M4I2 Introduction
Summary
TLDRThe video discusses solving a system of equations related to water levels in two tanks, A and B. Tank A starts with 8 gallons and fills at 1.5 gallons per minute, while Tank B starts with 6 gallons and fills at 2 gallons per minute. The goal is to find a time when both tanks hold the same volume of water. Through graph analysis and algebraic methods, the solution shows that at 4 minutes, both tanks contain 14 gallons of water, making the pair (4, 14) a solution to the system.
Takeaways
- 📊 The investigation focuses on finding a pair of values satisfying relationships in both scenarios (Tank A and Tank B).
- 🚰 Tank A starts with 8 gallons of water and fills at 1.5 gallons per minute, represented by the function A(T).
- 🚱 Tank B starts with 6 gallons of water and fills at 2 gallons per minute, represented by the function B(T).
- 🔍 The goal is to find a time-volume pair that satisfies the relationship for both Tank A and Tank B.
- 📉 The graph shows that the point (4,14) is on both A(T) and B(T), meaning after 4 minutes, both tanks contain 14 gallons of water.
- ✏️ This intersection of graphs represents a solution to the system of equations.
- 🔢 Finding a solution algebraically involves setting A(T) equal to B(T) and solving for T, leading to the equation 8 + 1.5T = 6 + 2T.
- ➗ By solving this equation, we find that T equals 4, meaning both tanks have the same volume at 4 minutes.
- 📏 Substituting T = 4 back into either equation shows that both Tank A and Tank B have 14 gallons of water at this time.
- ✔️ The pair (4, 14) is a solution to this system, confirming that both tanks reach the same water volume at 4 minutes.
Q & A
What is the main focus of this investigation?
-The main focus is to determine if there is a pair of values that satisfies the time-volume relationship for both tank A and tank B.
What are the initial conditions for tank A?
-Tank A initially has 8 gallons of water and fills at a constant rate of 1.5 gallons per minute.
What does A of T represent?
-A of T represents the volume of water in tank A in gallons, T minutes after it started filling.
What are the initial conditions for tank B?
-Tank B initially has 6 gallons of water and fills at a constant rate of 2 gallons per minute.
What does B of T represent?
-B of T represents the volume of water in tank B in gallons, T minutes after it started filling.
What does the point (4, 14) on the graph represent?
-The point (4, 14) means that four minutes after each tank started filling, both tanks have 14 gallons of water in them. This point is a solution to the system.
How can we algebraically find a solution to the system?
-We can find a solution by solving the equation where A of T equals B of T, which involves setting 8 + 1.5T equal to 6 + 2T and solving for T.
What is the value of T when the volumes of tank A and tank B are equal?
-The value of T is 4, meaning four minutes after the tanks started filling, their volumes are equal.
What is the volume of water in both tanks when T equals 4?
-When T equals 4, both tank A and tank B have 14 gallons of water.
Why is the pair (4, 14) considered a solution to the system?
-The pair (4, 14) is considered a solution because it represents the time (4 minutes) when both tanks have equal volumes of water (14 gallons), satisfying the relationships for both tanks.
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