Use the exponential decay model, $$ \mathrm{A}=\mathrm{A}_{0} e^{\mathrm{kt}} $$, to solve the following. The half-life of a certain substance is 16 years. How long will it take for a sample of this substance to decay to $$ 75 \% $$ of its original amount? It will take approximately $$ \square $$ original amount. (Round the final answer to on places as needed.)

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We start with the exponential decay formula:

$$
A = A_0 e^{kt}.
$$

**Step 1. Find the decay constant $$ k $$ from the half-life**

The half-life is 16 years, meaning that when $$ t = 16 $$, the amount is $$ A = \frac{A_0}{2} $$:

$$
\frac{A_0}{2} = A_0 e^{k \cdot 16}.
$$

Divide both sides by $$ A_0 $$:

$$
\frac{1}{2} = e^{16k}.
$$

Take the natural logarithm of both sides:

$$
\ln\left(\frac{1}{2}\right) = 16k.
$$

Solve for $$ k $$:

$$
k = \frac{\ln(1/2)}{16} = -\frac{\ln 2}{16}.
$$

**Step 2. Determine the time $$ t $$ for the substance to decay to $$ 75\% $$ of its original amount**

We want $$ A = 0.75 A_0 $$. Substitute into the decay formula:

$$
0.75 A_0 = A_0 e^{kt}.
$$

Divide both sides by $$ A_0 $$:

$$
0.75 = e^{kt}.
$$

Take the natural logarithm of both sides:

$$
\ln(0.75) = kt.
$$

Substitute $$ k = -\frac{\ln 2}{16} $$:

$$
\ln(0.75) = -\frac{\ln 2}{16} \, t.
$$

Solve for $$ t $$:

$$
t = \frac{\ln(0.75)}{-\frac{\ln 2}{16}} = -\frac{16 \ln(0.75)}{\ln 2}.
$$

**Step 3. Evaluate the expression**

Compute the needed logarithms:

$$
\ln(0.75) \approx -0.2877, \quad \ln 2 \approx 0.6931.
$$

Substitute these values:

$$
t \approx -\frac{16 \times (-0.2877)}{0.6931} \approx \frac{4.6032}{0.6931} \approx 6.64.
$$

Rounded to one decimal place,

$$
t \approx 6.6 \text{ years}.
$$

Thus, it will take approximately $$ \boxed{6.6} $$ years for the substance to decay to $$ 75\% $$ of its original amount.
It will take approximately 6.6 years for the substance to decay to 75% of its original amount.

Use the exponential decay model, , to solve the following.
The half-life of a certain substance is 16 years. How long will it take for a sample of this substance to decay to
of its original amount?
It will take approximately
original amount.
(Round the final answer to on
places as needed.)

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Use the exponential decay model, , to solve the following. The half-life of a certain substance is 16 years. How long will it take for a sample of this substance to decay to of its original amount? It will take approximately original amount. (Round the final answer to on places as needed.)