Each day that a library book is kept past its due date, a [tex]$\$[/tex]0.30[tex]$ fee is charged at midnight. Which ordered pair is a viable solution if $[/tex]x[tex]$ represents the number of days that a library book is late and $[/tex]y[tex]$ represents the total fee?

A. $[/tex](-3, -0.90)[tex]$
B. $[/tex](-2.5, -0.75)[tex]$
C. $[/tex](4.5, 1.35)[tex]$
D. $[/tex](8, 2.40)$



Answer :

To determine which ordered pairs are viable solutions where [tex]\( x \)[/tex] represents the number of days a library book is late and [tex]\( y \)[/tex] represents the total fee charged at a rate of [tex]$0.30 per day, we need to evaluate each pair. First, let's recall the given fee per day: \[ \text{Fee per day} = \$[/tex]0.30 \]

We will calculate the total fee [tex]\( y \)[/tex] for each [tex]\( x \)[/tex] by using the equation:
[tex]\[ y = x \times 0.30 \][/tex]

Let's evaluate each ordered pair provided in the question:

1. Ordered pair [tex]\((-3, -0.90)\)[/tex]:
[tex]\[ x = -3, \quad y = -3 \times 0.30 = -0.90 \][/tex]
The calculated [tex]\( y \)[/tex] value is [tex]\(-0.90\)[/tex]. While this calculation is correct, note that a viable solution should have a positive [tex]\( x \)[/tex] value since the number of days cannot be negative.

2. Ordered pair [tex]\((-2.5, -0.75)\)[/tex]:
[tex]\[ x = -2.5, \quad y = -2.5 \times 0.30 = -0.75 \][/tex]
Again, the calculated [tex]\( y \)[/tex] value is [tex]\(-0.75\)[/tex]. However, like the previous case, a negative [tex]\( x \)[/tex] value is not viable since it is not possible to have negative days.

3. Ordered pair [tex]\((4.5, 1.35)\)[/tex]:
[tex]\[ x = 4.5, \quad y = 4.5 \times 0.30 = 1.35 \][/tex]
The calculated [tex]\( y \)[/tex] value is [tex]\( 1.35 \)[/tex]. This pair is viable since both [tex]\( x \)[/tex] (number of days late) and [tex]\( y \)[/tex] (total fee) are positive and correctly calculated based on the fee per day.

4. Ordered pair [tex]\((8, 2.40)\)[/tex]:
[tex]\[ x = 8, \quad y = 8 \times 0.30 = 2.40 \][/tex]
The calculated [tex]\( y \)[/tex] value is [tex]\( 2.40 \)[/tex]. This pair is also viable as both the number of days and the total fee are positive and correctly calculated.

Therefore, the viable solutions are:
[tex]\[ (4.5, 1.35) \quad \text{and} \quad (8, 2.40) \][/tex]