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Difficulty: MediumInterpreting Linear Relationships in Context

A vertical farm uses an automated nutrient delivery system for a crop of lettuce. The total volume of nutrient solution, VV, in liters, remaining in the system's reservoir is modeled as a linear function of the time tt, in hours, since the system started running for the day. The table below shows several values of tt and the corresponding values of VV.

Time, tt (hours)Volume, VV (liters)
3415
5385
8340
12280

According to the model, what was the initial volume of nutrient solution, in liters, in the reservoir when the system started running?

Answer: 460 liters

Answer

460
The correct answer is 460. The remaining volume of nutrient solution, VV, is a linear function of time, tt, which can be written in the form V=mt+bV = mt + b, where mm is the rate of change (slope) and bb is the initial volume (y-intercept). The rate of change can be found using any two points from the table, for example, (3,415)(3, 415) and (5,385)(5, 385): m=385−4155−3=−302=−15m = \frac{385 - 415}{5 - 3} = \frac{-30}{2} = -15 liters per hour. Using the point (3,415)(3, 415) and substituting m=−15m = -15, t=3t = 3, and V=415V = 415 into the equation V=mt+bV = mt + b yields 415=−15(3)+b415 = -15(3) + b. Simplifying this expression gives 415=−45+b415 = -45 + b, and adding 45 to both sides gives b=460b = 460. Therefore, the initial volume of nutrient solution in the reservoir was 460 liters.

Step-by-Step Solution

1
Calculate the constant rate of consumption (slope) of the nutrient solution.
The rate of consumption is 15 liters per hour.
The relationship between volume and time is linear. The slope mm can be calculated from two coordinates from the table, (3,415)(3, 415) and (5,385)(5, 385), as m=385−4155−3=−15m = \frac{385 - 415}{5 - 3} = -15 liters per hour.
2
Determine the initial volume of nutrient solution (the y-intercept) using the rate and a data point.
The initial volume is 460 liters.
Substitute the slope m=−15m = -15, the time t=3t = 3, and the remaining volume V=415V = 415 into the slope-intercept equation V=mt+bV = mt + b. Solving 415=−15(3)+b415 = -15(3) + b gives b=460b = 460.

Key Concept

Interpreting the y-intercept of a linear function in context as the initial value of the dependent variable when the independent variable is 0.
Estimated Time:1m 30s
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Topics

Interpreting Linear Relationships in Context

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