Question

A sample of a radioactive substance has an initial mass of 27.1 mg. This substance follows a continuous exponential decay model and has a half-life of 8

minutes. (a) Let + be the time (in minutes) since the start of the experiment, and
let v be the amount of the substance at time t.
Write a formula relatina v to t.
Use exact expressions to fill in the missing parts of the formula.
Do not use approximations.

Answer 1

Step-by-step explanation:

This may help

In a continuous exponential decay model, the amount of a radioactive substance can be described by the formula:

v = v0 * e^(-kt)

Where:

v is the amount of the substance at time t,

v0 is the initial amount of the substance,

e is the base of the natural logarithm (approximately 2.71828),

k is the decay constant,

t is the time.

To determine the missing parts of the formula, we need to find the decay constant (k) using the half-life (t1/2).

Given that the half-life of the substance is 8 minutes, we know that after 8 minutes, the amount of the substance will be reduced to half its initial value. Using this information, we can set up the following equation:

(1/2) * v0 = v0 * e^(-k * 8)

Simplifying the equation, we can cancel out v0:

1/2 = e^(-k * 8)

To solve for k, we'll take the natural logarithm (ln) of both sides of the equation:

ln(1/2) = ln(e^(-k * 8))

ln(1/2) = -k * 8

Now we can solve for k by dividing both sides of the equation by -8:

k = ln(1/2) / -8

Now that we have the value of k, we can substitute it back into the original formula:

v = v0 * e^(-kt)

v = 27.1 mg * e^(-(ln(1/2) / -8) * t)

Therefore, the formula relating v to t is:

v = 27.1 mg * e^(-(ln(1/2) / -8) * t)