Digital SATMath: Advanced MathMedium
A biologist is tracking the growth of a bacterial colony. The number of bacteria, N(t), after t hours, is given by N(t) = 100 * 2^(0.5t). How many hours will it take for the colony to reach 1,600 bacteria?
- A8 hours
- B10 hours
- C4 hours
- D6 hours
Show answer & explanationAnswer & explanation
Correct answer: A. 8 hours
Set N(t) = 1600: 1600 = 100 * 2^(0.5t). Divide by 100: 16 = 2^(0.5t). Recognize that 16 = 2^4. So, 2^4 = 2^(0.5t). Equate the exponents: 4 = 0.5t. Solve for t: t = 4 / 0.5 = 8 hours.
Why the other options are wrong
- B. Incorrect calculation of the exponent, possibly 0.5t=5.
- C. Incorrect calculation of the exponent, possibly 0.5t=2.
- D. Incorrect calculation of the exponent, possibly 0.5t=3.
Solving Exponential Equations by Matching Bases
A technique to solve exponential equations where both sides of the equation can be expressed with the same base. Once the bases are equal, the exponents can be set equal to each other and solved.
- Requires rewriting numbers as powers of a common base.
- If b^x = b^y, then x = y.
- Useful when logarithms are not directly applicable or simpler.
Memory trick: Make the bases 'match' then cut off the 'tops'!