Practise the method.
Example 1
Two cultures each grow exponentially with a constant hourly multiplier. A has 128 cells at hour 1 and 512 at hour 3. B has 16 at hour 1 and 64 at hour 2. What is the first whole-number hour when B has more cells than A?
Show worked answer
Answer: 5
- A doubles each hour
- A(t)=64×2t.
- B quadruples
- B(t)=4×4t.
- Equality requires 2t=16, giving t=4.
- B is strictly larger at the next whole hour, 5.
Example 2
An investment grows by a constant positive multiplier each year. A reference balance is $60 after year 1 and $375 after year 3. A second account uses the same multiplier, withdraws $12 immediately after year 2 growth, and has $1345 after year 3 growth. What was the second account's initial balance?
Show worked answer
Answer: 88
- The two-year factor is 425
- the yearly multiplier is 25.
- Reverse year 3 growth, restore the withdrawal
- reverse two more years: ((1345×52)+12)×254=88.
Example 3
Two cultures each grow exponentially with a constant hourly multiplier. A has 24 cells at hour 1 and 96 at hour 3. B has 12 at hour 1 and 48 at hour 2. What is the first whole-number hour when B has more cells than A?
Show worked answer
Answer: 3
- A doubles each hour
- A(t)=12×2t.
- B quadruples
- B(t)=3×4t.
- Equality requires 2t=4, giving t=2.
- B is strictly larger at the next whole hour, 3.