Let's break down the given problem step by step to determine the total number of credit cards opened at both locations.
Firstly, we are given the functions for the number of credit cards opened at two different locations:
1. The first location \( A(t) \):
[tex]\[
A(t) = 10 + 2t
\][/tex]
2. The second location \( B(t) \):
[tex]\[
B(t) = 25 - t
\][/tex]
Next, our goal is to find an expression for the total number of credit cards opened at both locations combined, which is given by \( A(t) + B(t) \).
Let’s add these two functions together:
[tex]\[
A(t) + B(t) = (10 + 2t) + (25 - t)
\][/tex]
Combine like terms:
[tex]\[
A(t) + B(t) = 10 + 25 + 2t - t
\][/tex]
Simplify the expression:
[tex]\[
A(t) + B(t) = 35 + t
\][/tex]
Thus, the total number of credit cards opened at both locations at any time \( t \) months since January can be represented by the expression \( 35 + t \).
Among the given multiple-choice options, the correct answer is:
[tex]\[
\boxed{(A+B)(t) = 35 + t}
\][/tex]
