QuestionThe half-life of silver-108m is approximately 130 years.
Step 1 of 3 : Determine $a$ so that $A(t)=A_{0} a^{t}$ describes the amount of silver-108m left after $t$ years, where $A_{0}$ is the amount at time $t=0$ . Round to six decimal places.

Studdy Solution

STEP 1

1. The half-life of silver-108m is 130 years.
2. The function $A(t) = A_0 a^t$ describes the amount of silver-108m left after $t$ years.
3. We need to determine the value of $a$ .

STEP 2

1. Understand the concept of half-life and its mathematical representation.
2. Set up the equation using the half-life information.
3. Solve for $a$ .

STEP 3

Understand that the half-life of a substance is the time it takes for half of the substance to decay. Mathematically, this means that after one half-life, the amount of substance remaining is half of the original amount.

STEP 4

Set up the equation using the half-life information. After 130 years, the amount of silver-108m remaining should be half of the initial amount $A_0$ . Therefore, we can write:
$A(130) = \frac{1}{2} A_0$
Substitute $A(t) = A_0 a^t$ into the equation:
$A_0 a^{130} = \frac{1}{2} A_0$

STEP 5

Solve for $a$ . Divide both sides of the equation by $A_0$ (assuming $A_0 \neq 0$ ):
$a^{130} = \frac{1}{2}$
To solve for $a$ , take the 130th root of both sides:
$a = \left( \frac{1}{2} \right)^{\frac{1}{130}}$
Calculate the value of $a$ and round to six decimal places:
$a \approx 0.994645$
The value of $a$ is approximately:
$\boxed{0.994645}$

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