A sample of a radioactive isotope decays such that its mass, in grams, is modeled by the f… — SAT

A sample of a radioactive isotope decays such that its mass, in grams, is modeled by the function $M(t) = A \cdot d^{\frac{t}{12}}$ , where $t$ is the time in hours since the measurement began, and $A$ and $d$ are positive constants. The mass of the sample decreases by $75\%$ every $24$ hours. If the mass of the sample is $15$ grams when $t = 36$ , what was the initial mass, in grams, of the sample?

Answer: 120 grams

Answer

The initial mass of the sample was 120 grams.

The correct answer is 120. A decrease of

75\%

over 24 hours means the mass at

t+24

0.25

times the mass at

t

. According to the function,

M(t+24) = A \cdot d^{(t+24)/12} = A \cdot d^{t/12} \cdot d^2 = M(t) \cdot d^2

. Equating the two yields

d^2 = 0.25

, so

d = 0.5

. Substituting

t = 36

and

M(36) = 15

gives

15 = A \cdot (0.5)^{36/12} = A \cdot (0.5)^3 = 0.125A

. Solving for

A

gives

A = 120

Step-by-Step Solution

Relate the 24-hour decay rate to the exponent in the function to set up an equation for

d

d^2 = 0.25

Every 24 hours (

t

increases by 24), the mass decreases by

75\%

, so it becomes

25\%

(

0.25

) of its previous value. The exponent increases by

24/12 = 2

, multiplying the mass by

d^2

Solve for the decay base

d

d = 0.5

Since

d

is a positive constant, taking the square root of

0.25

gives

0.5

Substitute the given mass at

t = 36

into the model to solve for the initial mass

A

A = 120

Plugging

t = 36

and

M(36) = 15

into

M(t) = A \cdot (0.5)^{t/12}

gives

15 = A \cdot (0.5)^3

, which simplifies to

15 = 0.125A

Key Concept

Interpreting and solving exponential decay functions with fractional exponents

Alternative Method

Instead of solving for

d

first, recognize that

36

hours is exactly

1.5

intervals of

24

hours. Since the mass is multiplied by

0.25

(or

\frac{1}{4}

) every

24

hours, after

36

hours it will be multiplied by

(\frac{1}{4})^{1.5} = (\frac{1}{4})^{3/2} = \frac{1}{8}

of its initial value. Therefore,

15 = A \cdot \frac{1}{8} \implies A = 120

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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