Question

the aquarium is draining a large fish tank. the pump they are using can drain 24 gallons per hour. after 3 hours, there are 78 gallons left in the tank.
a) write an equation to model the water left in the tank, y, after x hours of draining.
b) what does the slope & y - intercept mean in terms of the problem?
c) how many gallons were originally in the tank?

Explanation:

Step1: Find the slope

The pump drains 24 gallons per hour, so the slope \(m=-24\) (negative because the amount of water is decreasing).

Step2: Use the point - slope form

We know that when \(x = 3\), \(y=78\). The point - slope form of a line is \(y - y_1=m(x - x_1)\). Substituting \(m=-24\), \(x_1 = 3\), and \(y_1=78\):

$$
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$$

Step3: Interpret the slope and y - intercept

The slope \(m=-24\) means that the amount of water in the tank is decreasing by 24 gallons per hour. The y - intercept \(b = 150\) means that when \(x = 0\) (at the start, before any draining), there were 150 gallons of water in the tank.

Step4: Answer part c

The original amount of water in the tank is the value of \(y\) when \(x = 0\). From the equation \(y=-24x + 150\), when \(x = 0\), \(y=150\)

Answer:

a) The equation is \(y=-24x + 150\)
b) The slope \(-24\) means the water is decreasing at a rate of 24 gallons per hour. The y - intercept \(150\) means the initial amount of water in the tank.
c) 150 gallons