Assume that the half-life of pennies is 1 shake. Show your work answering the following questions: If you start with 600 pennies, how many would be left unchanged after 4 shakes? You come across an unfinished lab that has 6 pennies with heads up and 94 with tails up. How many shakes did the unfinished group do?
The Correct Answer and Explanation is:
Part 1: How many pennies remain unchanged after 4 shakes?
Given:
- Half-life = 1 shake
- Starting pennies = 600
The concept of half-life means that after each shake, half of the remaining pennies stay unchanged (heads up). We can use the formula:N=N0×(12)tN = N_0 \times \left( \frac{1}{2} \right)^tN=N0×(21)t
Where:
- NNN = number of unchanged pennies after ttt shakes
- N0N_0N0 = initial number of pennies = 600
- ttt = number of shakes = 4
Calculation:N=600×(12)4=600×116=37.5N = 600 \times \left( \frac{1}{2} \right)^4 = 600 \times \frac{1}{16} = 37.5N=600×(21)4=600×161=37.5
Since we cannot have half a penny, we approximate to the nearest whole number. So, 38 pennies remain unchanged after 4 shakes.
Part 2: How many shakes did the unfinished group do?
Given:
- Heads up (unchanged) = 6 pennies
- Total pennies = 6 heads + 94 tails = 100 pennies
- Half-life = 1 shake
We use the same formula:N=N0×(12)tN = N_0 \times \left( \frac{1}{2} \right)^tN=N0×(21)t
Where:
- NNN = 6 unchanged pennies
- N0N_0N0 = 100 pennies
- ttt = unknown number of shakes
Solving for ttt:6=100×(12)t6 = 100 \times \left( \frac{1}{2} \right)^t6=100×(21)t
Divide both sides by 100:0.06=(12)t0.06 = \left( \frac{1}{2} \right)^t0.06=(21)t
Take logarithms of both sides:log(0.06)=t×log(0.5)\log(0.06) = t \times \log(0.5)log(0.06)=t×log(0.5)
Solve for ttt:t=log(0.06)log(0.5)≈−1.2218−0.3010≈4.06t = \frac{\log(0.06)}{\log(0.5)} \approx \frac{-1.2218}{-0.3010} \approx 4.06t=log(0.5)log(0.06)≈−0.3010−1.2218≈4.06
So, approximately 4 shakes were completed.
Conclusion:
- After 4 shakes, 38 pennies remain unchanged out of 600.
- The unfinished group completed approximately 4 shakes, based on 6 out of 100 pennies remaining unchanged.
This follows the principle of exponential decay where each shake reduces the unchanged group by half.
