Question
upstudy study bank question image url

The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 3691 kg and decreases continuously at a relative rate of \( 9 \% \) per day. Find the mass of the sample after six days. Do not round any intermediate computations, and round your answer to the nearest tenth.

Ask by Ellis Bowman. in the United States
Feb 21,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

After six days, the mass of the radioactive sample is approximately 2148.2 kg.

Solution

To solve the problem of finding the mass of a radioactive substance after six days, we can use the continuous exponential decay formula: \[ M(t) = M_0 e^{-kt} \] where: - \( M(t) \) is the mass at time \( t \), - \( M_0 \) is the initial mass, - \( k \) is the decay constant (relative rate of decay), - \( t \) is the time in days, - \( e \) is the base of the natural logarithm. ### Step 1: Identify the known values - Initial mass, \( M_0 = 3691 \) kg - Relative rate of decay, \( k = 0.09 \) (since \( 9\% = 0.09 \)) - Time, \( t = 6 \) days ### Step 2: Substitute the values into the formula We need to calculate \( M(6) \): \[ M(6) = 3691 e^{-0.09 \times 6} \] ### Step 3: Calculate the exponent First, calculate the exponent: \[ -0.09 \times 6 = -0.54 \] ### Step 4: Calculate \( e^{-0.54} \) Now we need to calculate \( e^{-0.54} \): \[ e^{-0.54} \approx 0.582 \] ### Step 5: Calculate the mass after 6 days Now substitute back into the mass formula: \[ M(6) = 3691 \times 0.582 \] ### Step 6: Perform the multiplication Now we calculate: \[ M(6) = 3691 \times 0.582 \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(3691\times 0.582\) - step1: Multiply the numbers: \(2148.162\) The calculation gives us: \[ M(6) \approx 2148.162 \text{ kg} \] ### Step 7: Round the result Rounding this to the nearest tenth, we get: \[ M(6) \approx 2148.2 \text{ kg} \] ### Final Answer The mass of the sample after six days is approximately **2148.2 kg**.

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Extra Insights

To find the mass of the sample after six days, we can use the formula for continuous exponential decay, which is given by: \[ M(t) = M_0 e^{-kt} \] where: - \( M(t) \) is the mass at time \( t \), - \( M_0 \) is the initial mass, - \( k \) is the decay constant (in this case, \( k = 0.09 \) for a \( 9\% \) decay rate), - \( t \) is the elapsed time (in days), - \( e \) is the base of the natural logarithm (approximately 2.71828). Given: - \( M_0 = 3691 \text{ kg} \), - \( k = 0.09 \), - \( t = 6 \text{ days} \). Plugging in the values, we have: \[ M(6) = 3691 e^{-0.09 \times 6} \] Calculating the exponent: \[ -0.09 \times 6 = -0.54 \] Now, finding \( e^{-0.54} \): Using a calculator, we find: \[ e^{-0.54} \approx 0.582 \] Now substituting back into the mass equation: \[ M(6) = 3691 \times 0.582 \] Calculating this gives: \[ M(6) \approx 2141.862 \] Rounding to the nearest tenth, the mass of the sample after six days is approximately: **2141.9 kg**.

Latest Pre Calculus Questions

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