To understand what [tex]\( f(15) \)[/tex] represents for the given cost function [tex]\( f(x) = 15 + 150x \)[/tex], let's break down the function and the evaluation process step by step:
1. Understand the Cost Function:
The function [tex]\( f(x) = 15 + 150x \)[/tex] describes the total cost in dollars to rent a houseboat for [tex]\( x \)[/tex] days. In this function:
- The constant term [tex]\( 15 \)[/tex] likely represents a fixed cost or initial fee associated with renting the houseboat.
- The term [tex]\( 150x \)[/tex] represents the variable cost, which depends on the number of days [tex]\( x \)[/tex] for which the houseboat is rented.
2. Evaluate [tex]\( f(15) \)[/tex]:
To find the total cost of renting the houseboat for 15 days, we substitute [tex]\( x = 15 \)[/tex] into the cost function:
[tex]\[
f(15) = 15 + 150 \times 15
\][/tex]
- Here, [tex]\( 150 \times 15 \)[/tex] represents the cost for renting the houseboat for 15 days.
- Adding 15 gives the total cost including the fixed cost and the variable cost.
3. Interpret [tex]\( f(15) \)[/tex]:
The result of [tex]\( f(15) \)[/tex] provides the total cost in dollars for renting the houseboat specifically for 15 days.
Given the multiple-choice options:
- The number of dollars it costs for 15 people to rent the houseboat (Incorrect, the function is about days, not people).
- The number of houseboats that can be rented for 15 days (Incorrect, the function calculates cost, not the number of houseboats).
- The number of dollars it costs to rent the houseboat for 15 days (Correct).
- The number of days the houseboat can be rented for a cost of [tex]$\$[/tex]150[tex]$ (Incorrect, this misinterprets the function and the given problem).
Therefore, \( f(15) \) represents the number of dollars it costs to rent the houseboat for 15 days.
Given the correct evaluation through the cost function, the final numerical result is:
\[
f(15) = 2265
\]
So, the correct interpretation is indeed "the number of dollars it costs to rent the houseboat for 15 days," which amounts to \(\$[/tex]2265\).
