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Difficulty: MediumLinear Functions and Graphs

A water tank is being filled at a constant rate. The volume of water in the tank, in gallons, is a linear function of the time, in minutes, since the filling process began. The volume of water in the tank was 2424 gallons after 33 minutes of filling, and it was 4040 gallons after 77 minutes of filling. What was the initial volume of water, in gallons, in the tank before the filling process began?

Answer: 12 gallons

Answer

The initial volume of water in the tank was 12 gallons.
To find the initial volume of water, we must determine the y-intercept of the linear relationship between the volume and time. First, find the rate of change (slope) using the formula m = \frac{V_2 - V_1}{t_2 - t_1}. Substituting the given values gives m = \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4. Using the slope-intercept form V(t) = mt + b, substitute one of the points, such as (3, 24), to solve for b: 24 = 4(3) + b, which simplifies to b = 12. Thus, the initial volume of water is 12 gallons.

Step-by-Step Solution

1
Calculate the slope (rate of change) of the volume with respect to time.
4
The slope is the change in volume divided by the change in time: \frac{40 - 24}{7 - 3} = \frac{16}{4} = 4.
2
Write the linear function V(t) = mt + b using the calculated slope, and substitute one of the given points to solve for b.
b = 12
Using the point (3, 24) and the slope m = 4, we substitute into V(t) = 4t + b to get 24 = 4(3) + b, which simplifies to 24 = 12 + b, meaning b = 12.
3
Identify the initial volume of water in the tank, which is the value of V(t) when t = 0.
12
When t = 0, V(0) = 4(0) + 12 = 12, which represents the initial volume.

Key Concept

Determining the y-intercept (initial value) of a linear function given two points.
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Topics

Linear Functions and Graphs

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