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

Studdy Solution

STEP 1

Assumptions1. The formula for the amount of a radioactive substance present at time $t$ is $y=8,000()^{-0.3 t}$ . The time $t$ 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 $t$ in months.
$t =2\, years =2 \\times12\, months$

STEP 3

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

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.
$y =8,000(2)^{-0.3 t}$

STEP 5

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

STEP 6

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

STEP 7

Calculate the value of $2^{-7.2}$ .

STEP 8

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

STEP 9

Calculate the value of $y$ .

STEP 10

If necessary, round the result to three decimal places.

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

Studdy Solution

STEP 1

Assumptions1. The formula for the amount of a radioactive substance present at time $t$ is $y=8,000()^{-0.3 t}$ . The time $t$ 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 $t$ in months.
$t =2\, years =2 \\times12\, months$

STEP 3

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

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.
$y =8,000(2)^{-0.3 t}$

STEP 5

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

STEP 6

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

STEP 7

Calculate the value of $2^{-7.2}$ .

STEP 8

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

STEP 9

Calculate the value of $y$ .

STEP 10

If necessary, round the result to three decimal places.

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Question 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}y=8,000(2)−0.3t, where ttt is in months. Round the result to three decimal places if needed.

Studdy Solution

STEP 1

Assumptions1. The formula for the amount of a radioactive substance present at time ttt is y=8,000()−0.3ty=8,000()^{-0.3 t}y=8,000()−0.3t . The time ttt 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 ttt in months.t=2 years=2times12 monthst =2\, years =2 \\times12\, monthst=2years=2times12months

STEP 3

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

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}y=8,000(2)−0.3t

STEP 5

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

STEP 6

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

STEP 7

Calculate the value of 2−7.22^{-7.2}2−7.2.

STEP 8

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

STEP 9

Calculate the value of yyy.

STEP 10

If necessary, round the result to three decimal places.

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

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

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

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

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

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

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

Calculate the value of $2^{-7.2}$ .

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

Calculate the value of $y$ .