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Difficulty: HardInterpreting Linear Relationships in Context

A commercial agricultural irrigation system distributes liquid fertilizer from a storage tank. The volume of fertilizer remaining in the tank, VV, in liters, mm minutes after the system is turned on is modeled by the equation:

V=1,800−15(m−3c)V = 1,800 - 15(m - 3c)

where cc is the number of times the system's nozzles are cleaned during the irrigation process. During each cleaning cycle, the flow of fertilizer is completely paused, and no fertilizer is distributed. Based on the model, what is the duration, in minutes, of a single cleaning cycle?

Answer: 3 minutes

Answer

The duration of a single cleaning cycle is 3 minutes.
In the model V=1,800−15(m−3c)V = 1,800 - 15(m - 3c), the term (m−3c)(m - 3c) represents the total time, in minutes, that the irrigation system is actively distributing fertilizer. Since the total elapsed time is mm minutes and the system pauses during cleaning cycles, the term 3c3c represents the total paused time for cc cleanings. Therefore, the duration of a single cleaning cycle is 3cc=3\frac{3c}{c} = 3 minutes.

Step-by-Step Solution

1
Analyze the structure of the equation to identify the meaning of each term.
The coefficient 15 is the active flow rate in liters per minute, and (m−3c)(m - 3c) is the active distribution time in minutes.
To understand how the time variables affect the volume of fertilizer remaining.
2
Relate the total elapsed time to the active time and the paused time.
The total elapsed time is mm minutes, and the active time is m−3cm - 3c minutes, meaning the system is paused for a total of 3c3c minutes.
To find the expression for the total duration of all cleaning cycles.
3
Determine the duration of a single cleaning cycle.
Since cc cleaning cycles result in a total pause of 3c3c minutes, each cycle lasts 3cc=3\frac{3c}{c} = 3 minutes.
To calculate the duration of one individual cleaning cycle.

Key Concept

Interpreting Linear Relationships in Context
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Topics

Interpreting Linear Relationships in Context

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