Math Problem Statement

Amanda is filling up her pool with a hose. The pool starts out empty. The hose pumps liters of water every minute that it runs. Let be the time (in minutes) that the hose runs, and let be the amount of water (in liters) in the pool. (a)Graph the line that shows the relationship between and . Amount of water (liters) y50100150200250300350400450500x1234567890

Solution

Let's break down the problem and identify the relationship between the variables:

  • Let xx represent time in minutes.
  • Let yy represent the amount of water in liters.
  • The hose pumps at a constant rate of liters per minute, meaning that the relationship between xx and yy is linear.

From the graph details given, it seems the hose pumps 50 liters of water per minute.

The relationship can be expressed by the equation: y=50xy = 50x Where:

  • yy is the amount of water in liters,
  • xx is the time in minutes.

Step-by-Step Explanation of the Graph:

  1. Start at the origin (0,0): When x=0x = 0 (the hose hasn't run yet), y=0y = 0 (no water in the pool).
  2. Rate of increase: For each minute xx increases, the amount of water increases by 50 liters. Hence, the slope of the line is 50.

For some sample points:

  • At x=1x = 1, y=50×1=50y = 50 \times 1 = 50 liters.
  • At x=2x = 2, y=50×2=100y = 50 \times 2 = 100 liters.
  • At x=3x = 3, y=50×3=150y = 50 \times 3 = 150 liters.
  • And so on, until x=10x = 10, where y=500y = 500 liters.

Graph:

You can now plot these points on the graph:

  • (0, 0)
  • (1, 50)
  • (2, 100)
  • (3, 150)
  • (4, 200)
  • (5, 250)
  • (6, 300)
  • (7, 350)
  • (8, 400)
  • (9, 450)
  • (10, 500)

These points form a straight line, which represents the relationship y=50xy = 50x.


Would you like further details about the graph or any particular concept?

Here are 5 relative questions that expand this problem:

  1. What happens if the hose pumps at a different rate, say 75 liters per minute?
  2. How would the graph change if the pool already had 100 liters of water at the start?
  3. How long will it take for the pool to reach 1000 liters with this hose?
  4. What would the equation be if the hose had to stop after 8 minutes?
  5. How would the graph change if Amanda used two hoses, each with the same rate of 50 liters per minute?

Tip: When dealing with linear relationships, the slope tells you how fast one variable is changing with respect to the other.

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Math Problem Analysis

Mathematical Concepts

Linear Relationships
Graphing
Slope-Intercept Form

Formulas

y = 50x

Theorems

Slope of a Line

Suitable Grade Level

Grades 6-8