The function $f(x)=2.15\left(2 x^2-4 x-6\right)$ models the cost, in dollars, of a rug with width $x$ feet. What is the cost of a rug that is 9 feet wide?
A. $120
B. $258
C. $606
D. $655
Answer (1)
Substitute x = 9 into the function f ( x ) = 2.15 ( 2 x 2 − 4 x − 6 ) .
Calculate f ( 9 ) = 2.15 ( 2 ( 9 ) 2 − 4 ( 9 ) − 6 ) .
Simplify the expression: f ( 9 ) = 2.15 ( 162 − 36 − 6 ) = 2.15 ( 120 ) .
The cost of the rug is $258 .
Explanation
Understanding the problem We are given the function f ( x ) = 2.15 ( 2 x 2 − 4 x − 6 ) that models the cost of a rug with width x feet. We need to find the cost of a rug that is 9 feet wide.
Substituting the value of x To find the cost of a rug that is 9 feet wide, we need to substitute x = 9 into the function f ( x ) . So we have: f ( 9 ) = 2.15 ( 2 ( 9 ) 2 − 4 ( 9 ) − 6 )
Simplifying the expression Now, we simplify the expression: f ( 9 ) = 2.15 ( 2 ( 81 ) − 36 − 6 ) f ( 9 ) = 2.15 ( 162 − 36 − 6 ) f ( 9 ) = 2.15 ( 162 − 42 ) f ( 9 ) = 2.15 ( 120 ) f ( 9 ) = 258
Final Answer Therefore, the cost of a rug that is 9 feet wide is $258 .
Examples
Imagine you are designing a rectangular garden where the length is dependent on the width. The cost of materials for the garden can be modeled by a quadratic function similar to the one in the problem. By substituting different width values into the function, you can determine the cost of materials for various garden sizes. This helps you plan your budget and optimize the dimensions of your garden to achieve the desired aesthetic within your financial constraints. Understanding how to use functions to model real-world costs allows you to make informed decisions in various design and construction projects.
