key term - Product Problems

Definition

Product problems are a type of word problem that involve finding the product of two or more quantities. These problems often require the use of quadratic equations to solve for an unknown variable that represents one of the factors in the product.

5 Must Know Facts For Your Next Test

1. Product problems often involve finding the dimensions of a rectangle or other geometric shape when the area (or product of the dimensions) is known.
2. Solving product problems typically requires setting up a quadratic equation and then using factoring or the quadratic formula to find the unknown dimension.
3. The solutions to product problems may involve both positive and negative values, which must be considered in the context of the problem.
4. Graphing the quadratic equation can provide additional insight into the problem and help verify the solutions.
5. Product problems can also involve the volume or surface area of three-dimensional shapes, requiring the use of additional formulas and concepts.

Review Questions

• Explain how the concept of a product relates to the solution of quadratic equations in the context of product problems.
  • In product problems, the key concept is that the product of two or more quantities is known, and the goal is to find the values of the individual quantities. This leads to the formulation of a quadratic equation, where one of the variables represents an unknown dimension or factor in the product. Solving the quadratic equation, often through factoring or the quadratic formula, allows the problem solver to determine the values of the unknown quantities that satisfy the given product.
• Describe the process of setting up and solving a product problem that involves the dimensions of a rectangle.
  • To solve a product problem involving the dimensions of a rectangle, the first step is to identify the known information, such as the area of the rectangle. This allows the problem solver to set up a quadratic equation in the form $A = lw$, where $A$ is the known area, and $l$ and $w$ represent the length and width of the rectangle, respectively. The next step is to rearrange the equation to isolate one of the variables, such as $w = A/l$. This results in a quadratic equation in the remaining variable, which can then be solved using factoring or the quadratic formula. The solutions obtained represent the possible dimensions of the rectangle that satisfy the given product (area).
• Analyze how the solutions to a product problem can be interpreted in the context of the original problem statement, and discuss the implications of both positive and negative solutions.
  • When solving a product problem, it is important to consider the context of the original problem statement and the physical meaning of the solutions. In the case of a rectangle, the length and width must be positive values, so any negative solutions obtained from the quadratic equation would not be valid in the real-world context. Additionally, there may be two positive solutions, each representing a different set of dimensions that result in the same product (area). The problem solver must carefully evaluate the solutions and determine which one(s) are meaningful and applicable to the original problem statement. This analysis helps ensure that the final answer is both mathematically correct and logically consistent with the given information and constraints.

Related terms

Quadratic Equation:

A polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

Factoring:

The process of expressing a polynomial as a product of simpler polynomials.

Word Problem: A problem presented in the form of a narrative that requires the solver to determine the appropriate mathematical operations to find the solution.