Problem 1. You ingest a certain dangerous bacteria at dinner by accident. When more than 1,000,000 of these organisms are in your gut, you will become violently ill. Their population doubles every 20 minutes. You ate dinner at 7pm and by 5am you are sick. At least how many of these bacteria did you ingest at dinner?

Problem 2. You have a sample of a certain radioactive substance. After two days, thirty percent of the sample has decayed. What is the half-life of the material and how much will have decayed after a week?

Please explain why. Thank you! :)

you doubled for ten hours

A=1E6 * (2)^10

.7=1e^(-.693 t/th)
t=2days,

take the ln of each side
ln(.7)=-.693*2/th

thalflife= -.693*2/ln(.7) I get about 3.89 days

Problem 1:

To solve this problem, we need to determine how many times the population of bacteria doubled from the time you ate dinner to the time you became sick. Since the population doubles every 20 minutes, we can calculate the number of 20-minute intervals between 7pm and 5am.

First, let's convert 7pm to minutes:
7pm = 7 hours * 60 minutes/hour = 420 minutes

Next, let's convert 5am to minutes:
5am = 5 hours * 60 minutes/hour = 300 minutes

The total time elapsed is:
300 minutes - 420 minutes = 780 minutes

Now, let's calculate the number of 20-minute intervals:
780 minutes / 20 minutes/interval = 39 intervals

Since the population doubles in each interval, we can express the number of bacteria in terms of the initial population. Let N be the initial number of bacteria.

After 39 intervals, the population would be:
N * 2^39

We know that once the population exceeds 1,000,000 bacteria, you become sick. Therefore, we can set up an equation:
N * 2^39 ≥ 1,000,000

Now we can solve for N:
N ≥ 1,000,000 / 2^39

Calculating this value gives us:
N ≥ 0.00155

So, you must have ingested at least 0.00155 (or approximately 0.16) of these bacteria at dinner.

Problem 2:

To determine the half-life of a radioactive substance and the amount decayed after a week, we need to use the information that after two days, thirty percent of the sample has decayed.

Let's assume the initial amount of the substance is X.

Since thirty percent of the substance has decayed after two days, we can set up an equation:
X - 0.3X = 0.7X

This means that 0.7X is the remaining amount after two days.

Since the half-life of a substance is the amount of time it takes for half of the initial amount to decay, we can say that:
0.7X = 0.5 * X

Simplifying this equation, we find that:
0.7 = 0.5

Now, we can find the half-life by solving for the time it takes for the substance to decay by a factor of 0.5.

To find the amount decayed after a week, we need to calculate how many half-lives have occurred in a week.

There are 7 days in a week, and since the half-life is the time it takes for half of the substance to decay, we can calculate the number of half-lives:
Number of half-lives = 7 days / 2 days/half-life = 3.5 half-lives

Since each half-life reduces the amount by a factor of 0.5, we can calculate the remaining amount after a week:
Remaining amount = 0.5^(Number of half-lives) * X

Plugging in the value of 3.5 for the number of half-lives, we get:
Remaining amount = 0.5^3.5 * X

Calculating this value gives us:
Remaining amount = 0.1768 * X

Thus, after a week, approximately 17.68% of the substance will have decayed.

To determine the exact half-life of the material, we would need to account for the decay constant specific to the substance being studied. Typically, the half-life is calculated using a decay constant and the formula: T(1/2) = ln(2)/λ, where λ is the decay constant. However, since the problem doesn't provide this information, we can't solve for the exact half-life.