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Difficulty: MediumInterpreting Linear Relationships in Context

A commercial cargo ship is unloading shipping containers at a port. The total mass of the ship and its remaining cargo, MM, in kilotonnes (kt), is a linear function of the number of hours, hh, since the unloading process began. After 33 hours of unloading, the total mass of the ship and its cargo is 116116 kt. After 88 hours of unloading, the total mass is 9898 kt. According to this model, what is the mass, in kilotonnes, of the cargo that is unloaded each hour?

Answer: 3.6 kilotonnes

Answer

The mass of the cargo unloaded each hour is 3.63.6 kilotonnes (or the equivalent fraction 185\frac{18}{5}).
The correct answer is 3.63.6. Since the relationship between the total mass of the ship and cargo, MM, and the elapsed time, hh, is linear, the rate at which cargo is unloaded corresponds to the magnitude of the slope of the linear function. Using the two data points (3,116)(3, 116) and (8,98)(8, 98), the slope mm can be calculated as 98−1168−3=−185=−3.6\frac{98 - 116}{8 - 3} = \frac{-18}{5} = -3.6. The negative sign indicates that the mass is decreasing, meaning that 3.63.6 kilotonnes of cargo are unloaded each hour.

Step-by-Step Solution

1
Set up the linear relationship model.
M=mh+bM = mh + b
Since the relationship between total mass MM and time hh is linear, it can be modeled by a linear equation where mm is the slope (unloading rate) and bb is the y-intercept (initial mass).
2
Determine the two data points from the given information.
(3,116)(3, 116) and (8,98)(8, 98)
After 33 hours of unloading, the mass is 116116 kt, and after 88 hours, the mass is 9898 kt.
3
Calculate the slope (rate of change) of the linear relationship.
m=−3.6m = -3.6
The slope is calculated as m=98−1168−3=−185=−3.6m = \frac{98 - 116}{8 - 3} = \frac{-18}{5} = -3.6.
4
Interpret the slope's value in context to find the amount of cargo unloaded per hour.
3.63.6
The slope is −3.6-3.6 kilotonnes per hour, which means the total mass decreases by 3.63.6 kilotonnes each hour. Therefore, the mass of the cargo unloaded each hour is 3.63.6 kilotonnes.

Key Concept

Interpreting the slope of a linear relationship in context
Estimated Time:1m 30s
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Topics

Interpreting Linear Relationships in Context

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