Question

A variable \(P\) is proportional to \(I^{2}\) (a) Use the measurements in Table \(7.4\) to determine an equation connecting \(P\) and \(I\). $$ \begin{array}{rrrrrr} \hline P & 24 & 54 & 96 & 150 & 216 \\ I & 2 & 3 & 4 & 5 & 6 \\ \hline \end{array} $$ If \(a\) is inversely proportional to \(b\) state which of the following are true and which are false: (a) \(a\) multiplied by \(b\) is a constant (b) \(a\) divided by \(b\) is a constant (c) \(a^{2}\) is inversely proportional to \(b^{2}\)

Short answer

Answer: The equation connecting P and I is $$P = 6I^2$$. The true statements about inverse proportion are (a) a multiplied by b is a constant and (c) a^2 is inversely proportional to b^2.

Step-by-step solution

Recognize the given proportional relationship

Given that the variable P is proportional to I^2, which can be written as: $$P \propto I^2$$

Convert the relationship into an equation

To convert the proportional relationship into an equation, we need a constant of proportionality, say k: $$P = kI^2$$

Use the table data to find the constant of proportionality k

We can use any row from the table, let's use the first row with P=24 and I=2: $$24 = k(2)^2$$ To find k, we divide both sides by 4: $$k = 6$$

Write the final equation connecting P and I

Now that we have the constant of proportionality k, we can write the final equation connecting P and I: $$P=6I^2$$ #b. Evaluating the statements about inverse proportion#

Understand inverse proportion

When a is inversely proportional to b, it can be expressed as: $$a \propto \frac{1}{b}$$ or: $$ab = C$$ where C is a constant.

Evaluate the truth of given statements

(a) \(a\) multiplied by \(b\) is a constant: Since we know that for inversely proportional quantities, \(ab=C\), this statement is True. (b) \(a\) divided by \(b\) is a constant: This statement is false, as we already know that \(ab = C\). If \(ab = C\), then \(\frac{a}{b}\neq C\) unless \(C = 1\). (c) \(a^{2}\) is inversely proportional to \(b^{2}\): Consider the relationship, \(ab = C\). If we square both sides, we get: $$(ab)^2 = C^2$$ $$a^2b^2 = C^2$$ Thus, we change the constant to C^2 and we can still conclude that \(a^2\) is inversely proportional to \(b^2\). So the statement is True.

Key concepts

Constant of Proportionality

When dealing with proportional relationships, the constant of proportionality plays a crucial role. It's like a glue that connects two variables in a mathematical dance. Imagine you have a variable, say \( P \), and it grows in relation to another variable squared, like \( I^2 \). This can be expressed as \( P \propto I^2 \), which means \( P \) is directly proportional to \( I^2 \).

But to handle this proportional relationship in real terms, we need more than just a squiggle. That’s where the constant of proportionality, denoted often by \( k \), steps in to form a bridge. So, instead of just \( P \propto I^2 \), we write \( P = kI^2 \).

Finding the Constant "k"
To find the constant \( k \), take a known set of values from your data. For example, if you know \( P = 24 \) when \( I = 2 \), plug these values into your equation:

• \( 24 = k(2)^2 \)
• Solve for \( k \) by simplifying: \( 24 = 4k \), thus \( k = 6 \)

With \( k \) in hand, your equation becomes clear: \( P = 6I^2 \). This simple equation gives you the full story on how \( P \) changes in relation to \( I \). Add more values and check to ensure \( k \) remains the same to verify accuracy.

Inverse Proportionality

Inverse proportionality can feel like the flip side of a coin. Instead of two things lifting each other up, they counterbalance each other. Consider two variables, \( a \) and \( b \), that refuse to hold hands in a straight line. When one grows, the other shrinks. This is expressed as \( a \propto \frac{1}{b} \).

In practical terms, it means their product is a constant: \( ab = C \). Here, "\( C \)" is similar to the constant of proportionality but applies in the inverse relationship.

Verifying Statements

• \( a \) multiplied by \( b \) is a constant: True! They swap efforts so the product, \( C \), stays solid.
• \( a \) divided by \( b \) isn't a constant: False! Only their product remains constant, not the quotient unless under special conditions.
• \( a^2 \) inversely proportional to \( b^2 \): Also True. Square both sides: \( a^2b^2 = C^2 \). It maintains the inverse relationship.

When dealing with inverse proportions, look for that balancing point just like a seesaw.

Mathematical Equations

Mathematical equations are the language of logic, forever translating relationships into symbols. In proportional relationships, equations guide us in making sense of connections among variables.

Creating Equations from Proportions

• Direct Proportion: If you have \( P \propto I^2 \), then the equation \( P = kI^2 \) maps it out. \( k \) makes the abstract concrete.
• Inverse Proportion: With \( a \propto \frac{1}{b} \), the equation \( ab = C \) does the talking.

Equations don't just give form to ideas; they allow calculations, predictions, and discoveries. Remember, they're tools to help us see patterns and consistencies in what might first appear random.

As you write or solve equations, look for constants. They anchor these relationships, guiding us to insights while honing our problem-solving skills. Equations are like blueprints in mathematics, revealing the harmony within chaos, one expression at a time.