A local municipal water utility charges a flat monthly service fee plus a constant rate fo… — SAT

A local municipal water utility charges a flat monthly service fee plus a constant rate for each hundred cubic feet (HCF) of water consumed. During one month, a household that consumed 12 HCF of water was charged 46.00 dollars. Another household that consumed 18 HCF of water was charged 61.00 dollars. What is the flat monthly service fee, in dollars, charged by the utility?

Cevap: 16 dollars

Cevap

The flat monthly service fee is 16 dollars.

By modeling the relationship as a linear function

y = mx + b

, where

x

is the consumption in HCF and

y

is the total charge in dollars, we use the points

(12, 46)

and

(18, 61)

to find the slope

m = \frac{61 - 46}{18 - 12} = 2.5

. Substituting

m = 2.5

and the point

(12, 46)

back into the linear equation gives

46 = 2.5(12) + b

, which simplifies to

46 = 30 + b

. Solving for

b

gives 16, representing the flat service fee.

Adım Adım Çözüm

Set up a system of points representing the linear relationship between water consumed and monthly charge.

Two coordinate points are established:

(12, 46)

and

(18, 61)

, where the first coordinate is the consumption in HCF and the second is the charge in dollars.

Since the utility charges a flat fee plus a constant rate, the relationship is linear and can be solved using coordinate points.

Find the constant rate per HCF (the slope of the line).

The slope

m

is calculated as

\frac{61 - 46}{18 - 12} = \frac{15}{6} = 2.5

dollars per HCF.

The slope represents the constant rate of change in total cost per HCF of water consumed.

Determine the flat monthly service fee (the y-intercept of the line).

Using the slope-intercept equation

y = mx + b

with point

(12, 46)

and

m = 2.5

yields

46 = 2.5(12) + b \implies 46 = 30 + b \implies b = 16

The y-intercept represents the cost when consumption is zero, which is the flat monthly service fee.

Anahtar Kavram

Finding the y-intercept of a linear function from two points.

Tahmini Süre:1m 30s

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