Math Help Please!!

Question

Scientists at the local university are studying a highly toxic radioactive compound. They have a 200-g sample. They will place this compound in a chamber to be tested. After 3 h, 140 g of the sample remains. a) Determine the half-life of this radioactive material.b) Determine the equation that represents the decay of the material from your answer to part a).c) Graph the amount of the material remaining as a function of time. Use your graph to determine the amount of time it will take for 85% of the material to decay.

Answer 1

(a)

 

An exponential decay law applies here.  For N(t) = amount present at time t,

 

N(t) = N(0)e^-kt

 

k = hours^-1

 

N(0)  = 200 g

N(3 hours) = N(3) = 140 g

 

140 = 200e^-k(3)

 

k = ln(140/200) / (-3) = 0.118892 hours^-1

 

To get half life, set

 

1/2 = e^{-0.118892 t}

 

t_1/2 = ln(1/2) / (-0.118892) = 5.83 hours

 

(b)

 

N(t) = (200 g) e^{-0.118892 t}

 

For parts (c) and (d), just plot N(t) and obtain the answers  graphically.  The medium of this web site is not suitable for that.

Answer 2

(a) The initial mass is M₀ = 200 g.

In general, the equation for radioactive decay is

 

M(t) = M₀(1/2)^t/τ

 

where

 

M(t) is the mass remaining as a function of time t, and τ is the half-life, of the time it takes for half of the substance to decay.

 

We know one data point. We know that M(t = 3h) = 140 g. Let's solve the above equation for τ and use this data to give τ a numerical value.

 

(1/2)^t/τ = M/M₀

 

Take the natural log of each side.

 

ln(1/2)^t/τ = ln(M/M₀)

 

Using the properties of logarithms, we can make the power a coefficient and write:

 

(t/τ)ln(1/2) = ln(M/M₀)

 

t/τ = ln(M/M₀) / ln(1/2)

 

We can flip this equation upside-down, and write

 

τ/t = ln(1/2) / ln(M/M₀)

 

Now, multiply both sides by t.

 

τ = t ln(1/2) / ln(M/M₀)

 

Now, we can use the given data point to solve for τ:

 

τ = (3 h) ln(1/2) / ln(140g/200g)

 

or

 

τ = (3 h) ln(1/2) / ln(7/10)

 

τ = 5.83 hour

 

(b) We can now specifically write the original equation now as:

 

M(t) = (200 g)(1/2)^t/5.83

 

 

Answer 3

140 = 200(0.5)^3/h

0.7 = (0.5)^3/h

ln 0.7 = (3/h) ln 0.5

-0.35667 ≈ (3/h)(-0.6931

0.5146 ≈ 3/h

0.5146h ≈ 3

h ≈ 5.8