Math Problem Statement

The​ Farm-4-Less tractor company produces a grain combine​ (GC) in addition to both a large​ (LT) and small size tractor​ (SM). Its production manager desires to produce to customer demand using a mixed model production line. The current sequence of​ production, which is repeated 24

times during a​ shift, is: SM ​-GC​-SM​-LT​-SM​-LTdashSMdashGCdashSMdashLT A new machine is produced every 2

minutes. The plant operates two​ eight-hour shifts. There is no downtime because the​ four-hours between each shift are dedicated to maintenance and restocking raw material. Based on this​ information, answer the following questions. Part 2 a. How long does it take the production cycle to be​ completed? The cycle sequence is completed in enter your response here

minutes. ​(Enter your response as a whole​ number.) Part 3 b. How many of each type of machine does​ Farm-4-Less produce in a​ shift? ​(Enter your responses as whole​ numbers.) Type

Quantity in a shift SM

enter your response here LT

enter your response here GC

enter your response here

Solution

Part 2: Production Cycle Duration
To find the length of the cycle, let’s analyze the given sequence:

The sequence is: SM-GC-SM-LT-SM-LT-SM-GC-SM-LT

Since each machine is produced every 2 minutes, we need to count the total number of machines in one complete cycle:

  1. The sequence consists of 6 SM, 3 LT, and 2 GC.

Adding them up, we have a total of 6+3+2=116 + 3 + 2 = 11 machines produced in one cycle.

The total time for one cycle is:
11 machines×2 minutes per machine=22 minutes11 \text{ machines} \times 2 \text{ minutes per machine} = 22 \text{ minutes}

Therefore, the production cycle is completed in 22 minutes.

Part 3: Quantity Produced in a Shift
The plant operates two 8-hour shifts. In one shift, the total time is: 8 hours×60 minutes per hour=480 minutes8 \text{ hours} \times 60 \text{ minutes per hour} = 480 \text{ minutes}

To find the number of cycles completed in one shift: Number of cycles per shift=480 minutes22 minutes per cycle=21.82\text{Number of cycles per shift} = \frac{480 \text{ minutes}}{22 \text{ minutes per cycle}} = 21.82

Since the number of cycles must be a whole number, we round down to 21 complete cycles in a shift. Now, let’s calculate the number of each machine type produced in one shift:

  1. SM in one cycle: 6
    SM in one shift: 21 cycles×6=12621 \text{ cycles} \times 6 = 126

  2. LT in one cycle: 3
    LT in one shift: 21 cycles×3=6321 \text{ cycles} \times 3 = 63

  3. GC in one cycle: 2
    GC in one shift: 21 cycles×2=4221 \text{ cycles} \times 2 = 42

Final Answer:

  • SM: 126
  • LT: 63
  • GC: 42

Would you like more details or have any questions about this calculation? Here are some follow-up questions to help you dive deeper:

  1. What impact would adding downtime between shifts have on total production?
  2. How would a change in production cycle (e.g., producing more LT or GC) affect the overall output?
  3. What if a machine took longer than 2 minutes to produce—how would this affect the cycle duration?
  4. How many total machines are produced in a full day (including both shifts)?
  5. What would happen to production counts if the shifts were extended to 10 hours?

Tip: When calculating production cycles, always verify the number of machines per cycle to avoid miscounts.

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Math Problem Analysis

Mathematical Concepts

Arithmetic
Multiplication
Division
Time and Rate Calculations

Formulas

Total production time = Number of machines per cycle × Time per machine
Production per shift = Shift time / Cycle time

Theorems

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Suitable Grade Level

Grades 10-12

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