A commercial 3D printer uses a polymer filament to print prototype parts. The remaining ma… — SAT

A commercial 3D printer uses a polymer filament to print prototype parts. The remaining mass of the filament spool, $M$ , in grams, is modeled by a linear function of the total printing time, $t$ , in minutes. When the printer has been running for $20$ minutes, the spool has $850$ grams of filament remaining. When the printer has been running for $50$ minutes, the spool has $685$ grams of filament remaining. According to the model, how many grams of filament does the printer use per minute?

Cevap: 5.5 grams

Cevap

The printer uses 5.5 grams of filament per minute.

The relationship between the printing time

t

and the remaining mass of the filament

M

is linear. The slope of this line represents the change in filament mass per unit of time. Based on the given coordinates

(20, 850)

and

(50, 685)

, the slope is calculated as

\frac{685 - 850}{50 - 20} = \frac{-165}{30} = -5.5

grams per minute. This means that the remaining mass decreases by

5.5

grams for each additional minute the printer runs, which corresponds to a usage rate of

5.5

grams of filament per minute.

Adım Adım Çözüm

Identify the data points representing time and remaining filament mass from the context.

The two points are

(20, 850)

and

(50, 685)

These points allow us to calculate the rate at which the mass of the filament is changing over time.

Calculate the change in remaining mass and the change in printing time.

Change in mass is

685 - 850 = -165

grams. Change in time is

50 - 20 = 30

minutes.

Finding the differences in the dependent variable (mass) and independent variable (time) is the standard method to calculate a rate of change.

Calculate the rate of filament usage per minute.

\frac{-165\text{ grams}}{30\text{ minutes}} = -5.5

grams per minute. The rate of usage is the positive magnitude, which is

5.5

grams per minute.

The slope of the linear relationship is negative because the mass is decreasing, but the rate of consumption/usage is represented as a positive quantity.

Anahtar Kavram

Interpreting the slope of a linear function in context as a rate of change.

Soru

Cevap

Adım Adım Çözüm

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