Question

13 mark for review the function $f(t) = 60,000(2)^{\frac{t}{48}}$ gives the number of bacteria in a population $t$ minutes after an initial observation. how much time, in minutes, does it take for the number of bacteria in the population to double?

Explanation:

Step1: Understand the doubling condition

We need to find \( t \) when \( f(t) = 2\times60000=120000 \). The function is \( f(t)=60000(2)^{\frac{t}{48}} \).

Step2: Set up the equation

Substitute \( f(t) = 120000 \) into the function:
\( 120000=60000(2)^{\frac{t}{48}} \)

Step3: Simplify the equation

Divide both sides by 60000:
\( \frac{120000}{60000}=(2)^{\frac{t}{48}} \)
\( 2=(2)^{\frac{t}{48}} \)

Step4: Solve for \( t \)

Since the bases are the same, we can set the exponents equal:
\( 1=\frac{t}{48} \)
Multiply both sides by 48:
\( t = 48 \)

Answer:

48