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.

Question

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.

Bytelearn · Accepted Answer

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: $50,000 \times \left(\frac{4}{3}\right)^t < 100,000$.Isolating Exponential Term: To solve for $t$, we can divide both sides of the inequality by 50,000 to isolate the exponential term: $\left(\frac{4}{3}\right)^t < \frac{100,000}{50,000}$.Simplify Right Side: Simplifying the right side of the inequality gives us: $\left(\frac{4}{3}\right)^t < 2$.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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