The rate of decay is proportional to the mass for radioactive material. For a certain radioactive isotope, this rate of decay is given by the differential equation dm/dt = -.022m, where m is the mass of the isotope in mg and t is the time in years.

A. If m(0)=20, write a function m(t) for the radioactive decay of the isotope. Show the steps in separating variables of the given differential equation and solving the equation for m.

B. The half-life of a radioactive substance is the time required for half of the substance to decay. What is the half-life of this radioactive isotope to the nearest tenth of a year?

Steve ur a savage

c'mon, you can do this.

dm/dt = -0.22m
dm/m = -0.22 dt
ln(m) = -0.22t + c
m = c e^(-0.22t)

c is the initial amount, so

m(t) = 20 e^(-0.22t)

I'm sure you can find the half-life now, ok?

Thank you

Do not cheat and use this website!

A. To separate variables in the given differential equation, we need to express dm and dt separately and rearrange the equation. Let's start:

Given differential equation: dm/dt = -0.022m

Step 1: Separate the variables by multiplying both sides by dt:
dm = -0.022m dt

Step 2: Divide both sides by m:
dm/m = -0.022 dt

Step 3: Integrate both sides with respect to their respective variables:
∫(dm/m) = ∫(-0.022 dt)

Step 4: Evaluate the integrals:
ln|m| = -0.022t + C

Here, C is the constant of integration.

Step 5: Solve for m by exponentiating both sides:
e^(ln|m|) = e^(-0.022t + C)

Simplifying this gives:
|m| = e^C * e^(-0.022t)

Step 6: Since e^C is a positive constant, let's denote it as k:
|m| = k * e^(-0.022t)

Step 7: To find the function m(t), we consider the initial condition m(0) = 20:
|m(0)| = k * e^(-0.022 * 0)
|20| = k * e^0
20 = k

Therefore, our final function m(t) is:
m(t) = 20 * e^(-0.022t)

B. To find the half-life of the radioactive isotope, we need to find the time at which the mass has decreased to half its initial value (m(0) = 20/2 = 10 mg).

Substituting this into the function m(t), we have:
10 = 20 * e^(-0.022t)

Divide both sides by 20:
0.5 = e^(-0.022t)

Take the natural logarithm of both sides to isolate t:
ln(0.5) = -0.022t

Solve for t:
t = ln(0.5) / -0.022

Using a calculator, we find t ≈ 31.5548 years.

Therefore, the half-life of this radioactive isotope is approximately 31.6 years (rounded to the nearest tenth of a year).