Question

A culture of bacteria starts with 50,050 bacteria and increases exponentially. The relationship between B, the number of bacteria in the culture, and d, the elapsed time in days, is modeled by the following equation:

B=50⋅10d^2
In how many days will the number of bacteria in the culture reach 800,000? Express your answer as a base-ten logarithm.
a. log₁₀(10)
b. log₁₀(8)
c. log₁₀(4)
d. log₁₀(2)

Answer 1

To determine the number of days for the bacteria population to reach 800,000, the given equation B=50·10^{d^2} is used and solved for d.

Step-by-step explanation:

The question involves an exponential growth model concerning bacteria, given by the equation B = 50 · 10d2. To find when the number of bacteria will reach 800,000, we set B to 800,000 and solve for d:

800,000 = 50 · 10d2

Divide both sides by 50:

16,000 = 10d2

We can write 16,000 as 104 × 1.6, and since 1.6 is relatively close to the base ten, we approximate 1.6 as 10log10(1.6):

104 × 10log10(1.6) = 104 + log10(1.6) = 10d2

This gives us the equation:

4 + log10(1.6) = d2

Since log10(1.6) is a bit less than 0.2, we have approximately:

4 + 0.2 = d2

4.2 = d2

d = log10(4.2)

Thus, the closest answer is:

c. log10(4)