At a municipal water desalination facility, the filtration rate of a reverse osmosis membr… — SAT

At a municipal water desalination facility, the filtration rate of a reverse osmosis membrane decreases linearly over time due to particle accumulation. The daily volume of purified water, $V$ , in thousands of gallons per day, can be modeled as a function of the number of days, $t$ , since the membrane was last serviced. The graph of this relationship in the $tV$ -plane has a $t$ -intercept of $100$ and passes through the point $(20, 64)$ . Which of the following is the best interpretation of the slope of the graph of this relationship?

The daily volume of purified water decreases by $0.8$ thousand gallons each day after the membrane is serviced.Answer
B
The daily volume of purified water decreases by $1.25$ thousand gallons each day after the membrane is serviced.
C
The daily volume of purified water decreases by $80$ thousand gallons each day after the membrane is serviced.
D
For every decrease of $1$ thousand gallons in the daily volume of purified water, $0.8$ days have passed since the membrane was serviced.

Answer

The daily volume of purified water decreases by

0.8

thousand gallons each day after the membrane is serviced.

The correct answer describes that the daily volume of purified water decreases by

0.8

thousand gallons each day. The

t

-intercept of

100

indicates the point

(100, 0)

is on the graph, and the problem states the graph passes through

(20, 64)

. The slope of the relationship is

m = \frac{0 - 64}{100 - 20} = -0.8

. Since the vertical axis represents the daily volume of purified water (in thousands of gallons) and the horizontal axis represents the number of days, the slope of

-0.8

represents a decrease of

0.8

thousand gallons of water per day.

Step-by-Step Solution

Identify the coordinates of two points on the line from the given context.

The

t

-intercept of

100

corresponds to the point

(100, 0)

. The second point is given directly as

(20, 64)

Two points on the line are needed to calculate the slope of the linear relationship.

Calculate the slope (

m

) using the slope formula

m = \frac{V_2 - V_1}{t_2 - t_1}

m = \frac{0 - 64}{100 - 20} = \frac{-64}{80} = -0.8

The slope of the line represents the rate of change of the daily volume of water (

V

) with respect to the elapsed days (

t

Interpret the meaning of the slope in context.

A slope of

-0.8

indicates that for every increase of

1

day in

t

, the daily volume of water

V

decreases by

0.8

units (thousands of gallons).

The slope value of

-0.8

represents a daily reduction of

0.8

thousand gallons of purified water.

Key Concept

Interpreting the slope of a linear model in a real-world context

Alternative Method

Instead of calculating the slope directly from the formula, one can write the equation of the line in point-slope form or slope-intercept form. Since

(100,0)

is the

t

-intercept, the line can be represented as

V = m(t - 100)

. Substituting the point

(20, 64)

gives

64 = m(20 - 100)

, which simplifies to

64 = -80m

, solving to

m = -0.8

. This slope represents the daily change in the volume

V

for each unit change in day

t

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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