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Algebra / Linear functions Difficulty: Hard

A window repair specialist charges $dollar sign 220$ for the first two hours of repair plus an hourly fee for each additional hour. The total cost for $5$ hours of repair is $dollar sign 400$ . Which function $f$ gives the total cost, in dollars, for $x$ hours of repair, where $x greater than or equals 2$ ?

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Explanation

Choice A is correct. It’s given that the window repair specialist charges $dollar sign 220$ for the first two hours of repair plus an hourly fee for each additional hour. Let $n$ represent the hourly fee for each additional hour after the first two hours. Since it’s given that $x$ is the number of hours of repair, it follows that the charge generated by the hourly fee after the first two hours can be represented by the expression $n left parenthesis x minus 2 right parenthesis$ . Therefore, the total cost, in dollars, for $x$ hours of repair is $f (x) = 220 + n (x - 2)$ . It’s given that the total cost for $5$ hours of repair is $dollar sign 400$ . Substituting $5$ for $x$ and $400$ for $f (x)$ into the equation $f (x) = 220 + n (x - 2)$ yields $400 = 220 + n (5 - 2)$ , or $400 = 220 + 3 n$ . Subtracting $220$ from both sides of this equation yields $180 equals 3 n$ . Dividing both sides of this equation by $3$ yields $n equals 60$ . Substituting $60$ for $n$ in the equation $f (x) = 220 + n (x - 2)$ yields $f (x) = 220 + 60 (x - 2)$ , which is equivalent to $f (x) = 220 + 60 x - 120$ , or $f (x) = 60 x + 100$ . Therefore, the total cost, in dollars, for $x$ hours of repair is $f (x) = 60 x + 100$ .

Choice B is incorrect. This function represents the total cost, in dollars, for $x$ hours of repair where the specialist charges $dollar sign 340$ , rather than $dollar sign 220$ , for the first two hours of repair.

Choice C is incorrect. This function represents the total cost, in dollars, for $x$ hours of repair where the specialist charges $dollar sign 160$ , rather than $dollar sign 220$ , for the first two hours of repair, and an hourly fee of $dollar sign 80$ , rather than $dollar sign 60$ , after the first two hours.

Choice D is incorrect. This function represents the total cost, in dollars, for $x$ hours of repair where the specialist charges $dollar sign 380$ , rather than $dollar sign 220$ , for the first two hours of repair, and an hourly fee of $dollar sign 80$ , rather than $dollar sign 60$ , after the first two hours.