Question

Writing Real-Life Problems In Exercises 47-50, solve the number problem and write a real-life problem that could be represented by this verbal model. For instance, an applied problem that could be represented by Exercise 47 is as follows. The sum of the length and width of a one-story house is 100 feet. The house has 2500 square feet of floor space. What are the length and width of the house? Find two numbers whose sum is 100 and whose product is 2500 .

Short answer

The two numbers are \(50 + 50*sqrt(3)\) and \(50 - 50*sqrt(3)\). These numbers might represent the length and the width of the house in feet.

Step-by-step solution

Define two variables

Let's represent the two numbers as \(x\) and \(y\). So we have \(x + y = 100\) and \(x * y = 2500\).

Express one variable in terms of the other

From the first equation \(x + y = 100\), let's express \(y\) in terms of \(x\). We get \(y = 100 - x\).

Substitute into the second equation

Now, substitute \(y = 100 - x\) into the equation \(x * y = 2500\) to get \(x * (100 - x) = 2500\).

Simplify and solve for x

Expanding the equation gives \(100x - x^2 = 2500\). Rearranging terms and simplifying gives quadratics equation as \(x^2 - 100x + 2500 = 0\). Solve this quadratic equation for \(x\). Using the quadratic formula, we find the values of \(x\) to be \(50 + 50*sqrt(3)\) and \(50 - 50*sqrt(3)\).

Solve for y

Substitute the value of \(x\) into \(y = 100 - x\) and solve for \(y\), get \(y = 50 + 50*sqrt(3)\) and \(y = 50 - 50*sqrt(3)\).

Interpret the solutions

The obtained solutions correspond to the length and width of the house. Note that a negative width is not physically meaningful, so you would chooses the solution that gives a positive width and length. Therefore, the two numbers are the pair that gives positive values.

Key concepts

Algebraic Models

Algebraic models are mathematical structures that use variables and equations to represent real-world scenarios. They transform concrete situations into abstract formulas, offering a simplified and calculable version of reality.

Consider the example of finding the dimensions of a house given the sum of the lengths and the total area. Instead of measuring each wall, which could be impractical, algebra allows us to set up an equation system to find the exact values. Equations like x + y = 100 and xy = 2500 are algebraic models of the real-world problem, where x and y are the unknown dimensions that we need to find. By using algebraic principles, these unknowns can be discovered in a systematic way.

Quadratic Equations

Quadratic equations are a part of algebra involving polynomials of the second degree, and they are typically written in the form ax^2 + bx + c = 0. They're integral in modeling phenomena where two quantities are multiplied together and have extensive applications ranging from physics to economics.

In the context of the house dimension problem, we obtained a quadratic equation x^2 - 100x + 2500 = 0. Solving this requires finding the factors of the equation or using the quadratic formula, which provides a systematic method to determine the roots or solutions. The roots represent possible dimensions that satisfy the given constraints of the total sum and product. In real-world scenarios, the solutions need to be evaluated for practicality—for instance, as dimensions of a house, they must be positive numbers.

Real-Life Math Problems

Mathematics isn't just for textbooks; it's deeply embedded in our everyday life and is crucial for solving practical problems. Real-life math problems allow people to apply mathematical reasoning to come up with tangible solutions in various fields such as architecture, engineering, finance, and more.

The house dimension example is a type of real-life math problem where we need to figure out realistic values for length and width that adhere to the constraints. By setting up equations based on these realistic constraints, we can use mathematical methods to solve for the unknown quantities. Educating students on how to set up and solve such problems prepares them for real-world challenges where mathematical thinking is essential.

Variable Substitution

Variable substitution is a technique used in algebra to simplify equations and make them solvable. It involves replacing one variable with another expression that's equivalent in the context of the given problem.

In our house dimensions problem, after defining the two variables x and y, the next step is to express y in terms of x to reduce the system of equations down to one equation in one variable. This step, which involves substituting y = 100 - x into the second equation, transforms the problem from a system of equations into a single quadratic equation. Substitution is a powerful algebraic technique that not only aids in solving equations but also enhances the understanding of the relationships between different variables in a given problem.