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Difficulty: MediumLinear Equations in Two Variables

A municipal swimming pool is being filled with water at a constant rate. After 22 hours of filling, the pool contains 14,50014,500 gallons of water. After 55 hours of filling, the pool contains 20,20020,200 gallons of water. If the relationship between the time the pool has been filling, tt, in hours, and the volume of water in the pool, VV, in gallons, is linear, which of the following equations represents this relationship?

  1. A
    V=1,900t+14,500V = 1,900t + 14,500
  2. B
    V=5,700t+3,100V = 5,700t + 3,100
  3. V=1,900t+10,700V = 1,900t + 10,700Answer
  4. D
    V=1,900t+18,300V = 1,900t + 18,300

Answer

The equation representing the relationship is V=1,900t+10,700V = 1,900t + 10,700.
The correct equation is V=1,900t+10,700V = 1,900t + 10,700. The rate of change of the water volume is the change in volume divided by the change in time: 20,200−14,5005−2=5,7003=1,900\frac{20,200 - 14,500}{5 - 2} = \frac{5,700}{3} = 1,900 gallons per hour. Using the point-slope form with the coordinate point (2,14,500)(2, 14,500) yields V−14,500=1,900(t−2)V - 14,500 = 1,900(t - 2). Simplifying this expression gives V−14,500=1,900t−3,800V - 14,500 = 1,900t - 3,800, which results in V=1,900t+10,700V = 1,900t + 10,700.

Step-by-Step Solution

1
Identify two data points from the problem context.
The two coordinate pairs representing (t,V)(t, V) are (2,14,500)(2, 14,500) and (5,20,200)(5, 20,200).
These points will allow us to calculate the slope and the y-intercept of the linear equation.
2
Calculate the slope (mm) using the slope formula m=V2−V1t2−t1m = \frac{V_2 - V_1}{t_2 - t_1}.
m=20,200−14,5005−2=5,7003=1,900m = \frac{20,200 - 14,500}{5 - 2} = \frac{5,700}{3} = 1,900.
The slope represents the constant rate, in gallons per hour, at which the pool is being filled.
3
Substitute the slope m=1,900m = 1,900 and the point (2,14,500)(2, 14,500) into the point-slope form equation V−V1=m(t−t1)V - V_1 = m(t - t_1) to solve for VV.
V−14,500=1,900(t−2)  ⟹  V−14,500=1,900t−3,800  ⟹  V=1,900t+10,700V - 14,500 = 1,900(t - 2) \implies V - 14,500 = 1,900t - 3,800 \implies V = 1,900t + 10,700.
This yields the equation representing the volume of water VV in the pool at any time tt.

Key Concept

Determining a linear equation in two variables given two points from a word problem context.

Alternative Method

Instead of solving the linear equation algebraically, you can test the coordinates of the two given points (2,14,500)(2, 14,500) and (5,20,200)(5, 20,200) in the answer choices. Substituting t=2t = 2 and t=5t = 5 into the correct equation V=1,900t+10,700V = 1,900t + 10,700 satisfies both conditions: 1,900(2)+10,700=14,5001,900(2) + 10,700 = 14,500 and 1,900(5)+10,700=20,2001,900(5) + 10,700 = 20,200. None of the other options satisfy both points.
Estimated Time:1m 30s
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Topics

Linear Equations in Two Variables

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