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Difficulty: MediumLinear Equations in Two Variables

A commercial printing press uses a continuous roll of paper to print newspapers at a constant rate. After 1010 minutes of operation, the remaining length of the paper roll is 12,50012,500 feet. After 2525 minutes of operation, the remaining length of the paper roll is 8,0008,000 feet. If the relationship between the printing time, in minutes, and the remaining length of the paper roll, in feet, is linear, how many minutes after the printing press starts operating will the remaining length of the paper roll be 2,0002,000 feet?

Answer: 45 minutes

Answer

The remaining length of the paper roll will be 2,0002,000 feet after 4545 minutes of operation.
We are given that the remaining length of the paper roll is a linear function of time, tt. Let L(t)L(t) be the remaining length of the paper roll, in feet, after tt minutes of operation. We can represent the given information as two coordinate points: (10,12500)(10, 12500) and (25,8000)(25, 8000). First, find the slope, which represents the constant rate at which the paper is consumed: m=8000−1250025−10=−450015=−300m = \frac{8000 - 12500}{25 - 10} = \frac{-4500}{15} = -300 feet per minute. Next, write the linear equation using the point-slope form: L(t)−12500=−300(t−10)L(t) - 12500 = -300(t - 10), which simplifies to L(t)=15500−300tL(t) = 15500 - 300t. To find the time when the remaining length is 2,0002,000 feet, set L(t)=2000L(t) = 2000 and solve for tt: 2000=15500−300t2000 = 15500 - 300t, which simplifies to 300t=13500300t = 13500, giving t=45t = 45.

Step-by-Step Solution

1
Calculate the rate of paper consumption (the slope of the linear equation) using the two given points, (10,12500)(10, 12500) and (25,8000)(25, 8000).
The rate of paper consumption is −300-300 feet per minute.
To establish the linear relationship, we first need the constant rate of change (slope) from the two known coordinate points.
2
Use the point-slope equation of a line, y−y1=m(x−x1)y - y_1 = m(x - x_1), with the point (10,12500)(10, 12500) and slope m=−300m = -300, to find the equation relating the remaining length, LL, to the time, tt.
L(t)=15500−300tL(t) = 15500 - 300t
We need the full linear model to calculate the remaining length at any specific time.
3
Substitute L(t)=2000L(t) = 2000 into the linear equation and solve for tt.
t=45t = 45
This gives the specific operating time in minutes when the remaining paper roll length is 2,0002,000 feet.

Key Concept

Writing and solving linear equations in two variables from two coordinate points.
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Topics

Linear Equations in Two Variables

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