Answer :
To determine how long it takes for the colony of bacteria to double in size when it grows by 5% every hour, we can use the concept of exponential growth. When a quantity grows by a certain percentage over time, we can apply the formula:
\[A = P \times (1 + r)^t\]
Where:
- \(A\) is the final amount (in this case, double the original size),
- \(P\) is the initial amount,
- \(r\) is the growth rate (5% expressed as 0.05 since it's in decimal form),
- \(t\) is the time in hours.
Since we want to find out how long it takes for the colony to double in size, the final amount will be 2 times the initial amount:
\[2P = P \times (1 + 0.05)^t\]
Simplify the equation:
\[2 = (1.05)^t\]
Now, we need to solve for \(t\). Taking the natural logarithm (ln) of both sides to solve for the exponent:
\[ln(2) = ln(1.05)^t\]
Use the property of logarithms to bring down the exponent:
\[ln(2) = t \times ln(1.05)\]
Now, divide by ln(1.05) to solve for \(t\):
\[t = \frac{ln(2)}{ln(1.05)}\]
Calculate the value of \(t\) using this formula, and you will find the number of hours it takes for the colony of bacteria to double in size.
