M4I2 Introduction

Summary

The 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.