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Difficulty: MediumLinear Functions and Graphs

A local coffee shop tracks its remaining syrup inventory at the end of each day. On day 3, the shop has 20 liters of syrup remaining. On day 7, the shop has 12 liters of syrup remaining. If the amount of syrup decreases at a constant daily rate, on which day will the shop have exactly 6 liters of syrup remaining?

  1. A
    4
  2. B
    7
  3. 10Answer
  4. D
    31

Answer

10
To find the day when 66 liters of syrup remain, we first find the constant rate of decrease (slope) of the syrup. The slope is calculated as the change in syrup volume divided by the change in days: m=12−207−3=−2m = \frac{12 - 20}{7 - 3} = -2 liters per day. Using the point-slope formula with the point (3,20)(3, 20), the relationship is y−20=−2(x−3)y - 20 = -2(x - 3), which simplifies to y=−2x+26y = -2x + 26. Setting the remaining syrup yy to 66 gives 6=−2x+266 = -2x + 26. Solving for xx yields 2x=202x = 20, so x=10x = 10. Thus, the shop will have exactly 66 liters of syrup remaining on day 1010.

Step-by-Step Solution

1
Calculate the constant daily rate of decrease (slope) of the syrup inventory.
The rate of decrease is −2-2 liters per day.
The slope mm of a linear relationship between day xx and remaining syrup yy is given by m=y2−y1x2−x1=12−207−3=−84=−2m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 20}{7 - 3} = \frac{-8}{4} = -2.
2
Determine the linear equation modeling the remaining syrup as a function of the day.
The linear equation is y=−2x+26y = -2x + 26.
Using the point-slope form with the point (3,20)(3, 20) and slope m=−2m = -2, we have y−20=−2(x−3)y - 20 = -2(x - 3). Distributing and simplifying gives y−20=−2x+6y - 20 = -2x + 6, which simplifies to y=−2x+26y = -2x + 26.
3
Find the day when the remaining syrup is exactly 66 liters.
The shop will have exactly 66 liters of syrup remaining on day 1010.
Substitute y=6y = 6 into the equation and solve for xx: 6=−2x+26⇒−20=−2x⇒x=106 = -2x + 26 \Rightarrow -20 = -2x \Rightarrow x = 10.

Key Concept

Linear Functions and Graphs

Alternative Method

Instead of finding the full linear equation, we can use the rate of change directly. The syrup decreases by 22 liters per day. From day 7, where there are 1212 liters remaining, we need to reach 66 liters remaining. This is a decrease of 12−6=612 - 6 = 6 liters. Since the rate of decrease is 22 liters per day, it will take 62=3\frac{6}{2} = 3 additional days. Adding 33 days to day 7 gives 7+3=107 + 3 = 10.
Estimated Time:1m 30s
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Topics

Linear Functions and Graphs

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