\begin{tabular}{l} Done \ Submit Question \ Question 4 Certain radioactive material decays in such a way \ that the mass remaining after $$ t $$ years is given by the \ function \ $$ m(t)=175(0.965)^{t} $$ \ where $$ m(t) $$ is measured in grams. \ (a) Find the mass at time $$ t=0 $$. \ Your answer is \ (b) How much of the mass remains after 40 years? \ Your answer is \ Give your answer to two decimal places. \ Submit Question \ \hline$$ \checkmark $$ Question 5\end{tabular}

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UpStudy AI · Accepted Answer

Let's solve both parts of the problem step by step.

### Given:
The mass remaining after $$ t $$ years is given by the function:
$$ m(t) = 175 \times (0.965)^t $$
where $$ m(t) $$ is measured in grams.

---

### (a) Find the mass at time $$ t = 0 $$.

**Solution:**
To find the mass at $$ t = 0 $$, substitute $$ t = 0 $$ into the function:

$$
m(0) = 175 \times (0.965)^0
$$

Since any number raised to the power of 0 is 1:

$$
m(0) = 175 \times 1 = 175 \text{ grams}
$$

**Answer:**
$$
\boxed{175}
$$

---

### (b) How much of the mass remains after 40 years?  
**Give your answer to two decimal places.**

**Solution:**
We need to calculate $$ m(40) $$:

$$
m(40) = 175 \times (0.965)^{40}
$$

First, compute $$ (0.965)^{40} $$:

1. **Calculate the natural logarithm:**
   $$
   \ln(0.965) \approx -0.03563
   $$
   
2. **Multiply by 40:**
   $$
   40 \times \ln(0.965) \approx 40 \times (-0.03563) = -1.4252
   $$
   
3. **Exponentiate to find $$ (0.965)^{40} $$:**
   $$
   e^{-1.4252} \approx 0.2405
   $$
   
4. **Multiply by 175:**
   $$
   m(40) = 175 \times 0.2405 \approx 42.08 \text{ grams}
   $$

**Answer:**
$$
\boxed{42.08 \text{ grams}}
$$
At time $$ t = 0 $$, the mass is 175 grams. After 40 years, approximately 42.08 grams remain.

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