A custom t-shirt printing company uses the function C(n) = 12n + 45 to determine the total… — SAT

A custom t-shirt printing company uses the function $C(n) = 12n + 45$ to determine the total cost, in dollars, for an order of $n$ t-shirts. What is the best interpretation of the value $45$ in this function?

A
The price, in dollars, charged for each printed t-shirt.
The setup fee, in dollars, charged for the entire order regardless of the number of t-shirts printed.Answer
C
The total cost, in dollars, for an order of exactly $12$ t-shirts.
D
The minimum number of t-shirts that must be purchased in a single order.

Answer

The setup fee, in dollars, charged for the entire order regardless of the number of t-shirts printed.

In the linear function

C(n) = 12n + 45

, the term

45

is a constant that does not depend on the variable

n

(the number of t-shirts). Therefore, it represents the initial or flat setup cost in dollars for placing an order, which is charged regardless of the number of t-shirts printed.

Step-by-Step Solution

Identify the structure of the linear function

C(n) = 12n + 45

The function is in the slope-intercept form

y = mx + b

, where

m = 12

is the slope and

b = 45

is the y-intercept.

Understanding the components of a linear function helps determine which part of the equation corresponds to which real-world quantity.

Determine the meaning of the constant term (y-intercept) in the context of the problem.

The constant term

45

represents the value of

C(n)

when

n = 0

, which is the cost before any t-shirts are added, representing a flat setup fee.

Evaluating the function at

n = 0

isolates the base cost or initial charge.

Key Concept

Interpreting Linear Relationships in Context

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