Question

after a time \\(t\\), the new value of an exponentially growing quantity with a doubling time of \\(t_{\text{double}}\\) is given by

options:

• \\(\text{new value} = \text{initial value} \times 2^{t/t_{\text{double}}}\\)
• \\(\text{new value} = \text{initial value} \times 2^{t_{\text{double}}/t}\\)

Explanation:

Identify the exponential growth formula with doubling time

The general formula for exponential growth given a doubling time \(T_{\text{double}}\) is:

$$ \text{new value} = \text{initial value} \times 2^{\frac{t}{T_{\text{double}}}} $$

Compare with the given options

The dropdown menu shows two visible options:

1. \(\text{new value} = \text{initial value} \times 2^{t/T_{\text{double}}}\)
2. \(\text{new value} = \text{initial value} \times 2^{T_{\text{double}}/t}\)

The correct mathematical representation is the first option, where the exponent is the ratio of elapsed time \(t\) to the doubling period \(T_{\text{double}}\).

Answer:

• new value = initial value \(\times 2^{t/T_{\text{double}}}\) (Correct answer)
• new value = initial value \(\times 2^{T_{\text{double}}/t}\)