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Jessica is selling books during the summer to earn money for college. She earns a commission on each sale but has to pay for her own expenses.

After a month of driving from neighborhood to neighborhood and walking door-to-door, she figures out that her weekly earnings are approximately a linear function of the number of doors she knocks on.

She writes the equation of the function like this: [tex]E(x) = 10x - 30[/tex], where [tex]x[/tex] is the number of doors she knocks on during the week and [tex]E(x)[/tex] is her earnings for the week in dollars.

Which of the following are reasonable interpretations of the [tex]y[/tex]-intercept of Jessica's function?

Check all that apply:
A. If she does not knock on any doors at all during the week, she will lose [tex]$\$[/tex]30[tex]$.
B. She can earn $[/tex]\[tex]$10$[/tex] per week even if she does not knock on any doors.
C. Her expenses are [tex]$\$[/tex]30[tex]$ per week.
D. She will lose $[/tex]\[tex]$10$[/tex] per week if she does not knock on any doors.

Sagot :

GinnyAnswer
To solve this problem, we need to analyze the given linear function for Jessica's earnings, [tex]\(E(x) = 10x - 30\)[/tex], where [tex]\(x\)[/tex] represents the number of doors she knocks on during the week, and [tex]\(E(x)\)[/tex] is her earnings for the week in dollars.

The [tex]\(y\)[/tex]-intercept of a linear function in the form [tex]\(E(x) = mx + b\)[/tex] is the constant term [tex]\(b\)[/tex]. The [tex]\(y\)[/tex]-intercept represents the value of [tex]\(E(x)\)[/tex] when [tex]\(x = 0\)[/tex].

Given the function [tex]\(E(x) = 10x - 30\)[/tex]:
- The term [tex]\(10x\)[/tex] indicates that Jessica earns [tex]$10 for each door she knocks on. - The term \(-30\) represents the \(y\)-intercept, which is the value of her earnings when she does not knock on any doors (\(x = 0\)). Let's go through each option to determine if they are reasonable interpretations of the \(y\)-intercept: A. If she does not knock on any doors at all during the week, she will lose $[/tex]\[tex]$30$[/tex].
- When [tex]\(x = 0\)[/tex], [tex]\(E(0) = 10(0) - 30 = -30\)[/tex].
- This means if Jessica does not knock on any doors, her earnings for the week will be [tex]\(-\$30\)[/tex], indicating a loss of \[tex]$30 due to her expenses. - This is a reasonable interpretation. B. She can earn \$[/tex]10 per week even if she does not knock on any doors.
- When [tex]\(x = 0\)[/tex], [tex]\(E(0) = -30\)[/tex], not \[tex]$10. - This suggests that she does not earn any money if she does not knock on any doors; instead, she incurs a loss of \$[/tex]30.
- This is not a reasonable interpretation.

C. Her expenses are \[tex]$30 per week. - The \(-\$[/tex]30\) indicates that Jessica has fixed expenses of \[tex]$30 per week. - No matter how many doors she knocks on, she will always have to cover her \$[/tex]30 expenses to break even.
- This is a reasonable interpretation.

D. She will lose \[tex]$10 per week if she does not knock on any doors. - When she does not knock on any doors, she loses \$[/tex]30, not \$10.
- This does not correctly reflect the given equation.
- This is not a reasonable interpretation.

Therefore, the reasonable interpretations of the [tex]\(y\)[/tex]-intercept are options A and C.
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