Question

the area covered by a lake is 11 square kilometers. it is decreasing exponentially at a rate of 2 percent each year and can be modeled by $a(t)=11\cdot(.98)^t$.
which equation models how much the area of the lake will change each month?
a $a(t)=$
b $a(t)=11\cdot(.98)^{\frac{t}{12}}$
c $a(t)=12\cdot(.98)^t$
d $a(t)=11\cdot(.98)^{12t}$

Explanation:

Step1: Understand annual to monthly conversion

We need to adjust the annual exponential decay model to a monthly one. Let \(t\) now represent months. There are 12 months in a year, so the number of years is \(\frac{t}{12}\).

Step2: Substitute into original formula

The original annual model is \(A(\text{years}) = 11 \cdot (0.98)^{\text{years}}\). Replace years with \(\frac{t}{12}\):
\(A(t) = 11 \cdot (0.98)^{\frac{t}{12}}\)

Answer:

B. \(A(t)=11 \cdot (.98)^{\frac{t}{12}}\)