Let's use exponential growth formula to determine the number of weeks it will take for the algae bloom to cover 100 square feet:
A = P * (1 + r)^t
where:
A = final area (100 square feet)
P = initial area (13 square feet)
r = growth rate (2.5% = 0.025)
t = time in weeks (unknown)
Substituting the values into the formula, we get:
100 = 13 * (1 + 0.025)^t
Dividing both sides by 13, we get:
7.6923 = (1 + 0.025)^t
Taking the logarithm of both sides, we get:
log(7.6923) = log[(1 + 0.025)^t]
Using the power rule of logarithms, we can simplify the right-hand side of the equation:
log(7.6923) = t * log(1 + 0.025)
Dividing both sides by log(1 + 0.025), we get:
t = log(7.6923) / log(1 + 0.025)
Using a calculator, we get:
t ≈ 18.6
Therefore, it will take about 19 weeks for the algae bloom to cover 100 square feet.