The radioactive substance uranium- 240 has a half-life of 14 hours. The amount $$ A(t) $$ of a sample of uranium- 240 remaining (in grams) after $$ t $$ hours is giv the following exponential function. $$ A(t)=2800\left(\frac{1}{2}\right)^{\frac{t}{14}} $$ Find the amount of the sample remaining after 9 hours and after 60 hours. Round your answers to the nearest gram as necessary. Amount after 9 hours: grams

Question

UpStudy AI · Accepted Answer

To determine the amount of uranium-240 remaining after a given time, we use the exponential decay formula:

$$
A(t) = 2800 \left(\frac{1}{2}\right)^{\frac{t}{14}}
$$

Where:
- $$ A(t) $$ is the amount remaining after $$ t $$ hours.
- 2800 grams is the initial amount.
- The half-life of uranium-240 is 14 hours.

### 1. Amount After 9 Hours

Let's calculate $$ A(9) $$:

$$
A(9) = 2800 \left(\frac{1}{2}\right)^{\frac{9}{14}}
$$

To simplify:
$$
\left(\frac{1}{2}\right)^{\frac{9}{14}} \approx 0.6403
$$

Now, multiply by the initial amount:
$$
A(9) = 2800 \times 0.6403 \approx 1793 \text{ grams}
$$

**Amount after 9 hours:** **1793 grams**

### 2. Amount After 60 Hours

Now, let's calculate $$ A(60) $$:

$$
A(60) = 2800 \left(\frac{1}{2}\right)^{\frac{60}{14}} = 2800 \left(\frac{1}{2}\right)^{4.2857}
$$

Breaking it down:
$$
\left(\frac{1}{2}\right)^{4} = \frac{1}{16} \approx 0.0625
$$
$$
\left(\frac{1}{2}\right)^{0.2857} \approx 0.820
$$
$$
A(60) = 2800 \times 0.0625 \times 0.820 \approx 144 \text{ grams}
$$

**Amount after 60 hours:** **144 grams**
After 9 hours, there are approximately 1793 grams remaining. After 60 hours, there are approximately 144 grams remaining.

Real Tutor Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Pre Calculus Questions