Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A biologist is studying the growth of a bacterial colony. The number of bacteria, N(t), after t hours is given by the function N(t) = N_0 * (1.2)^t, where N_0 is the initial number of bacteria. If the biologist observes 500 bacteria after 3 hours, what was the initial number of bacteria (N_0)?
- A500
- B347.22
- C289.35
- D416.67
Show answer & explanationAnswer & explanation
Correct answer: C. 289.35
We are given N(t) = 500 when t = 3. Substitute these values into the function: 500 = N_0 * (1.2)^3. Calculate (1.2)^3 = 1.2 * 1.2 * 1.2 = 1.44 * 1.2 = 1.728. So, 500 = N_0 * 1.728. To find N_0, divide 500 by 1.728: N_0 = 500 / 1.728 ≈ 289.3518. Rounding to two decimal places gives 289.35.
Why the other options are wrong
- A. This would imply no growth or that N_0 was 500, which is not true if N(3) = 500.
- B. This might be a result of dividing by (1.2)^2 instead of (1.2)^3, or a calculation error.
- D. This is 500 / 1.2, which is an incorrect application of the formula.
Solving for Initial Value in Exponential Growth
To find the initial value (N_0 or P_0) in an exponential growth model (N(t) = N_0 * b^t), given a future value N(t) at a specific time t, rearrange the formula to N_0 = N(t) / b^t.
- The initial value is the amount at time t=0.
- The base 'b' is the growth factor.
- Requires knowing a value at a time t > 0.
Memory trick: Go BACK in Time, DIVIDE by the growth!