To determine what the value [tex]\(5\)[/tex] represents in the expression [tex]\(5 \cdot 1.15^t\)[/tex], let's analyze the components of the expression.
The expression [tex]\(5 \cdot 1.15^t\)[/tex] is used to model the area damaged by fire over time, where:
- [tex]\(5\)[/tex] is a constant factor.
- [tex]\(1.15\)[/tex] is the base of the exponential function indicating a growth rate of 15% per minute.
- [tex]\(t\)[/tex] represents time in minutes since the fire was discovered.
In an exponential growth model of the form [tex]\(A(t) = A_0 \cdot b^t\)[/tex]:
- [tex]\(A(t)\)[/tex] is the amount at time [tex]\(t\)[/tex].
- [tex]\(A_0\)[/tex] is the initial amount.
- [tex]\(b\)[/tex] is the growth factor.
- [tex]\(t\)[/tex] is the time.
Here, [tex]\(A(t) = 5 \cdot 1.15^t\)[/tex], where:
- [tex]\(A_0 = 5\)[/tex] (the initial amount, in this case, the initial area damaged by fire when [tex]\(t = 0\)[/tex]) and
- [tex]\(b = 1.15\)[/tex] (the growth factor, indicating a 15% increase per minute).
Therefore, the term [tex]\(5\)[/tex] represents the initial area in square meters that had been damaged by the fire by the time it was discovered.
So, the correct choice is:
A. An area of 5 square meters had been damaged by the fire by the time it was discovered.
