Answer:
The time it would take the element to decay to 28 grams is approximately 22.5 minutes
Step-by-step explanation:
The give half life of element X = 6 minutes
The given initial mass of the radioactive element X = 390 grams
The mass of the element X after decay = 28 grams
The function that represent the decay of element X is given as follows here;
[tex]y = a \cdot (0.5)^{\dfrac{t}{h} }[/tex]
Which is of the form;
[tex]N(t) = N_0 \left (\dfrac{1}{2} \right )^{\dfrac{t}{t_{1/2}}[/tex]
Therefore;
h = 6 minutes = The time duration of the half life
a = N₀ = The initial mass = 390 g
N(t) = The final mass = 28 g
t = The time it takes the element to decay
Plugging in the values gives;
[tex]28 = 390 \times(0.5)^{\dfrac{t}{6} }[/tex]
[tex](0.5)^{\dfrac{t}{6} } = \dfrac{28}{390}[/tex]
Therefore;
[tex]\dfrac{t}{6} \times ln\left(0.5} \right) = ln \left(\dfrac{28}{390} \right)[/tex]
t/6 = ln(29/390)/ln(0.5)
t = 6 × ln(29/390)/ln(0.5) ≈ 22.496
Given the answer to the nearest tenth of a minute, the time it would take the element to decay to 28 grams, t ≈ 22.5 minutes.
