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Element X decays radioactively with a half life of 6 minutes. If there are 900 grams ofElement X, how long, to the nearest tenth of a minute, would it take the element todecay to 121 grams?

Element X decays radioactively with a half life of 6 minutes. If there are 900 grams-example-1
User Marlina
by
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1 Answer

5 votes

The equation for the half life is expressed as

y = a(0.5)^t/h

where

y represents the final amount after t minutes

a represents the initial amount

0.5 is the decay rate which is half

t is the time

h is the time taken for half of the element to decay.

From the information given,

a = 900

y = 121

h = 6

We want to find t.

By substituting the values into the equation, we have

121 = 900(0.5)^t/6

Dividing both sides of the equation by 900, we have

121/900 = 900/900(0.5)^t/6

0.134 = 0.5^0.17t

Taking the natural log of both sides of the equation, we have

ln 0.134 = ln 0.5^0.17t

On the rightm we would apply one of the rules of logarithm which is expressed as

lna^x = xlna

Thus, we have

ln 0.134 = 0.17t ln0.5

Dividing both sides of the equation by ln 0.5, we have

ln 0.134/ln 0.5 = 0.17t ln0.5/ln 0.5

0.17t = 2.8997

t = 2.8997/0.17

t = 17.1

The time it will take the element to decay to 121 grams is 17.1 minutes

User Jenilyn
by
8.8k points
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