - 𝑥 * log(1 - 0.06) > log(33,000) - log(10,000)
We can now simplify the equation further:
𝑥 * (log(1 + 0.08) - log(1 - 0.06)) > log(33,000) - log(10,000)
Finally, we can calculate the value of 𝑥:
𝑥 > (log(33,000) - log(10,000)) / (log(1 + 0.08) - log(1 - 0.06))
Using a calculator, we can evaluate the right side of the equation to find the value of 𝑥. The result will give us the number of hours it takes for the number of cells of Bacteria A to surpass the number of cells of Bacteria B.