Problem 6 - Algebra Form and Function

Question

If \(A\) is inversely proportional to the cube of \(B,\) and \(A=20.5\) when \(B=-4,\) write \(A\) as a power function of \(B\)

Step-by-Step Solution

Answer

Question: Express the function A in terms of B, if A is inversely proportional to the cube of B, and A = 20.5 when B = -4. Answer: A = -1312/B^3

Step 1: Write the equation for inverse proportionality

For an inverse proportionality, A can be written as A=k/B^3

Step 2: Substitute given values A and B

We are given A = 20.5 and B = -4. Substitute these values into the equation: 20.5 = k/(-4)^3

Step 3: Solve for k

Now we need to find the value of k: 20.5 = k/(-64) Solve for k: k = 20.5 * (-64) k = -1312

Step 4: Write the power function of A in terms of B

Now that we have the value of k, we can express A in terms of B: A = -1312/B^3

Key Concepts

Power Function

A power function is a type of mathematical function where a variable, such as \( B \), is raised to a specific power or exponent, and often multiplied by a coefficient, like \( k \). In the given exercise, this was demonstrated through the relationship \( A = \frac{k}{B^3} \), which can be rewritten as \( A = k \cdot B^{-3} \). Here, \( B^{-3} \) indicates a power function where the variable \( B \) is raised to the negative third power, representing an inverse proportionality. Power functions are often used in diverse fields including physics and economics to model real-world phenomena. An important aspect of power functions is identifying the role of the exponent:

• If the exponent is positive, the function indicates direct proportionality.
• If the exponent is negative, the function denotes inverse proportionality, as observed in our example.

Understanding the nature of power functions aids in interpreting relationships between variables efficiently.

Cube of a Variable

The cube of a variable, such as \( B^3 \), represents the variable \( B \) multiplied by itself three times. When dealing with cubes, it's important to consider the effect of negative numbers. For instance, the cube of \( -4 \) is calculated as \[ (-4) imes (-4) imes (-4) = -64 \]. Cubing a negative number yields a negative result, as multiplying an odd number of negative factors results in a negative product. Cubing is fundamental because it not only affects the scale but also the sign of the result, especially when dealing with inverse proportionality. A cube can significantly change the outcome compared to squaring a variable or leaving it linear. Key points about the cube of a variable include:

• It amplifies changes more dramatically than squaring or linear relationships.
• Negative inputs result in negative outputs, affecting the directionality of proportionality when used in equations like \( \frac{1}{B^3} \).

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. In our exercise, algebra plays a crucial role in expressing inverse proportionality and solving for constants like \( k \). When solving \[ 20.5 = \frac{k}{(-4)^3} \], algebraic principles guide us in isolating \( k \). To solve for \( k \), the steps involve:

• First, recognizing the inverse operation needed is multiplication, as dividing \( k \) by \( (-64) \) yields \( 20.5 \).
• Then, performing the necessary multiplication: \( k = 20.5 \times (-64) \).
• This calculates \( k = -1312 \), allowing us to rewrite the expression \( A = -\frac{1312}{B^3} \).

Algebraic manipulation provides the tools required to adjust equations and understand relationships between variables. By mastering algebra, deciphering complex relationships in equations like inverse proportionality becomes approachable and straightforward.