Question

A polymerase chain reaction yields 1.2 billion copies of DNA in 30 cycles. How many cycles would be needed to obtain 300 million copies?

• A. 7 cycles
• B. 8 cycles
• C. 15 cycles
• D. 28 cycles

Hint:

For exponential growth, always remember the relationship between the number of cycles and the doubling of the quantity. In this case, every cycle doubles the number of copies.

Answer 1

The Correct Option is D

We are given that in 30 cycles, 1.2 billion copies are obtained, and we need to determine how many cycles are required to obtain 300 million copies. Since the reaction is exponential, the number of copies after \(n\) cycles can be expressed as: \[ {Copies} = {Initial copies} \times 2^n \] Let \( x \) be the number of cycles needed to obtain 300 million copies. We know that after 30 cycles, the number of copies is 1.2 billion, so we can write: \[ 1.2 \times 10^9 = {Initial copies} \times 2^{30} \] We also know that after \( x \) cycles, we want the number of copies to be 300 million, so: \[ 3.0 \times 10^8 = {Initial copies} \times 2^x \] Dividing the two equations: \[ \frac{3.0 \times 10^8}{1.2 \times 10^9} = \frac{2^x}{2^{30}} \] Simplifying: \[ \frac{1}{4} = 2^{x-30} \] \[ 2^{-2} = 2^{x-30} \] \[ x - 30 = -2 \quad \Rightarrow \quad x = 28 \] Thus, 28 cycles would be required to obtain 300 million copies.