Question

The mass of a substance, which follows a continuous exponential growth model, is being studied in a lab. The doubling time for this substance was observed to be 4 minutes. There were 24.6mg of the substance present at the beginning of the study.

Let t be the time (in days) since the beginning of the study, and let y be the amount of the substance at time t. Write a formula relating y to t. Do not use approximations.

Answer 1

To write a formula relating the amount of substance to time in a continuous exponential growth model, we can use the formula: y = a * e^(kt), where a is the initial amount, e is the base of natural logarithms, k is the growth rate constant, and t is the time. Substituting the values given in the question, the formula becomes: y = 24.6 * e^(ln(2)/4 * t).

Step-by-step explanation:

To write a formula relating the amount of substance, y, to time, t, we can use the formula for exponential growth:

y = a * e^(kt)

Where a is the initial amount, e is the base of natural logarithms (approximately 2.71828), k is the growth rate constant, and t is the time.

In this case, the doubling time is 4 minutes, which means the growth rate constant k is given by the equation:

e^(k*4) = 2

Solving for k, we find k = ln(2)/4.

Substituting the initial amount a = 24.6 mg and the growth rate constant into the formula, we get the final formula relating y to t:

y = 24.6 * e^(ln(2)/4 * t)