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A specific radioactive substance follows a continuous exponential decay model. It has a half-life of 12 days. At the start of the experiment,
33.7g is present.
(a) Let
t be the time (in days) since the start of the experiment, and let
y be the amount of the substance at time
t.
Write a formula relating
y to
t.
Use exact expressions to fill in the missing parts of the formula.
Do not use approximations.

y=◻e^((◻)t)
(b) How much will be present in 22 days?
Do not round any intermediate computations, and round your answer to the nearest tenth.

◻ g

A specific radioactive substance follows a continuous exponential decay model. It has a half-life of $12$ days. At the start of the experiment, $33.7 \mathrm{~g}$ is present. $\newline$ (a) Let $t$ be the time (in days) since the start of the experiment, and let $y$ be the amount of the substance at time $t$ . $\newline$ Write a formula relating $y$ to $t$ . $\newline$ Use exact expressions to fill in the missing parts of the formula. $\newline$ Do not use approximations. $\newline$ $y=\square e^{(\square) t}$ $\newline$ (b) How much will be present in $22$ days? $\newline$ Do not round any intermediate computations, and round your answer to the nearest tenth. $\newline$ $\square$ g

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Find a value using two-variable equations: word problems

Full solution

Q. A specific radioactive substance follows a continuous exponential decay model. It has a half-life of

12

days. At the start of the experiment,

33.7 \mathrm{~g}

is present.

\newline

(a) Let

t

be the time (in days) since the start of the experiment, and let

y

be the amount of the substance at time

t

.

\newline

Write a formula relating

y

to

t

.

\newline

Use exact expressions to fill in the missing parts of the formula.

\newline

Do not use approximations.

\newline

y=\square e^{(\square) t}

\newline

(b) How much will be present in

22

days?

\newline

Do not round any intermediate computations, and round your answer to the nearest tenth.

\newline

\square

g

Write Decay Formula: First, we need to write the decay formula using the half-life of $12$ days. $\newline$ The general formula for exponential decay is $y = y_0 \cdot e^{(-kt)}$ , where $y_0$ is the initial amount, $k$ is the decay constant, and $t$ is the time. $\newline$ Since the half-life is $12$ days, we can find $k$ using the formula $0.5 = e^{(-k\cdot12)}$ .
Find Decay Constant: Solve for $k$ by taking the natural logarithm of both sides: $\ln(0.5) = \ln(e^{-k\cdot 12})$ . This simplifies to $\ln(0.5) = -k\cdot 12$ .
Calculate $k$ Value: Divide both sides by $-12$ to isolate $k$ : $k = \frac{\ln(0.5)}{-12}$ .
Plug k into Formula: Now we can plug the value of $k$ back into the decay formula. The initial amount $y_0$ is $33.7$ g, so the formula becomes $y = 33.7 \times e^{(\ln(0.5) / -12)t}$ .
Substitute $t$ Value: To find the amount present in $22$ days, substitute $t$ with $22$ in the formula: $y = 33.7 \times e^{(\ln(0.5) / -12)\times22}$ .
Calculate Exponent: Calculate the exponent first: $(\ln(0.5) / -12)\cdot 22$ .
Calculate Final Result: Now calculate $e$ to the power of the result from the previous step.

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