A financial analyst is comparing two investment options. Option A's value, A(t), after t years is given by A(t) = 1000(1.05)^t. Option B's value, B(t), after t years is given by B(t) = 1000 + 60t. Which statement about the investments is true?
- AOption B always grows faster than Option A.
- BOption B will always be worth more than Option A.
- COption A will eventually surpass Option B in value.
- DOption A always grows faster than Option B.
Show answer & explanationAnswer & explanation
Correct answer: C. Option A will eventually surpass Option B in value.
Option A is an exponential function, while Option B is a linear function. Exponential functions with a base greater than 1 (like 1.05) will always eventually surpass linear functions, even if the linear function initially grows faster. In the short term, B might be higher, but A's growth rate accelerates. For instance, at t=0, A(0)=1000, B(0)=1000. At t=1, A(1)=1050, B(1)=1060. At t=10, A(10) ≈ 1628.89, B(10) = 1600. At t=20, A(20) ≈ 2653.30, B(20) = 2200.
Why the other options are wrong
- A. Exponential growth eventually overtakes linear growth, so 'always' is incorrect.
- B. Exponential growth will eventually make Option A surpass Option B, so 'always' is incorrect.
- D. Option B (linear) might grow faster initially, so 'always' is incorrect.
Comparing Exponential and Linear Functions
Exponential functions (with base > 1) exhibit growth that increases over time, while linear functions exhibit constant growth. Exponential growth ultimately surpasses linear growth.
- Linear functions change by a constant amount per unit interval.
- Exponential functions change by a constant ratio (or percentage) per unit interval.
- For positive growth, exponential functions always eventually exceed linear functions.
Memory trick: Exponential ALWAYS Wins the Long Race!