Let's break down the steps to answer the question about the function representing the relationship between the number of days ([tex]$t$[/tex]) and the number of orders remaining.
Given:
- The store has 3000 orders initially.
- Each day, the store completes one-fourth of the remaining orders.
To find the function, let’s denote the number of orders remaining as [tex]$f(t)$[/tex] where [tex]$t$[/tex] is the number of days.
1. Initial Condition (Day 0):
At [tex]$t = 0$[/tex], the number of orders is 3000.
2. Day 1:
By the end of the first day, one-fourth of the initial orders are completed. So, [tex]\( \frac{3}{4} \)[/tex] of the orders remain.
Therefore, the number of orders remaining, [tex]$f(1)$[/tex], is:
[tex]\[
f(1) = 3000 \times \left( \frac{3}{4} \right) = 2250
\][/tex]
3. Day 2:
At [tex]$t = 2$[/tex], again one-fourth of the remaining orders from day 1 is completed. So, [tex]\( \frac{3}{4} \)[/tex] of the orders from day 1 remain.
[tex]\[
f(2) = 2250 \times \left( \frac{3}{4} \right) = 1687.5
\][/tex]
4. Day 3:
Similarly, at [tex]$t = 3$[/tex], one-fourth of the remaining orders from day 2 is completed. Thus, [tex]\( \frac{3}{4} \)[/tex] of the orders from day 2 remain.
[tex]\[
f(3) = 1687.5 \times \left( \frac{3}{4} \right) = 1265.625
\][/tex]
Following this pattern, for any day [tex]$t$[/tex], the function that describes the number of orders remaining can be written as an exponential decay function:
[tex]\[
f(t) = 3000 \times \left( \frac{3}{4} \right)^t
\][/tex]
This matches with one of the provided options. The correct function is:
[tex]\[
f(t) = 3000 \cdot\left(\frac{3}{4}\right)^t
\][/tex]
Thus, the correct answer for the given question is:
[tex]\[
f(t) = 3.000 \cdot\left(\frac{3}{4}\right)^t
\][/tex]
