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allsm [11]
3 years ago
13

Element X decays radioactively with a half life of 13 minutes. If there are 260 grams of Element X, how long, to the nearest ten

th of a minute, would it take the element to decay to 15 grams?
y=a(.5)^t/h

​
Mathematics
1 answer:
liraira [26]3 years ago
4 0

Answer:

It would take 54 minutes to the element X to decay to 15 grams.

Step-by-step explanation:

A radioactive half-life refers to the amount of time it takes for half of the original isotope to decay and its given by

                                                N(t)=N_0(\frac{1}{2})^\frac{t}{t_{1/2}}

where,

N(t) = quantity of the substance remaining

N_0 = initial quantity of the substance

t = time elapsed

t_{1/2} = half life of the substance

We know that the element X decays radioactively with a half life of 13 minutes (t_{1/2}), there are 260 grams of it (N_0) and we want to find how long (t) would it take the element to decay to 15 grams (N(t)).

Using the above formula and solving for t, we get that

15=260(\frac{1}{2})^\frac{t}{13}\\\\260\left(\frac{1}{2}\right)^{\frac{t}{13}}=15\\\\\left(\frac{1}{2}\right)^{\frac{t}{13}}=\frac{3}{52}\\\\\ln \left(\left(\frac{1}{2}\right)^{\frac{t}{13}}\right)=\ln \left(\frac{3}{52}\right)\\\\\frac{t}{13}\ln \left(\frac{1}{2}\right)=\ln \left(\frac{3}{52}\right)\\\\t=-\frac{13\ln \left(\frac{3}{52}\right)}{\ln \left(2\right)} \approx 54 \:min

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