Exercise 21 In Exercise Set 3.6 Can Be Solved Mentally ✓ Solved

Exercise 21 in Exercise Set 3.6 can be solved mentally

Exercise 21 in Exercise Set 3.6 can be solved mentally after careful reading of the problem. Explain how this can be done. When Sight-Rite's three polishing machines, A, B, and C, are all working, 5700 lenses can be polished in one week. When only A and B are working, 3400 lenses can be polished in one week. When only B and C are working, 4200 lenses can be polished in one week. How many lenses can be polished in a week by each machine alone?

Paper For Above Instructions

The problem presented involves three polishing machines, A, B, and C, each contributing to the total production of lenses in a specified time frame. The goal is to determine the individual output of each machine in terms of lenses polished per week. We can approach this problem logically and mathematically to deduce the answer without extensive computations. Analyzing the information provided allows us to systematically break down the contributions of each machine based on their collective output.

Understanding the Problem

From the problem, we know three key scenarios of machine operation:

  • All three machines (A, B, and C) working together can polish 5700 lenses per week.
  • Machines A and B together can polish 3400 lenses per week.
  • Machines B and C together can polish 4200 lenses per week.

We need to find out how many lenses each machine can polish when operating individually. Let's denote the weekly output of each machine as follows:

  • A = output of machine A (lenses per week)
  • B = output of machine B (lenses per week)
  • C = output of machine C (lenses per week)

Setting Up the Equations

We can set up the following system of equations based on the given scenarios:

  • (1) A + B + C = 5700
  • (2) A + B = 3400
  • (3) B + C = 4200

These equations represent the combined outputs of the machines when they are working together. To solve these equations systematically, we can derive the output of individual machines by manipulating the equations.

Solving the Equations

Starting with equations (2) and (3), we can express A and C in terms of B:

From equation (2):

A = 3400 - B

Now, substitute A into equation (1):

(3400 - B) + B + C = 5700

By simplifying this equation, we have:

3400 + C = 5700

C = 5700 - 3400 = 2300

Now that we know C, we can substitute this value back into equation (3) to find B:

B + 2300 = 4200

B = 4200 - 2300 = 1900

Finally, we can substitute the value of B back into the equation for A:

A = 3400 - 1900 = 1500

Summary of Results

Through systematic deduction, we find the individual outputs of the machines:

  • Machine A can polish 1500 lenses per week.
  • Machine B can polish 1900 lenses per week.
  • Machine C can polish 2300 lenses per week.

This method shows that by carefully reading and understanding the problem, we can simplify our calculations and reach a solution without needing complex mental arithmetic. Instead, the focus is on establishing relationships between the outputs of the machines through a series of logical steps.

Conclusion

In conclusion, Exercise 21 in Exercise Set 3.6 can be approached mentally by developing a clear understanding of the relationships between the three machines and establishing an effective method for solving linear equations. By following through the relationships derived from collective outputs, we can easily discern how many lenses each machine can polish individually. This structured approach not only aids in solving this particular problem but also enhances problem-solving skills in similar scenarios.

References

  • Larson, R., & Edwards, B. H. (2014). Calculus: An Introduction. Cengage Learning.
  • Blitzer, R. (2013). Algebra and Trigonometry. Pearson.
  • Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
  • Anton, H., & Rorres, C. (2010). Elementary Linear Algebra. Wiley.
  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
  • Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  • Weiss, N. A. (2014). Introductory Statistics. Pearson.
  • Riley, K. F., Hobson, M. P., & Bence, S. J. (2006). Mathematical Methods for Physics and Engineering. Cambridge University Press.
  • Harris, E. M. (2013). Applied Mathematics for Business, Economics, and the Social Sciences. McGraw-Hill.
  • Rosen, K. H. (2012). Discrete Mathematics and Its Applications. McGraw-Hill.

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