Short Answer 
The amount of mould at 5:00 AM is approximately 0.136g, calculated using the exponential growth formula and the growth constant derived from the data at 9 PM.
Step 1: Understand the Formula
The problem involves exponential growth, where the growth rate is proportional to the current quantity. The key formula used here is:
- M(t): the amount of mould at time t
- M0: the initial amount of mould
- k: the growth constant
- t: the elapsed time
This equation can be expressed as: M(t) = M0 * e^(kt). Understanding this will guide us to solve for the mould quantities at different times.
Step 2: Solve for Growth Constant (k)
To find the growth constant k, we will use the data given at t = 9 hours (which corresponds to 9 PM). The equation becomes:
- Start with 1.6 = 0.4 * e^(9k).
- Divide to isolate the exponential: 4 = e^(9k).
- Take the natural logarithm (ln) of both sides: ln(4) = 9k.
- Thus, we can solve for k: k = ln(4) / 9.
Step 3: Calculate Mould at 5:00 AM
Now that we have k, we need to find the amount of mould at t = -7 hours (5 AM). We substitute k back into the equation:
- Use the formula: M(-7) = 0.4 * e^(-7k).
- Substitute for k to get: M(-7) = 0.4 * e^(-7 * (ln(4) / 9)).
- Calculate the result to find M(-7) ≈ 0.136079g.
Thus, the amount of mould at 5:00 AM is approximately 0.136g.