Solved on Nov 03, 2023

Find the amount of radioactive substance present (in grams) after 2 years, given the formula y=8,000(2)0.3ty=8,000(2)^{-0.3 t}, where tt is in months. Round the result to three decimal places if needed.

STEP 1

Assumptions1. The formula for the amount of a radioactive substance present at time tt is y=8,000()0.3ty=8,000()^{-0.3 t} . The time tt is in months3. We are asked to find the amount of the substance present in years4.1 year is equivalent to12 months

STEP 2

First, we need to convert the time from years to months since the formula uses tt in months.
t=2years=2times12monthst =2\, years =2 \\times12\, months

STEP 3

Calculate the time tt in months.
t=2times12=24monthst =2 \\times12 =24\, months

STEP 4

Now that we have the time in months, we can substitute this into the formula to find the amount of the substance present.
y=8,000(2)0.3ty =8,000(2)^{-0.3 t}

STEP 5

Substitute t=24t =24 into the formula.
y=8,000(2)0.3times24y =8,000(2)^{-0.3 \\times24}

STEP 6

Calculate the exponent.
0.3times24=.2-0.3 \\times24 = -.2So the formula becomesy=8,000(2).2y =8,000(2)^{-.2}

STEP 7

Calculate the value of 27.22^{-7.2}.

STEP 8

Multiply the result from step7 by8,000 to find the amount of the substance present.
y=8,000times27.2y =8,000 \\times2^{-7.2}

STEP 9

Calculate the value of yy.

STEP 10

If necessary, round the result to three decimal places.

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