A population of insects triples every 10 days

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A population of insects triples every 10 days. Its population is 400 today. When will its population be 1000? When was its population 100?

The correct answer and explanation is:

To solve this, we need to use the formula for exponential growth: P(t)=P0×(1+r)tP(t) = P_0 \times (1 + r)^t

Where:

  • P(t)P(t) is the population at time tt,
  • P0P_0 is the initial population,
  • rr is the growth rate,
  • tt is the time in days.

Step 1: Determine the growth rate

Since the population triples every 10 days, the growth factor is 3. Thus, we can say that every 10 days, the population multiplies by 3.

Step 2: Write the exponential growth equation

The initial population P0P_0 is 400, and the population triples every 10 days, so the growth rate every 10 days is 3. The formula for the population at any given time tt (in days) would be: P(t)=400×3(t/10)P(t) = 400 \times 3^{(t/10)}

Step 3: When will the population reach 1000?

Set P(t)=1000P(t) = 1000 and solve for tt: 1000=400×3(t/10)1000 = 400 \times 3^{(t/10)}

First, divide both sides by 400: 2.5=3(t/10)2.5 = 3^{(t/10)}

Take the natural logarithm of both sides: ln⁡(2.5)=t10ln⁡(3)\ln(2.5) = \frac{t}{10} \ln(3)

Now, calculate ln⁡(2.5)\ln(2.5) and ln⁡(3)\ln(3): ln⁡(2.5)≈0.9163,ln⁡(3)≈1.0986\ln(2.5) \approx 0.9163, \quad \ln(3) \approx 1.0986

So, 0.9163=t10×1.09860.9163 = \frac{t}{10} \times 1.0986

Solve for tt: t≈0.9163×101.0986≈8.34 dayst \approx \frac{0.9163 \times 10}{1.0986} \approx 8.34 \text{ days}

So, the population will reach 1000 in approximately 8.34 days.

Step 4: When was the population 100?

Set P(t)=100P(t) = 100 and solve for tt: 100=400×3(t/10)100 = 400 \times 3^{(t/10)}

First, divide both sides by 400: 0.25=3(t/10)0.25 = 3^{(t/10)}

Take the natural logarithm of both sides: ln⁡(0.25)=t10ln⁡(3)\ln(0.25) = \frac{t}{10} \ln(3)

Calculate ln⁡(0.25)\ln(0.25): ln⁡(0.25)≈−1.3863\ln(0.25) \approx -1.3863

So, −1.3863=t10×1.0986-1.3863 = \frac{t}{10} \times 1.0986

Solve for tt: t≈−1.3863×101.0986≈−12.61 dayst \approx \frac{-1.3863 \times 10}{1.0986} \approx -12.61 \text{ days}

So, the population was 100 approximately 12.61 days ago.

Conclusion:

  • The population will reach 1000 in about 8.34 days.
  • The population was 100 about 12.61 days ago.

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