Chapter 7: Problem 26

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

Expert verified

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.

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Short Answer

Step by step solution

Recognize the given proportional relationship

Convert the relationship into an equation

Use the table data to find the constant of proportionality k

Write the final equation connecting P and I

Understand inverse proportion

Evaluate the truth of given statements

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