Question
upstudy study bank question image url

The radioactive substance cesium-137 has a half-life of 30 years. The amount \( A(t) \) (in grams) of a sample of cesium- 137 remaining after \( t \) years is given by the following exponential function. \( A(t)=621\left(\frac{1}{2}\right)^{\frac{t}{30}} \) Find the amount of the sample remaining after 20 years and after 50 years. Round your answers to the nearest gram as necessary. Amount after 20 years: Amount after 50 years:

Ask by Morrison Lang. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

After 20 years, there are approximately 391 grams remaining. After 50 years, there are approximately 196 grams remaining.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Beyond the Answer

To find the amount of cesium-137 remaining after 20 and 50 years, we'll use the given exponential function \( A(t) = 621\left(\frac{1}{2}\right)^{\frac{t}{30}} \). **For 20 years:** \[ A(20) = 621\left(\frac{1}{2}\right)^{\frac{20}{30}} \] Calculating \( \left(\frac{1}{2}\right)^{\frac{2}{3}} \) gives approximately \( 0.39685 \). Now multiply: \[ A(20) \approx 621 \times 0.39685 \approx 246.39 \] Rounding to the nearest gram, we get **246 grams** remaining after 20 years. **For 50 years:** \[ A(50) = 621\left(\frac{1}{2}\right)^{\frac{50}{30}} \] Calculating \( \left(\frac{1}{2}\right)^{\frac{5}{3}} \) gives approximately \( 0.25000 \). Now multiply: \[ A(50) \approx 621 \times 0.25000 \approx 155.25 \] Rounding to the nearest gram, we get **155 grams** remaining after 50 years. In summary, after 20 years, there are **246 grams** of cesium-137, and after 50 years, there are **155 grams** remaining.

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy