To determine the correct model for the cost of repair [tex]\( C \)[/tex] in terms of the number of hours of labor [tex]\( x \)[/tex]:
1. Identify fixed costs:
- The cost of materials is fixed at \[tex]$2700. This is a constant value and does not depend on the number of labor hours.
2. Identify variable costs:
- The labor cost is \$[/tex]50 per hour. This means for every hour of labor [tex]\( x \)[/tex], the cost will increase by \$50.
3. Formulate the equation:
- The total cost [tex]\( C \)[/tex] is the sum of the fixed material cost and the variable labor cost.
- The variable labor cost component can be expressed as [tex]\( 50x \)[/tex], where [tex]\( x \)[/tex] represents the number of hours.
- Adding the fixed material cost and the variable labor cost gives us the equation: [tex]\( C = 50x + 2700 \)[/tex].
4. Verify the options:
- (a) [tex]\( C = 2700x + 50 \)[/tex] is incorrect because the units are wrong: 2700 is the fixed cost and should not be multiplied by [tex]\( x \)[/tex]; 50 should be multiplied by [tex]\( x \)[/tex].
- (b) [tex]\( C = 2700x - 50 \)[/tex] is incorrect because 2700 should be added as a constant, not multiplied by [tex]\( x \)[/tex], and there should be no subtraction.
- (c) [tex]\( C = 50x + 2700 \)[/tex] matches our formulated equation and correctly represents the fixed and variable costs.
- (d) [tex]\( C = 50x - 2700 \)[/tex] is incorrect because we cannot subtract the material cost.
Therefore, the correct model that represents the cost of repairs is:
(c) [tex]\( C = 50x + 2700 \)[/tex]
The answer is [tex]\( \mathbf{3} \)[/tex].
