Suppose the quantity​ (in grams) of a radioactive substance present at time t is Upper Q left parenthesis t right parenthesis equals 10 comma 000 left parenthesis 5 Superscript negative 0.4 t Baseline right parenthesis​, where t is measured in months.
a. How much will be present in 4 ​months?
b. How long will it take to reduce the substance to 16 ​grams?
Question content area bottom
Part 1
a. There will be 
  
enter your response here grams of the substance present in 4 months.
​(Round to the nearest gram as​ needed.) 
Part 2
b. It will take 
  
10 months to reduce the substance to 16 grams.

We are given the function that models the quantity of a radioactive substance as:

\[
Q(t) = 10,000 \times 5^{-0.4t}
\]

where \( t \) is the time in months.

### Part 1: How much will be present in 4 months?
We need to find \( Q(4) \), the amount present at \( t = 4 \).

Substitute \( t = 4 \) into the formula:

\[
Q(4) = 10,000 \times 5^{-0.4 \times 4}
\]

Now let's calculate this.

### Part 2: How long will it take to reduce the substance to 16 grams?
We need to find \( t \) when \( Q(t) = 16 \). This leads to the equation:

\[
16 = 10,000 \times 5^{-0.4t}
\]

To solve for \( t \), follow these steps:
1. Divide both sides by 10,000:
   \[
   \frac{16}{10,000} = 5^{-0.4t}
   \]
2. Simplify:
   \[
   0.0016 = 5^{-0.4t}
   \]
3. Take the logarithm of both sides (base 5):
   \[
   \log_5(0.0016) = -0.4t
   \]
4. Solve for \( t \):
   \[
   t = \frac{\log_5(0.0016)}{-0.4}
   \]

Let’s calculate both values.### Part 1: How much will be present in 4 months?
The amount of the radioactive substance present in 4 months is approximately 761 grams.

### Part 2: How long will it take to reduce the substance to 16 grams?
It will take approximately 10 months for the substance to reduce to 16 grams.

Would you like more details or have any questions? 

Here are 5 related questions for further practice:
1. What is the decay factor in this exponential model?
2. How much substance will remain after 6 months?
3. If the initial amount of substance was 15,000 grams, how would the time to reduce to 16 grams change?
4. What percentage of the substance remains after 5 months?
5. Can you express the decay model using a natural logarithmic base?

**Tip:** In exponential decay problems, the base of the exponential function determines how quickly the quantity decreases over time.

Suppose the quantity​ (in grams) of a radioactive substance present at time t is Q(t) = 10,000 * (5^(-0.4t)), where t is measured in months. a. How much will be present in 4 ​months? b. How long will it take to reduce the substance to 16 ​grams?

Learn how to calculate the amount of a radioactive substance after 4 months using exponential decay. Also, find out how long it takes for the substance to reduce to 16 grams using logarithmic functions. This step-by-step guide covers algebraic concepts and exponential decay suitable for Grades 10-12.

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