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Difficulty: MediumAlgebraic Word Problems and Modeling

A IT consultant charges a flat setup fee of $150\$150 plus a standard hourly rate of $80\$80 for regular hours worked on a project. For any rush hours worked, the hourly rate increases by 50%50\%. On a recently completed project, the consultant worked a total of tt hours, of which rr hours were rush hours (where 0≤r≤t0 \leq r \leq t). Which of the following statements must be true regarding the total charge CC, in dollars, for this project? Select all such statements.

  1. The total charge, in dollars, can be expressed as C=150+80t+40rC = 150 + 80t + 40r.Answer
  2. If the consultant worked a total of 2020 hours and the total charge was $2,150\$2,150, then exactly 1010 of those hours were rush hours.Answer
  3. C
    The total charge, in dollars, can be expressed as C=150+120t−40rC = 150 + 120t - 40r.
  4. If no rush hours were worked (r=0r = 0), the average charge per hour, including the setup fee, is 150t+80\frac{150}{t} + 80 dollars per hour.Answer
  5. E
    If all hours worked were rush hours (r=tr = t), the total charge is given by C=150+160tC = 150 + 160t.

Answer

The correct statements are those expressing the total charge as C=150+80t+40rC = 150 + 80t + 40r, determining that 1010 rush hours were worked when total cost is $2,150\$2,150 for 2020 hours, and calculating the average hourly cost as 150t+80\frac{150}{t} + 80 dollars per hour when no rush hours are worked.
The model correctly partitions total hours tt into (t−r)(t - r) regular hours at $80\$80/hr and rr rush hours at $120\$120/hr, which simplifies algebraically to C=150+80t+40rC = 150 + 80t + 40r. Solving this equation for t=20t = 20 and C=2150C = 2150 yields r=10r = 10. Furthermore, when r=0r = 0, dividing total charge 150+80t150 + 80t by tt gives 150t+80\frac{150}{t} + 80.

Step-by-Step Solution

1
Calculate the hourly rush rate from the given percentage increase.
The standard rate is $80\$80/hr. The rush rate is $80×(1+0.50)=$120\$80 \times (1 + 0.50) = \$120/hr.
Rush hours cost 50%50\% more than standard regular hours.
2
Set up the algebraic model for total charge CC using standard hours (t−r)(t - r) and rush hours rr.
C=150+80(t−r)+120r=150+80t−80r+120r=150+80t+40rC = 150 + 80(t - r) + 120r = 150 + 80t - 80r + 120r = 150 + 80t + 40r.
This combines the fixed setup fee with variable costs from regular and rush hours.
3
Evaluate the specific case where total time t=20t = 20 and total cost C=2150C = 2150.
2150=150+80(20)+40r  ⟹  2150=1750+40r  ⟹  40r=400  ⟹  r=102150 = 150 + 80(20) + 40r \implies 2150 = 1750 + 40r \implies 40r = 400 \implies r = 10.
Plugging given values into the algebraic model allows solving for the unknown number of rush hours.
4
Calculate average hourly cost when r=0r = 0.
Average cost =150+80tt=150t+80= \frac{150 + 80t}{t} = \frac{150}{t} + 80.
Average hourly rate is total charge divided by total hours worked.

Key Concept

Linear algebraic modeling of rate problems with multiple rate components
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Topics

Algebraic Word Problems and Modeling

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