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(Exponential Functions and Function Combinations MC)

A retail clothing store offers customers an opportunity to open up a credit card during checkout. One location of the retail clothing store states that the number of credit cards, [tex]\(A\)[/tex], that are opened [tex]\(t\)[/tex] months since January can be modeled by the function [tex]\(A(t) = 10 + 2t\)[/tex]. The number of credit cards opened at another location, [tex]\(B\)[/tex], is defined by the function [tex]\(B(t) = 25 - t\)[/tex].

What is an expression that can be used to determine the total number of credit cards opened at the two locations?

A. [tex]\((A+B)(t) = 35 + 3t\)[/tex]
B. [tex]\((A-B)(t) = -15 + t\)[/tex]
C. [tex]\((A-B)(t) = -15 + 3t\)[/tex]

(Note: The first option is the correct expression. The extra option was removed to correct the nonsense.)

Sagot :

GinnyAnsweridn
To determine the expressions and values for the number of credit cards opened at different locations based on given functions, we need to define the combined functions and perform the necessary calculations.

1. Given Functions:

- [tex]\(A(t) = 10 + 2t\)[/tex]
- [tex]\(B(t) = 25 - t\)[/tex]

2. Expression for [tex]\((A + B)(t)\)[/tex]:

To find the combined number of credit cards opened at both locations, we sum the respective functions:

[tex]\[ (A + B)(t) = A(t) + B(t) \][/tex]

Substituting the given functions:

[tex]\[ (A + B)(t) = (10 + 2t) + (25 - t) \][/tex]

Simplifying:

[tex]\[ (A + B)(t) = 10 + 2t + 25 - t \][/tex]

Combine like terms:

[tex]\[ (A + B)(t) = 35 + t \][/tex]

3. Expression for [tex]\((A - B)(t)\)[/tex]:

To find the difference between the number of credit cards opened at the two locations, we subtract the respective functions:

[tex]\[ (A - B)(t) = A(t) - B(t) \][/tex]

Substituting the given functions:

[tex]\[ (A - B)(t) = (10 + 2t) - (25 - t) \][/tex]

Simplifying:

[tex]\[ (A - B)(t) = 10 + 2t - 25 + t \][/tex]

Combine like terms:

[tex]\[ (A - B)(t) = -15 + 3t \][/tex]

4. Evaluating [tex]\((A + B)(0)\)[/tex]:

To find the total number of credit cards opened at both locations at [tex]\( t = 0 \)[/tex]:

[tex]\[ (A + B)(0) = (35 + t) \big|_{t=0} = 35 + 0 = 35 \][/tex]

5. Summary of the Results:

- The expression for the total amount of credit cards opened at the two locations is [tex]\((A + B)(t) = 35 + t\)[/tex].
- The expression for the difference in the amount of credit cards opened at the two locations is [tex]\((A - B)(t) = -15 + 3t\)[/tex].
- The value for the total number of credit cards opened at [tex]\( t = 0 \)[/tex] is [tex]\((A + B)(0) = 35\)[/tex].

Based on the given options, the correct expressions are:

1. [tex]\((A + B)(t) = 35 + t\)[/tex]
2. [tex]\((A - B)(t) = -15 + 3t\)[/tex]

Therefore, the correct answers are:
- [tex]\((A + B)(t) = 35 + t\)[/tex]
- [tex]\((A - B)(t) = -15 + 3t\)[/tex]

However, the option provided in the prompt that corresponds to [tex]\( (A + B)(0) = 35 + t \)[/tex] is incorrect because it should just be [tex]\( 35 \)[/tex]. Thus, the accurate options considering the given choices are:

- [tex]\((A + B)(t) = 35 + t\)[/tex]
- [tex]\((A - B)(t) = -15 + 3t\)[/tex]
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