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On his way home from the laboratory, Louie realized that he left a test tube containing 2560 yeast cells in the lab. Every two minutes, the number of cells in the test tube increases by 50%. If the number of cells reaches 98,415 , the test tube will explode! Naturally, he turned around and rushed back to the lab. It took Louie t minutes to return to the lab, and he found the test tube intact. Write an inequality in terms of t that models the situation.

On his way home from the laboratory, Louie realized that he left a test tube containing 25602560 yeast cells in the lab. Every two minutes, the number of cells in the test tube increases by 50% 50 \% . If the number of cells reaches 9898,415415 , the test tube will explode! Naturally, he turned around and rushed back to the lab.\newlineIt took Louie t t minutes to return to the lab, and he found the test tube intact.\newlineWrite an inequality in terms of t t that models the situation.

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Q. On his way home from the laboratory, Louie realized that he left a test tube containing 25602560 yeast cells in the lab. Every two minutes, the number of cells in the test tube increases by 50% 50 \% . If the number of cells reaches 9898,415415 , the test tube will explode! Naturally, he turned around and rushed back to the lab.\newlineIt took Louie t t minutes to return to the lab, and he found the test tube intact.\newlineWrite an inequality in terms of t t that models the situation.
  1. Denote Initial Number: Let's denote the initial number of yeast cells as N0N_0, which is 25602560. The number of cells increases by 50%50\% every two minutes. This means that after every two minutes, the number of cells is multiplied by 1.51.5 (since 50%50\% increase is the same as multiplying by 1+0.51 + 0.5).
  2. Express Number of Cells: We need to express the number of yeast cells, NN, after tt minutes. Since the cells increase by 50%50\% every two minutes, we can say that for every two minutes that pass, we multiply the initial number of cells by 1.51.5. The number of two-minute intervals in tt minutes is t2\frac{t}{2}. Therefore, the formula to calculate the number of cells after tt minutes is N=N0×(1.5)t2N = N_0 \times (1.5)^{\frac{t}{2}}.
  3. Set Inequality Condition: We are given that the test tube will explode if the number of cells reaches 98,41598,415. Therefore, we want the number of cells, NN, to be less than 98,41598,415. We can write this as an inequality: N_0 \times (1.5)^{\frac{t}{2}} < 98,415.
  4. Substitute Initial Number: Substitute the initial number of cells, N0N_0, with 25602560 into the inequality: 2560 \times (1.5)^{(t/2)} < 98,415.
  5. Model Situation with Inequality: Now we have an inequality that models the situation. The inequality in terms of tt is 2560 \times (1.5)^{\frac{t}{2}} < 98,415.

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