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On her way home from the laboratory, Duru realized that she left a test tube containing 50,000 bacteria in the lab. Each minute that passes,
(1)/(3) of the total number of bacteria duplicate. If the number of bacteria reaches 100,000 , the test tube will explode! Naturally, she turned around and rushed back to the lab.
It took Duru
t minutes to return to the lab, and she found the test tube intact.
Write an inequality in terms of
t that models the situation.

On her way home from the laboratory, Duru realized that she left a test tube containing $50$ , $000$ bacteria in the lab. Each minute that passes, $\frac{1}{3}$ of the total number of bacteria duplicate. If the number of bacteria reaches $100$ , $000$ , the test tube will explode! Naturally, she turned around and rushed back to the lab. $\newline$ It took Duru $t$ minutes to return to the lab, and she found the test tube intact. $\newline$ Write an inequality in terms of $t$ that models the situation.

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Write and solve equations for proportional relationships

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Q. On her way home from the laboratory, Duru realized that she left a test tube containing

50

,

000

bacteria in the lab. Each minute that passes,

\frac{1}{3}

of the total number of bacteria duplicate. If the number of bacteria reaches

100

,

000

, the test tube will explode! Naturally, she turned around and rushed back to the lab.

\newline

It took Duru

t

minutes to return to the lab, and she found the test tube intact.

\newline

Write an inequality in terms of

t

that models the situation.

Given Information: We are given that the number of bacteria duplicates by $\frac{1}{3}$ of its total every minute. This means that if we start with $N_0$ bacteria, after one minute, we will have $N_0 + \frac{1}{3}N_0 = \frac{4}{3}N_0$ bacteria. This is a geometric progression where each term is $\frac{4}{3}$ times the previous term.
Initial Bacteria Count: Let's denote the initial number of bacteria as $N_0 = 50,000$ . After $t$ minutes, the number of bacteria will be $N_0 \times \left(\frac{4}{3}\right)^t$ . We want this number to be less than $100$ , $000$ to prevent the test tube from exploding.
Inequality Setup: We can now set up the inequality that models the situation: $\newline$ $50,000 \times \left(\frac{4}{3}\right)^t < 100,000$ .
Isolating Exponential Term: To solve for $t$ , we can divide both sides of the inequality by $50$ , $000$ to isolate the exponential term: $\newline$ $\left(\frac{4}{3}\right)^t < \frac{100,000}{50,000}$ .
Simplify Right Side: Simplifying the right side of the inequality gives us: $\newline$ $\left(\frac{4}{3}\right)^t < 2$ .
Applying Logarithm: Now, we need to find the value of $t$ that satisfies this inequality. Since $\frac{4}{3}$ is greater than $1$ , the function $\left(\frac{4}{3}\right)^t$ is increasing, and we can apply the logarithm to both sides of the inequality to solve for $t$ . However, since we are only asked to write the inequality in terms of $t$ , we do not need to solve for $t$ explicitly.

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