Digital SATMath: AlgebraHard
A financial analyst is modeling the growth of an investment. The value of the investment, V, in dollars, after t years is given by the function V(t) = P(1 + r)^t, where P is the principal and r is the annual interest rate. If P = $1000, r = 0.05, and the investment grows to $1215.51, approximately how many years, t, has the investment been held?
- A2 years
- B4 years
- C3 years
- D5 years
Show answer & explanationAnswer & explanation
Correct answer: B. 4 years
Substitute the given values into the formula: 1215.51 = 1000(1 + 0.05)^t. This simplifies to 1215.51 = 1000(1.05)^t. Divide both sides by 1000: 1.21551 = (1.05)^t. To solve for t, we can use logarithms or test the options. Testing the options: (1.05)^2 = 1.1025; (1.05)^3 = 1.157625; (1.05)^4 = 1.21550625. So, t is approximately 4 years.
Why the other options are wrong
- A. 1000 * (1.05)^2 = 1000 * 1.1025 = 1102.50. Incorrect.
- C. 1000 * (1.05)^3 = 1000 * 1.157625 = 1157.63. Incorrect.
- D. 1000 * (1.05)^5 = 1000 * 1.2762815625 = 1276.28. Incorrect.
Solving Exponential Equations
An exponential equation is an equation where the variable appears in the exponent. To solve it, one typically isolates the exponential term and then uses logarithms to bring the exponent down, or by finding a common base.
- If bases are the same, equate the exponents.
- If bases are different, take the logarithm of both sides.
- Use log properties: log(b^x) = x * log(b).
Memory trick: Logarithms 'unlock' the exponent from its high perch.