A plumber charges $40 per hour for labor with a minimum labor charge of $30. For example, if she
does her repairs in only 0.2 hours, then $40*0.2=$8, which is less than the minimum, so she would
charge $30 for labor. We will assume she can repair any problem in one day – a maximum of 8
hours work. In addition, she charges $1.50 per mile for every mile over 15 miles that she has to
travel to the work site. To answer the following questions, assume you live 27 miles away from the
plumber. (She also charges for parts, but we won’t consider that for now.)
a. Find a piecewise defined function to describe the possible charges for the plumber to repair a
leaky faucet at your home. (There will only be one input variable.)
b. What is the domain of this function?
c. What is the range of this function?
d. What is the total charge if she takes 2.5 hours to complete the repairs?
a. Y = Cost.
X = # of hrs.
Y = 40x + 1.5(27-15),
Y = 40x + 18.
b. Domain = All real values of X.
c. Range = All real values of Y.
d. Y = 40*2.5 + 18 = $118.=Tot. charges
a. To find a piecewise defined function to describe the possible charges for the plumber, we can break it down into different cases:
Case 1: If the plumber completes the repairs in less than 0.2 hours (minimum labor charge):
In this case, the total charge for labor would be $30. There would be no additional charge for mileage.
Case 2: If the plumber completes the repairs in 0.2 to 8 hours (normal labor charge):
In this case, the total charge for labor would be $40 per hour multiplied by the number of hours. Additionally, there would be an additional mileage charge of $1.50 per mile for every mile over 15 miles.
Case 3: If the plumber takes more than 8 hours to complete the repairs:
In this case, the total charge for labor would be $40 per hour multiplied by 8 hours (maximum labor charge). Additionally, there would be an additional mileage charge of $1.50 per mile for every mile over 15 miles.
Combining all the cases, the piecewise defined function to describe the possible charges for the plumber to repair a leaky faucet at your home would be:
f(x) = $30 if x < 0.2
f(x) = $40x + $1.50(x - 15) if 0.2 ≤ x ≤ 8
f(x) = $40(8) + $1.50(x - 15) if x > 8
b. The domain of this function is the set of all possible values for the input variable, which in this case represents the time (in hours) taken to complete the repairs. The domain would be: x ≥ 0
c. The range of this function is the set of all possible charges for the plumber, considering the labor and mileage charges. The range would vary depending on the specific time taken to complete the repairs, but it can be determined by evaluating the function for different values of x.
d. If she takes 2.5 hours to complete the repairs, we can plug that value into the piecewise defined function to find the total charge. Using the second case, where 0.2 ≤ x ≤ 8:
f(2.5) = $40(2.5) + $1.50(2.5 - 15)
= $100 + $1.50(-12.5)
= $100 - $18.75
= $81.25
Therefore, the total charge would be $81.25 if she takes 2.5 hours to complete the repairs.