For the given $$ x $$ and $$ y $$ variables, decide whether $$ y $$ is or is not proportional to $$ x $$. (a) $$ y $$ is the cost of $$ x $$ gallons of gasoline. (b) $$ y $$ is the perimeter of a square whose sides have length $$ x $$. (c) $$ y $$ is the area of a square whose sides have length $$ x $$. (d) $$ y $$ is the sales tax in Idaho of an item that costs $$ x $$ dollars. (e) $$ y $$ is the taxi fare for a ride of $$ x $$ miles. (a) $$ y $$ is the cost of $$ x $$ gallons of gasoline. $$ y $$

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**(a) $$ y $$ is the cost of $$ x $$ gallons of gasoline.**

If gasoline is sold at a fixed price per gallon, then the total cost is obtained by multiplying the number of gallons $$ x $$ by a constant unit price $$ k $$. This relationship can be expressed as  
$$
y = kx.
$$  
Since $$ y $$ is given by a constant multiple of $$ x $$, $$ y $$ is proportional to $$ x $$.

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**(b) $$ y $$ is the perimeter of a square whose sides have length $$ x $$.**

The perimeter $$ P $$ of a square with side length $$ x $$ is  
$$
P = 4x.
$$  
This is a direct linear relationship where $$ k = 4 $$. Thus, $$ y = 4x $$ demonstrates that $$ y $$ is proportional to $$ x $$.

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**(c) $$ y $$ is the area of a square whose sides have length $$ x $$.**

The area $$ A $$ of a square with side length $$ x $$ is calculated by  
$$
A = x^2.
$$  
Since the relationship involves squaring $$ x $$, it is not of the form $$ y = kx $$ for a constant $$ k $$. Therefore, $$ y $$ is not proportional to $$ x $$.

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**(d) $$ y $$ is the sales tax in Idaho of an item that costs $$ x $$ dollars.**

Sales tax is typically calculated as a fixed percentage of the cost. If the sales tax rate is $$ r $$ (expressed as a decimal), then the sales tax is given by  
$$
y = rx.
$$  
This is a proportional relationship because $$ y $$ is a constant multiple of $$ x $$.

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**(e) $$ y $$ is the taxi fare for a ride of $$ x $$ miles.**

Many taxi fare structures include a flat base fee plus a cost per mile. For example, if the base fee is $$ b $$ and the charge per mile is $$ k $$, the fare would be  
$$
y = b + kx.
$$  
Because of the additional base fee $$ b $$ (which is not multiplied by $$ x $$), the relationship is not of the form $$ y = kx $$. Hence, $$ y $$ is not proportional to $$ x $$.
- **(a)** $$ y $$ is proportional to $$ x $$. - **(b)** $$ y $$ is proportional to $$ x $$. - **(c)** $$ y $$ is not proportional to $$ x $$. - **(d)** $$ y $$ is proportional to $$ x $$. - **(e)** $$ y $$ is not proportional to $$ x $$.

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