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Question

 The tap on a tank has been left on and water is running out.  The volume of water in the tank after one hour is 100 L and after five hours the volume is 20 L.  Assuming the relationship is linear, find a rule and then state the initial volume of water in the tank 

Answer 1

The equation of a line is:

 

      y= mx + b

 

where

• y is the volume of water in the tank at time x
• x is the time since the tap was opened and the tank started draining
• m is the slope of the line (rate at which the water is running out of the tank)
•  b is the y-intercept, or initial amount of water in the tank

 

To solve the problem, we need to find the values of m and b.  The formula for the slope is:

 

m = (y₂-y₁)/(x₂-x₁)

 

where (x₁,y₁) and (x₂,y₂) are any two points on the line.  You have two points on the line:

 

      (x₁,y₁) = (1,100)

      (x₂,y₂) = (5,20)

 

Hence m = (20-100)/(5-1) = -80/4 = -20 L/hr.  So the equation of the line so far is y = -20x + b.  To find b, the initial volume of water in the tank, plug in either known point.  I'll use (1,100):

 

      y = -20x + b

      100 = -20*1 + b

      100 = -20 + b

      120 = b

 

The full linear equation (rule) is y = -20x + 120, where 120 L is the initial volume of water in the tank.