Question

the value of a boat after t years is represented by the function $f(t) = 11,200(0.93)^t$. what does the value 11,200 represent in this situation? - the initial value of the boat is $11,200. - the value of the boat after t years is $11,200. - the value of the boat increases by $11,200 each year. - the value of the boat decreases by $11,200 each year.

Explanation:

The function \( f(t) = 11200(0.93)^t \) is an exponential decay function. In the general form of an exponential function \( f(t)=a(b)^t \), \( a \) represents the initial value (when \( t = 0 \)). Substituting \( t = 0 \) into the function, we get \( f(0)=11200(0.93)^0 = 11200\times1=11200 \), so 11200 is the initial value of the boat. The other options are incorrect: the second option would require \( f(t)=11200 \) for all \( t \), which is not true; the third and fourth options suggest linear change, but the function is exponential, so the value does not increase or decrease by a constant $11200 each year.

Answer:

The initial value of the boat is $11,200.