Questions: Question 9, 5.1 VQ-2 HW Score: 43.33%, 6.5 of 15 points Points: 0 of 1 Save Watch the video and then solve the problem given below. Click here to watch the video An exponential decay function can be used to model the number of grams of a radioactive material that remain after a period of time. A radioactive isotope has given by y = 100e^(-0.0258t), where t is the number of years after 1 year and y is the amount remaining after t years. (a) The amount of a radioactive isotope remaining after 3500 years is approximately grams. (Type an integer or decimal rounded to the nearest tenth as needed.) Clear all Check Part Ask my Instructor Question list Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9

Question 9, 5.1 VQ-2 HW Score: 43.33%, 6.5 of 15 points Points: 0 of 1 Save Watch the video and then solve the problem given below. Click here to watch the video An exponential decay function can be used to model the number of grams of a radioactive material that remain after a period of time. A radioactive isotope has given by y = 100e^(-0.0258t), where t is the number of years after 1 year and y is the amount remaining after t years. (a) The amount of a radioactive isotope remaining after 3500 years is approximately grams. (Type an integer or decimal rounded to the nearest tenth as needed.) Clear all Check Part Ask my Instructor Question list Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9
Transcript text: Question 9, 5.1 VQ-2 HW Score: 43.33%, 6.5 of 15 points Points: 0 of 1 Save Watch the video and then solve the problem given below. Click here to watch the video An exponential decay function can be used to model the number of grams of a radioactive material that remain after a period of time. A radioactive isotope has given by y = 100e^(-0.0258t), where t is the number of years after 1 year and y is the amount remaining after t years. (a) The amount of a radioactive isotope remaining after 3500 years is approximately ______ grams. (Type an integer or decimal rounded to the nearest tenth as needed.) Clear all Check Part Ask my Instructor Question list Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9
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Solution

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Solution Steps

Step 1: Identify the given exponential decay function

The given exponential decay function is: \[ y = 100e^{-0.0258t} \]

Step 2: Substitute the given time into the function

We need to find the amount of the radioactive isotope remaining after 3500 years. Substitute \( t = 3500 \) into the function: \[ y = 100e^{-0.0258 \times 3500} \]

Step 3: Calculate the exponent

First, calculate the exponent: \[ -0.0258 \times 3500 = -90.3 \]

Step 4: Evaluate the exponential function

Now, evaluate the exponential function: \[ y = 100e^{-90.3} \]

Step 5: Approximate the value of the exponential term

Using a calculator, we find: \[ e^{-90.3} \approx 1.233 \times 10^{-39} \]

Step 6: Multiply by the initial amount

Finally, multiply by the initial amount (100 grams): \[ y = 100 \times 1.233 \times 10^{-39} \approx 1.233 \times 10^{-37} \]

Final Answer

The amount of a radioactive isotope remaining after 3500 years is approximately: \[ \boxed{0.0 \text{ grams}} \]

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