Problem 3 - College Algebra

Question

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(a\) is directly proportional to the square of \(b\)

Step-by-Step Solution

Answer

The equation expressing the relationship is \(a = k*b^2\).

Step 1: Identify the Variables

In the given exercise, \(a\) varies directly as the square of \(b\). Therefore, \(a\) can be identified as \(y\) and \(b\) can be identified as \(x\).

Step 2: Understand the type of variation

The exercise states that \(a\) is directly proportional to the square of \(b\). This can be interpreted as a direct variation where \(n=2\), as it varies with the square of \(b\).

Step 3: Formulate the Equation

Using the formula for direct variation \(y = kx^n\), replace \(y\) with \(a\), \(x\) with \(b\) and \(n\) with \(2\). This gives the equation \(a = k*b^2\).

Key Concepts

Square of a Number

The concept of the square of a number is fundamental in mathematics. When you "square" a number, you multiply the number by itself. For example, the square of 3 is written as 3² and calculated as 3 multiplied by 3, which equals 9. This is an important operation because it helps in understanding geometric shapes, solving equations, and performing various calculations in algebra and calculus.

Squaring is often used in mathematical formulas and is a key component in direct variation equations, as you'll see with proportional relationships. It is essential to note that squaring a number always results in a non-negative number. Hence, when you see something like the square of -4 (written as (-4)^2), the result is still 16, as negative times negative yields a positive.

Proportionality

Proportionality is a crucial concept that describes the relationship between two quantities. When two variables are directly proportional, it means as one variable increases, the other does so at a constant rate. This is represented with equations where a constant of proportionality, usually denoted by "k," is involved.

For direct proportionality, the relationship can be expressed as: - If variable "a" is directly proportional to the square of variable "b", the equation will be: \( a = k \cdot b^2 \).
Learn more about solving proportional relationships.

To put it simply, if you know the value of "b" and the constant "k," you can determine "a." Direct proportionality lays the foundation for many practical applications, such as physics and engineering, enabling us to predict and calculate behaviors of systems in a controlled and calculable manner.

Mathematical Equations

Mathematical equations are statements that express the equality between two expressions. Equations are the language of mathematics and serve as a powerful tool to model and solve real-world problems.

In the context of direct variation and proportionality, mathematical equations can take several forms. For the relationship described in the original problem where "a" is directly proportional to the square of "b," the equation becomes:

• \( a = k \cdot b^2 \)

Here, "k" is the constant of variation, "a" is dependent on "b^2," meaning as "b" changes, "a" changes according to the square of "b." This equation is fundamental in understanding how changes in one variable can affect another in a predictable manner.

Solving equations like these allows for deeper insights into mathematical relationships and lays the groundwork for solving more complex problems in mathematics, science, and engineering.