The rate of decay in the mass, M, of a radioactive substance is given by the differential equation dM dt equals negative 1 times k times M, where k is a positive constant. If the initial mass was 100g, then find the expression for the mass, M, at any time t.

dM/dt = -k M

separate variables, M left, t right
dM/M = -k dt
integrate
ln M = -k t + c

note e^log a = a
ao
e^ln M = M = e^(-kt+c) = e^-kt e^c
or since e^c is some constant call it C
M = C e^-kt
note that when t = 0
e^-kt = 1
so
C = initial amount
so here
M = 100 e^-kt

Damon is right

To find the expression for the mass, M, at any time t, we will integrate the given differential equation.

Given: dM/dt = -kM

Let's separate the variables and integrate:

1/M dM = -k dt

Integrating both sides:

∫ (1/M) dM = -k ∫ dt

ln|M| = -kt + C

Where C is the constant of integration.

Now, let's solve for M:

M = e^(ln|M|) = e^(ln(M)) = e^(ln(e^(-kt + C)))

Since e^ln(x) = x, we have:

M = e^(-kt + C)

To find the value of C, we'll use the initial condition that the initial mass was 100g, which means when t = 0, M = 100.

Therefore:

100 = e^(-k * 0 + C)
100 = e^C

Taking the natural logarithm of both sides:

ln(100) = C

So the expression for the mass, M, at any time t is:

M = e^(-kt + ln(100))

This is the desired expression for the mass, M, at any time t.

To find the expression for the mass, M, at any time t, we need to solve the given differential equation.

The differential equation is: dM/dt = -kM

To solve this type of differential equation, we can use separation of variables method.

Step 1: Rewrite the equation in differential form:
(1/M) dM = -k dt

Step 2: Integrate both sides of the equation:
∫(1/M) dM = ∫(-k dt)

Step 3: Integrate:
ln|M| = -kt + C

Here, C is the constant of integration.

Step 4: Solve for M:
|M| = e^(-kt + C)

Since e^C is just another constant, let's rename it as A:
|M| = A * e^(-kt)

Step 5: Remove the absolute value sign:
M = ±A * e^(-kt)

However, since the initial mass is given as 100g, we can determine the sign and the value of A.

Given the initial mass was 100g, at t = 0, M = 100g.

Plugging these values into the equation:
100 = ±A * e^(0)

Simplifying,
100 = ±A * 1
±A = 100

So, A can be either 100 or -100.

Thus, the expression for the mass, M, at any time t is:
M = ±100 * e^(-kt)

Note: The ± sign accounts for the possibility of decay (negative values) or growth (positive values) depending on the nature of the radioactive substance.