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Answer :
Sure! Let's go through this step-by-step.
1. Understanding the problem:
- A window repair specialist has a charge of \[tex]$220 for the first two hours of work.
- After the first two hours, there is an additional hourly rate for each hour worked.
- The total cost for 5 hours of repair is \$[/tex]400.
- We need to find a function [tex]\( f \)[/tex] that gives the total cost for [tex]\( x \)[/tex] hours of repair where [tex]\( x \geq 2 \)[/tex], from the given choices.
2. Identifying our target function form:
- Let [tex]\( f(x) \)[/tex] be the function that gives the cost for [tex]\( x \)[/tex] hours of repair.
- We know that for the first two hours, the charge is \[tex]$220.
- After the first two hours, there is an additional hourly fee.
3. Setting up the equation based on the given information:
- For 5 hours of repair, the total cost is \$[/tex]400. Since the first two hours cost \$220, the remaining cost for the additional 3 hours (from the 5 total hours) is:
[tex]\[
400 - 220 = 180 \text{ dollars}
\][/tex]
- Therefore, the additional hourly fee can be calculated as:
[tex]\[
\text{hourly fee} = \frac{180 \text{ dollars}}{3 \text{ hours}} = 60 \text{ dollars per hour}
\][/tex]
4. Constructing the function:
- The total cost function [tex]\( f(x) \)[/tex] will include the initial charge for the first two hours and then the additional hourly fee for hours beyond the first two.
- Thus, for [tex]\( x \geq 2 \)[/tex], the cost function can be written as:
[tex]\[
f(x) = 220 + (x - 2) \times 60
\][/tex]
5. Simplifying the function:
- Distribute and combine like terms:
[tex]\[
f(x) = 220 + 60x - 120
\][/tex]
[tex]\[
f(x) = 60x + 100
\][/tex]
6. Finding the correct choice:
- The function we derived is [tex]\( f(x) = 60x + 100 \)[/tex].
- We compare this with the given options.
[tex]\[
(A) \; f(x) = 60x + 100
\][/tex]
[tex]\[
(B) \; f(x) = 60x + 220
\][/tex]
[tex]\[
(C) \; f(x) = 80x
\][/tex]
[tex]\[
(D) \; f(x) = 80x + 220
\][/tex]
The correct choice is [tex]\( \boxed{A} \)[/tex].
1. Understanding the problem:
- A window repair specialist has a charge of \[tex]$220 for the first two hours of work.
- After the first two hours, there is an additional hourly rate for each hour worked.
- The total cost for 5 hours of repair is \$[/tex]400.
- We need to find a function [tex]\( f \)[/tex] that gives the total cost for [tex]\( x \)[/tex] hours of repair where [tex]\( x \geq 2 \)[/tex], from the given choices.
2. Identifying our target function form:
- Let [tex]\( f(x) \)[/tex] be the function that gives the cost for [tex]\( x \)[/tex] hours of repair.
- We know that for the first two hours, the charge is \[tex]$220.
- After the first two hours, there is an additional hourly fee.
3. Setting up the equation based on the given information:
- For 5 hours of repair, the total cost is \$[/tex]400. Since the first two hours cost \$220, the remaining cost for the additional 3 hours (from the 5 total hours) is:
[tex]\[
400 - 220 = 180 \text{ dollars}
\][/tex]
- Therefore, the additional hourly fee can be calculated as:
[tex]\[
\text{hourly fee} = \frac{180 \text{ dollars}}{3 \text{ hours}} = 60 \text{ dollars per hour}
\][/tex]
4. Constructing the function:
- The total cost function [tex]\( f(x) \)[/tex] will include the initial charge for the first two hours and then the additional hourly fee for hours beyond the first two.
- Thus, for [tex]\( x \geq 2 \)[/tex], the cost function can be written as:
[tex]\[
f(x) = 220 + (x - 2) \times 60
\][/tex]
5. Simplifying the function:
- Distribute and combine like terms:
[tex]\[
f(x) = 220 + 60x - 120
\][/tex]
[tex]\[
f(x) = 60x + 100
\][/tex]
6. Finding the correct choice:
- The function we derived is [tex]\( f(x) = 60x + 100 \)[/tex].
- We compare this with the given options.
[tex]\[
(A) \; f(x) = 60x + 100
\][/tex]
[tex]\[
(B) \; f(x) = 60x + 220
\][/tex]
[tex]\[
(C) \; f(x) = 80x
\][/tex]
[tex]\[
(D) \; f(x) = 80x + 220
\][/tex]
The correct choice is [tex]\( \boxed{A} \)[/tex].
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