A 28,000-gallon Swimming Pool Is Being Drained Using A Pump That Empties 700 Gallons Per Hour. Which Equation Models This Situation If \[$ G \$\] Is The Number Of Gallons Remaining In The Pool And \[$ T \$\] Is The Amount Of Time In
Introduction
In this scenario, we are presented with a situation where a 28,000-gallon swimming pool is being drained using a pump that empties 700 gallons per hour. The goal is to determine the equation that models this situation, where { g $}$ represents the number of gallons remaining in the pool and { t $}$ represents the amount of time in hours. This problem involves the application of mathematical concepts, specifically the concept of rate of change and the use of differential equations.
Understanding the Situation
To begin, let's break down the information provided in the problem. We know that the pool initially contains 28,000 gallons of water. The pump is capable of emptying 700 gallons per hour. This means that the rate at which the water is being drained is 700 gallons per hour. We can represent this rate as a function of time, denoted as { \frac{dg}{dt} $}$, where { g $}$ is the number of gallons remaining in the pool and { t $}$ is the amount of time in hours.
Modeling the Situation with a Differential Equation
To model this situation, we can use a differential equation. A differential equation is a mathematical equation that describes how a quantity changes over time. In this case, we want to describe how the number of gallons remaining in the pool changes over time. We can represent this as:
{ \frac{dg}{dt} = -700 $}$
This equation states that the rate at which the water is being drained is -700 gallons per hour. The negative sign indicates that the water level is decreasing over time.
Solving the Differential Equation
To solve this differential equation, we need to find the function { g(t) $}$ that represents the number of gallons remaining in the pool at time { t $}$. We can do this by integrating both sides of the equation with respect to time:
{ \int \frac{dg}{dt} dt = \int -700 dt $}$
{ g(t) = -700t + C $}$
where { C $}$ is the constant of integration. This equation represents the number of gallons remaining in the pool at time { t $}$.
Applying the Initial Condition
We know that the pool initially contains 28,000 gallons of water. This means that at time { t = 0 $}$, the number of gallons remaining in the pool is 28,000. We can use this information to determine the value of the constant { C $}$:
{ g(0) = 28000 $}$
{ -700(0) + C = 28000 $}$
{ C = 28000 $}$
The Final Equation
Now that we have determined the value of the constant { C $}$, we can write the final equation that models this situation:
{ g(t) = -700t + 28000 $}$
This equation represents the number of gallons remaining in the pool at time { t $}$.
Conclusion
In this scenario, we were presented with a situation where a 28,000-gallon swimming pool is being drained using a pump that empties 700 gallons per hour. We used mathematical concepts, specifically the concept of rate of change and the use of differential equations, to model this situation. We derived the equation { g(t) = -700t + 28000 $}$ that represents the number of gallons remaining in the pool at time { t $}$. This equation can be used to determine the number of gallons remaining in the pool at any given time.
Applications of the Equation
This equation has several applications in real-world scenarios. For example, it can be used to determine the number of gallons remaining in a pool after a certain amount of time, or to calculate the time it takes to drain a pool to a certain level. It can also be used to model other situations where a quantity is changing over time, such as the depletion of a resource or the growth of a population.
Limitations of the Equation
While this equation is a useful tool for modeling the situation, it has several limitations. For example, it assumes that the pump is always emptying 700 gallons per hour, which may not be the case in reality. Additionally, it does not take into account any external factors that may affect the rate at which the water is being drained, such as changes in the pump's efficiency or the presence of obstacles in the pool. Therefore, it is essential to consider these limitations when using this equation in real-world applications.
Future Directions
Q&A: A 28,000-gallon Swimming Pool Draining Situation
Q: What is the initial amount of water in the pool?
A: The initial amount of water in the pool is 28,000 gallons.
Q: How many gallons of water is the pump capable of emptying per hour?
A: The pump is capable of emptying 700 gallons of water per hour.
Q: What is the rate at which the water is being drained?
A: The rate at which the water is being drained is 700 gallons per hour.
Q: What is the equation that models this situation?
A: The equation that models this situation is { g(t) = -700t + 28000 $}$, where { g(t) $}$ is the number of gallons remaining in the pool at time { t $}$.
Q: What is the meaning of the constant { C $}$ in the equation?
A: The constant { C $}$ represents the initial amount of water in the pool, which is 28,000 gallons.
Q: How can the equation be used in real-world scenarios?
A: The equation can be used to determine the number of gallons remaining in the pool at any given time, or to calculate the time it takes to drain a pool to a certain level. It can also be used to model other situations where a quantity is changing over time, such as the depletion of a resource or the growth of a population.
Q: What are some limitations of the equation?
A: Some limitations of the equation include the assumption that the pump is always emptying 700 gallons per hour, and the failure to take into account any external factors that may affect the rate at which the water is being drained.
Q: Can the equation be used to model other situations?
A: Yes, the equation can be used to model other situations where a quantity is changing over time. For example, it can be used to model the depletion of a resource or the growth of a population.
Q: How can the equation be solved numerically?
A: The equation can be solved numerically using methods such as the Euler method or the Runge-Kutta method.
Q: What are some potential applications of the equation in other fields?
A: Some potential applications of the equation in other fields include modeling the growth of a population in biology, modeling the depletion of a resource in economics, or modeling the flow of a fluid in engineering.
Q: Can the equation be used to model more complex situations?
A: Yes, the equation can be used to model more complex situations by incorporating additional variables or parameters into the model.
Q: How can the equation be used to make predictions about the future?
A: The equation can be used to make predictions about the future by using it to model the behavior of a system over time. For example, it can be used to predict the number of gallons remaining in the pool at a future time, or to predict the time it will take to drain a pool to a certain level.
Q: What are some potential challenges in using the equation to make predictions?
A: Some potential challenges in using the equation to make predictions include the need to accurately model the behavior of the system, the need to account for any external factors that may affect the system, and the need to make accurate assumptions about the initial conditions of the system.