A water utility company charges a flat monthly connection fee plus a constant rate per gal… — SAT

A water utility company charges a flat monthly connection fee plus a constant rate per gallon of water consumed. In a certain month, a household that consumed $2,800$ gallons of water was billed a total of $62.00$ dollars, and a household that consumed $4,500$ gallons of water was billed a total of $87.50$ dollars. What is the flat monthly connection fee, in dollars, charged by the water utility company?

Answer: 20 dollars

Answer

The flat monthly connection fee charged by the water utility company is 20 dollars.

The flat monthly connection fee is 20 dollars. A linear relationship can be modeled by the equation

y = mx + b

, where

x

represents the water consumed in gallons,

y

represents the total bill in dollars,

m

represents the rate per gallon, and

b

represents the flat monthly connection fee. Using the data for the two households, we get the points

(2800, 62)

and

(4500, 87.5)

. The rate per gallon is the slope of the line passing through these points:

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon. Substituting this slope and the point

(2800, 62)

into the equation gives

62 = 0.015(2800) + b

, which simplifies to

62 = 42 + b

. Solving for

b

yields

20

dollars.

Step-by-Step Solution

Define the linear relationship using the variables

x

for water consumed in gallons and

y

for the total bill in dollars.

The relationship can be written as

y = mx + b

, where

m

represents the cost per gallon of water and

b

represents the flat monthly connection fee.

This establishes the linear framework needed to solve for the unknown constants using the provided data points.

Calculate the constant rate of change (slope

m

) using the coordinates of the two households:

(2800, 62)

and

(4500, 87.5)

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon.

The slope of a linear function represents the constant rate of change between the two variables.

Substitute the calculated slope

m = 0.015

and the point

(2800, 62)

into the equation

y = mx + b

to find the flat fee

b

62 = 0.015(2800) + b \implies 62 = 42 + b \implies b = 20

The flat monthly connection fee corresponds to the vertical intercept (

y

-intercept) of the linear equation.

Key Concept

Linear Equations in Two Variables

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