Charlie is installing new carpet in his rectangular living room, which is 5 feet longer than it is wide. He has received quotes for the total installation costs from two different contractors for the same style of carpet.
Contractor A charges an installation fee of $125 plus $2 per square foot for the carpet.
Contractor B charges an installation fee of $75 plus $2.50 per square foot for the carpet.
Which system of equations could be used to determine x, the width of the living room in feet, for which the quote for the installation costs, y, is the same for both companies?
A.
[tex]
\begin{array}{l}
y=2 x^2+130 \\
y=2.5 x^2+80
\end{array}
[/tex]
B.
[tex]
\begin{array}{l}
y=2 x^2+5 x+125 \\
y=2.5 x^2+5 x+75
\end{array}
[/tex]
C.
[tex]
\begin{array}{l}
y=2 x^2+10 x+250 \\
y=2.5 x^2+12.5 x+150
\end{array}
[/tex]
D.
[tex]
\begin{array}{l}
y=2 x^2+10 x+125 \\
y=2.5 x^2+12.5 x+75
\end{array}
[/tex]
Answer (2)
The equations for the installation costs from both contractors have been formulated based on the living room's dimensions. For Contractor A, the cost is y = 2 x 2 + 10 x + 125 and for Contractor B, it is y = 2.5 x 2 + 12.5 x + 75 . Therefore, the correct answer is choice D.
;
Let x be the width, then the length is x + 5 , and the area is x ( x + 5 ) = x 2 + 5 x .
Contractor A's cost: y = 125 + 2 ( x 2 + 5 x ) = 2 x 2 + 10 x + 125 .
Contractor B's cost: y = 75 + 2.5 ( x 2 + 5 x ) = 2.5 x 2 + 12.5 x + 75 .
The correct equations are y = 2 x 2 + 10 x + 125 y = 2.5 x 2 + 12.5 x + 75 .
Explanation
Problem Analysis Let's analyze the problem. We are given that Charlie is installing new carpet in his rectangular living room, which is 5 feet longer than it is wide. We have quotes from two contractors. Contractor A charges an installation fee of $125 plus $2 per square foot. Contractor B charges an installation fee of $75 plus 2.50 p ers q u a re f oo t . W e n ee d t o f in d t h ee q u a t i o n s t ha t re p rese n tt h e t o t a l in s t a ll a t i o n cos t , y$, for both companies.
Defining Variables and Area Let x be the width of the rectangular living room. Then the length of the living room is x + 5 . The area of the living room is x ( x + 5 ) = x 2 + 5 x .
Contractor A's Cost For Contractor A, the total cost y is the installation fee plus the cost per square foot times the area: y = 125 + 2 ( x 2 + 5 x ) = 125 + 2 x 2 + 10 x = 2 x 2 + 10 x + 125 .
Contractor B's Cost For Contractor B, the total cost y is the installation fee plus the cost per square foot times the area: y = 75 + 2.5 ( x 2 + 5 x ) = 75 + 2.5 x 2 + 12.5 x = 2.5 x 2 + 12.5 x + 75 .
Finding the Correct Answer Comparing the derived equations with the given options, we find that the correct answer is:
y = 2 x 2 + 10 x + 125 y = 2.5 x 2 + 12.5 x + 75
Examples
Imagine you're planning to carpet your living room and want to compare quotes from different contractors. This problem helps you model the total cost based on the room's dimensions and each contractor's pricing structure. By understanding how to formulate these equations, you can make informed decisions about which contractor offers the best deal for your specific needs. This type of problem is useful in budgeting and comparing costs for home improvement projects.
