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Real-life examples of direct and inverse variation:

1. Direct Variation Example:

Scenario: Water flow rate and time to fill a tank.

• Explanation: In this case, the time it takes to fill a tank is directly proportional to the amount of water required to fill it. If the flow rate increases, the tank fills faster, and if it decreases, it takes longer. Mathematically, this means that time $t$ varies directly with the amount of water $V$.

a. Identify the variables:

• Flow rate $r$ (liters per minute)
• Time $t$ (minutes)

b. Direct variation equation: $V = r \cdot t$ Where $V$ is the total volume of water, $r$ is the flow rate, and $t$ is the time.

c. Problem: If the flow rate is 5 liters per minute, how long will it take to fill a 200-liter tank?

Solution: Using the equation $V = r \cdot t$: $200 = 5 \cdot t$ Solving for $t$: $t = \frac{200}{5} = 40 \text{ minutes}$ It will take 40 minutes to fill the tank.

2. Inverse Variation Example:

Scenario: Pressure and volume of a gas (Boyle's Law).

• Explanation: In a closed container, the pressure $P$ of a gas is inversely proportional to its volume $V$, assuming constant temperature. If the volume decreases, the pressure increases, and vice versa.

a. Identify the variables:

• Pressure $P$ (in atmospheres)
• Volume $V$ (in liters)

b. Inverse variation equation: $P \cdot V = k$ Where $k$ is a constant.

c. Problem: If the pressure of a gas is 2 atmospheres when the volume is 6 liters, what will the pressure be if the volume decreases to 3 liters?

Solution: Using the equation $P \cdot V = k$, we first find $k$ when $P = 2$ and $V = 6$: $2 \cdot 6 = 12$ Now, find the new pressure when the volume is 3 liters: $P \cdot 3 = 12$ Solving for $P$: $P = \frac{12}{3} = 4 \text{ atmospheres}$ The pressure will increase to 4 atmospheres when the volume is reduced to 3 liters.

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Related Questions:

1. How does the concept of direct variation apply to speed and distance?
2. Can you provide another real-world example of inverse variation?
3. How does the constant $k$ change in direct and inverse variation problems?
4. What happens when both variables in an inverse variation change simultaneously?
5. How does temperature affect the relationship between pressure and volume of a gas?

Tip:

Always verify whether the relationship between variables is linear or nonlinear to properly categorize it as direct or inverse variation.