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Chapter 9: Problem 54

Writing Write and solve a problem that can be modeled by a rational equation.

Short Answer

Expert verified
It would take approximately 800 minutes to fill the swimming pool with the garden hose.

Step by step solution

01

Formulate the Problem

Assume the hose fills at a rate of 7 gallons per minute, the swimming pool holds 5600 gallons, how long would it take to fill? This can be modeled by the rational equation \(\frac{5600}{7} = t\), with \(t\) representing time in minutes.
02

Solve the Equation

To solve for \(t\), you just need to divide 5600 by 7. So, \(t = \frac{5600}{7}\).
03

Interpret the Result

Upon simplification, \(t\) equals approximately 800 minutes. This means it would take approximately 800 minutes for the garden hose to fill the swimming pool.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations, especially rational equations, involves finding an unknown value. In our case, it is the time, denoted by \(t\), for water to fill a swimming pool using a hose. Rational equations have variables in the denominator or can be expressed as fractions. By rewriting this equation with fractions or rational expressions, you can determine the unknown value. To solve our equation, we have to isolate \(t\) by performing actions on the equation, such as division.
  • Understand the equation format \(\frac{a}{b} = c\), where \(a\) is the total volume of water, \(b\) is the rate of water flow, and \(c\) represents time.
  • To solve, apply inverse operations to isolate \(t\), resulting in simple division of 5600 by 7.
Make sure to double-check your division accuracy for precise solutions.
Problem Formulation
Problem formulation is the first step in dealing with mathematical modeling. You translate a real-world scenario into an equation that can be mathematically solved. We start by identifying the known quantities:
  • The pool's capacity: 5600 gallons.
  • Water flow rate: 7 gallons per minute.
These values are used to establish the relationship expressed in the equation \(\frac{5600}{7} = t\). It's vital to ensure all units and quantities are correctly expressed and relatable through the context of the problem. By framing the problem this way, we simplify the complex task into manageable mathematical components.
Mathematical Modeling
Mathematical modeling involves creating mathematical representations of real-world scenarios. In our exercise, modeling the problem using a rational equation converts the physical process (filling a pool) into a solvable mathematical task.
The model, \(\frac{5600}{7} = t\), follows the principle that the total amount of water divided by the rate gives the time needed.
  • This approach ensures clarity in solving, as each variable represents specific elements of the real-world situation.
  • The simplicity of using equations helps articulate such situations into structured problems easily approached with basic math skills.
Through modeling, complex or vague situations become understandable and mathematically tangible.
Division
Division is a fundamental mathematical operation used to distribute quantities evenly. In rational equations, division helps to find unknown variables by isolating them. In our pool example, dividing 5600 (total gallons) by 7 (gallons per minute) determines the time required to fill the pool.
  • Ensure accurate division: 5600 divided by 7 equals 800.
  • This confirms that it takes 800 minutes to fill.
Place emphasis on precision as division errors can skew results. Calculators or verifying through repeated addition are useful methods to ensure the accuracy of division processes in rational equations.

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