A municipal recycling facility processes plastic waste. The total mass of unprocessed plas… — SAT

A municipal recycling facility processes plastic waste. The total mass of unprocessed plastic, $M$ , in tons, remaining at the facility $h$ hours after the facility opens on a given day can be modeled by the equation $M = 142 - 8.5h$ . Which of the following is the best interpretation of the number $8.5$ in this context?

A
The facility has $142$ tons of unprocessed plastic when it opens.
The mass of unprocessed plastic at the facility decreases by $8.5$ tons each hour.Answer
C
The facility takes $8.5$ hours to process each ton of plastic.
D
The mass of unprocessed plastic at the facility decreases by $133.5$ tons each hour.

Answer

The mass of unprocessed plastic at the facility decreases by

8.5

tons each hour.

In the equation

M = 142 - 8.5h

, the total mass of unprocessed plastic

M

decreases by

8.5

tons for each hour

h

that passes. The number

8.5

is the magnitude of the slope of the linear relationship, which represents the rate of change. Since the coefficient of

h

is negative, the amount of unprocessed plastic decreases by

8.5

tons per hour.

Step-by-Step Solution

Identify the structure of the linear equation

M = 142 - 8.5h

The equation is in the form

y = b + mx

, where

b

is the y-intercept (

142

) and

m

is the slope (

-8.5

This helps separate the initial value from the rate of change.

Analyze the coefficient of the independent variable

h

The coefficient is

-8.5

The coefficient of

h

represents the rate of change of the mass

M

per hour.

Interpret the meaning of the rate of change in context.

A rate of change of

-8.5

means that the mass of unprocessed plastic decreases by

8.5

tons for each hour that passes.

The negative sign indicates a decrease, and the unit is tons per hour.

Key Concept

Interpreting the slope of a linear relationship in a real-world context.

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