Question

the function (f(h)=m(\frac{1}{2})^h) gives the mass, m, of a radioactive substance remaining after h half - lives. cobalt - 60 has a half - life of about 5.3 years. which equation gives the mass of a 50 mg cobalt - 60 sample remaining after 10 years, and approximately how many milligrams remain?
(f(x)=50(0.185)^{10};0 mg)
(f(x)=50(0.5)^{10};0.05 mg)
(f(x)=50(0.877)^{10};13.5 mg)
(f(x)=50(0.933)^{10};25 mg)

Explanation:

Step1: Calculate number of half - lives

First, find the number of half - lives $h$ in 10 years. Given the half - life of Cobalt - 60 is 5.3 years, so $h=\frac{10}{5.3}\approx1.887$.

Step2: Use the decay formula

The decay formula is $f(h)=m(\frac{1}{2})^h$, with $m = 50$ mg. Substituting $m = 50$ and $h\approx1.887$ into the formula, we get $f(x)=50\times(0.5)^{1.887}$. Now, $(0.5)^{1.887}\approx0.27$. So $f(x)=50\times0.27 = 13.5$ mg. Also, $(0.5)^{\frac{10}{5.3}}\approx0.27\approx0.877^{\ 10}$ (by taking the 10th root of both sides and doing some exponent manipulations).

Answer:

C. $f(x)=50(0.877)^{10};13.5$ mg