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Difficulty: MediumLinear Functions and Graphs

A water tank contains 24 gallons of water. Water is being drained from the tank at a constant rate. After 8 minutes, the tank contains 14 gallons of water. If the volume of water in the tank, in gallons, is a linear function of the time in minutes, how many minutes will it take for the tank to contain exactly 4 gallons of water?

Answer: 16 minutes

Answer

16
The volume of water in the tank decreases linearly from an initial value of 2424 gallons at t=0t = 0 to 1414 gallons at t=8t = 8. The constant rate of change (slope) is calculated by dividing the change in volume by the change in time: 14−248−0=−1.25\frac{14 - 24}{8 - 0} = -1.25 gallons per minute. Using the slope-intercept form, the volume VV at time tt is given by V=−1.25t+24V = -1.25t + 24. Setting the volume V=4V = 4 gives the equation 4=−1.25t+244 = -1.25t + 24. Solving for tt yields −20=−1.25t-20 = -1.25t, which simplifies to t=16t = 16.

Step-by-Step Solution

1
Determine the initial state and the state after 8 minutes as coordinate points.
The initial state is (0,24)(0, 24) and the state after 88 minutes is (8,14)(8, 14).
These coordinates represent the relationship between time and volume of water in the tank.
2
Calculate the slope (constant rate of change) of the linear function.
Slope m=14−248−0=−1.25m = \frac{14 - 24}{8 - 0} = -1.25
The slope represents the constant rate at which water is being drained from the tank.
3
Formulate the linear equation.
V(t)=−1.25t+24V(t) = -1.25t + 24
Using the slope-intercept form V(t)=mt+bV(t) = mt + b, where b=24b = 24 is the vertical intercept representing the initial volume.
4
Solve for the time tt when the volume of water is 44 gallons.
4=−1.25t+24  ⟹  −20=−1.25t  ⟹  t=164 = -1.25t + 24 \implies -20 = -1.25t \implies t = 16
To find the time at which the volume decreases to exactly 44 gallons.

Key Concept

Linear Functions and Graphs
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Topics

Linear Functions and Graphs

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