A certain mold spreads in such a way that it doubles in the number of spores present every three days. The initial sample contained 2,500 spores. How many spores will be present in six weeks?
The correct answer and explanation is:
To solve this problem, we will use the formula for exponential growth, which is represented as: N(t)=N0×2t/TN(t) = N_0 \times 2^{t/T}
Where:
- N(t)N(t) is the number of spores at time tt,
- N0N_0 is the initial number of spores,
- TT is the doubling period (in days),
- tt is the time passed in days.
Step 1: Identify the values
From the problem:
- Initial number of spores, N0=2500N_0 = 2500,
- Doubling period, T=3T = 3 days,
- Time, t=6t = 6 weeks = 6 × 7 = 42 days.
Step 2: Apply the formula
Substitute the values into the exponential growth formula: N(42)=2500×242/3N(42) = 2500 \times 2^{42/3}
Simplify the exponent: 423=14\frac{42}{3} = 14
So the formula becomes: N(42)=2500×214N(42) = 2500 \times 2^{14}
Now calculate 2142^{14}: 214=163842^{14} = 16384
Step 3: Calculate the final number of spores
Now, multiply the initial number of spores by 2142^{14}: N(42)=2500×16384=40960000N(42) = 2500 \times 16384 = 40960000
Thus, after 6 weeks, there will be 40,960,000 spores.
Explanation:
The mold growth follows an exponential pattern, meaning it doubles every fixed period of time, in this case, every 3 days. The doubling process means the number of spores increases rapidly over time. By substituting the values into the exponential growth equation, the correct number of spores after 42 days (6 weeks) can be calculated. The result, 40,960,000 spores, represents the rapid spread of the mold during this period.