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Difficulty: MediumAlgebraic Word Problems and Modeling

Two automated assembly robots, Robot P and Robot Q, produce identical components. Robot P operates at a constant rate of pp components per hour, and Robot Q operates at a constant rate of qq components per hour, where p>q>0p > q > 0. During a shift, Robot P worked for 44 hours and Robot Q worked for 66 hours to produce a combined total of 360360 components. If TT represents the total number of components produced when Robot P works for 77 hours and Robot Q works for 33 hours, which of the following statements must be true? Select all such statements.

  1. The rate of Robot P, pp, must be greater than 3636 components per hour.Answer
  2. The rate of Robot Q, qq, must be less than 3636 components per hour.Answer
  3. The total number of components TT must be greater than 360360 and less than 630630.Answer
  4. D
    The rate of Robot P, pp, must be less than 6060 components per hour.
  5. E
    The total number of components TT could be equal to 320320.

Answer

The correct statements are: the rate of Robot P, pp, must be greater than 3636 components per hour; the rate of Robot Q, qq, must be less than 3636 components per hour; and the total number of components TT must be greater than 360360 and less than 630630.
The system of equations 2p+3q=1802p + 3q = 180 combined with p>q>0p > q > 0 strictly bounds pp between 3636 and 9090, and qq between 00 and 3636. Substituting these boundary conditions into the total expression T=5p+180T = 5p + 180 yields the strict range 360<T<630360 < T < 630. Consequently, the three statements asserting p>36p > 36, q<36q < 36, and 360<T<630360 < T < 630 are all mathematically required.

Step-by-Step Solution

1
Set up the linear equation from the initial production shift and simplify.
4p+6q=360  ⟹  2p+3q=180  ⟹  q=60−23p4p + 6q = 360 \implies 2p + 3q = 180 \implies q = 60 - \frac{2}{3}p
This establishes the exact relationship between the production rates pp and qq.
2
Apply the given constraints p>q>0p > q > 0 to determine the domain bounds for pp and qq.
p>60−23p  ⟹  53p>60  ⟹  p>36p > 60 - \frac{2}{3}p \implies \frac{5}{3}p > 60 \implies p > 36. Also, q>0  ⟹  60−23p>0  ⟹  p<90q > 0 \implies 60 - \frac{2}{3}p > 0 \implies p < 90. Thus, 36<p<9036 < p < 90 and 0<q<360 < q < 36.
Determining extreme bounds for pp automatically constrains both individual rates.
3
Formulate TT in terms of pp and evaluate its numerical boundaries.
T=7p+3q=7p+(180−2p)=5p+180T = 7p + 3q = 7p + (180 - 2p) = 5p + 180. Substituting 36<p<9036 < p < 90 gives 360<T<630360 < T < 630.
Substituting 3q=180−2p3q = 180 - 2p simplifies TT into a single-variable linear modeling equation.

Key Concept

Linear word problem modeling, variable elimination, and system inequality constraint analysis
Estimated Time:1m 45s
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Topics

Algebraic Word Problems and Modeling

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