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Chapter 2: Problem 37

New designs for a wastewater treatment tank have proposed three possible shapes, four possible sizes, three locations for input valves, and four locations for output valves. How many different product designs are possible?

Short Answer

Expert verified
There are 144 different product designs possible.

Step by step solution

01

Understand the components of the design

We need to determine the number of possible designs for a wastewater treatment tank based on different combinations of components: shapes, sizes, input valve locations, and output valve locations. The components available are 3 shapes, 4 sizes, 3 locations for input valves, and 4 locations for output valves.
02

Calculate the number of combinations for each component

Identify the total combinations for each design component using multiplication since each choice is independent of the others. For shapes, sizes, input valves, and output valves, the combinations can be calculated using the formula \[ \text{Total Combinations} = \text{Shapes} \times \text{Sizes} \times \text{Input Valve Locations} \times \text{Output Valve Locations} \]
03

Multiply the combinations

Using the formula, substitute the given values:\[ \text{Total Combinations} = 3 \times 4 \times 3 \times 4 \]Calculate the total number of designs by multiplying these numbers step by step.
04

Perform the multiplication

Calculate\[ 3 \times 4 = 12 \] Then calculate\[ 12 \times 3 = 36 \] Finally, calculate\[ 36 \times 4 = 144 \]Thus, the total number of different product designs possible is 144.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
In the realm of combinatorics, probability often plays a crucial role. However, in the exercise of calculating tank design possibilities, we're dealing more with the certainty of combinations rather than chance.
The probability of an event is determined by the ratio of favorable outcomes to total possible outcomes when randomness is involved.
  • If you imagine selecting tank designs randomly, then each design has an equal chance of being selected out of the total of 144 designs.
  • This is because each combination is equally possible and no design is favored over another.
Understanding probability helps to grasp that in certain scenarios, all outcomes are equally likely, which aligns well with the deterministic approach to combinatorial problems like this one.
Statistics
Statistics is all about collecting, analyzing, and interpreting data. When applying statistics to combinatorics, it's often about evaluating data patterns or trends.
In this exercise, if we collected data about preferences for different tank designs, statistics could help make sense of the most popular combinations.
  • For instance, if certain shapes or sizes are more preferred, statistical analysis could reveal these patterns through frequency distribution and mode calculations.
  • This data might guide future design decisions or improve understanding of client needs.
Although this problem focuses on counting possible combinations, combining it with statistical analysis allows us to apply these concepts in practical, data-driven decision-making.
Mathematical Calculations
Mathematical calculations are at the heart of this exercise.
In particular, we focus on counting the possible combinations using basic multiplication, a process grounded in the principle of the fundamental counting theorem.
  • This theorem states that if you can choose one event in 'm' ways and another independent event in 'n' ways, the total number of choices is \( m \times n \).
  • Here, we multiply the numbers of choices for each component (shape, size, input, and output locations): \( 3 \times 4 \times 3 \times 4 = 144 \).
This efficient strategy allows us to handle large datasets or complex systems, breaking them down into simpler, calculable parts.
Engineering Applications
The principles of combinatorics that we applied to find the number of possible tank designs are highly relevant in engineering applications.
Engineers frequently deal with problems requiring optimization of designs through combinatorial analyses, much like choosing the best configuration for a wastewater treatment tank.
  • Each combination of components can be considered a potential solution or design that meets specific functional requirements.
  • By calculating all possible combinations, engineers can ensure they explore all design possibilities and select the most efficient option.
  • Moreover, computational methods can leverage these calculations to perform simulations, ensuring each design's feasibility and cost-effectiveness before any physical prototyping.
Such comprehensive combinatorial approaches not only enhance design strategies but also drive innovation across various engineering fields by maximizing resources and creativity.

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Most popular questions from this chapter

Magnesium alkyls are used as homogenous catalysts in the production of linear low-density polyethylene (LLDPE), which requires a finer magnesium powder to sustain a reaction. Redox reaction experiments using four different amounts of magnesium powder are performed. Each result may or may not be further reduced in a second step using three different magnesium powder amounts. Each of these results may or may not be further reduced in a third step using three different amounts of magnesium powder. (a) How many experiments are possible? (b) If all outcomes are equally likely, what is the probability that the best result is obtained from an experiment that uses all three steps? (c) Does the result in the previous question change if five or six or seven different amounts are used in the first step? Explain.

A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Let \(A\) denote the event that the design color is red and Let \(B\) denote the event that the font size is not the smallest one. Are \(A\) and \(B\) independent events? Explain why or why not.

If \(A, B,\) and \(C\) are mutually exclusive events with \(P(A)=0.2, P(B)=0.3,\) and \(P(C)=0.4,\) determine the following probabilities: (a) \(P(A \cup B \cup C)\) (b) \(P(A \cap B \cap C)\) (c) \(P(A \cap B)\) (d) \(P[(A \cup B) \cap C]\) (e) \(P\left(A^{\prime} \cap B^{\prime} \cap C^{\prime}\right)\)

Counts of the Web pages provided by each of two computer servers in a selected hour of the day are recorded. Let \(A\) denote the event that at least 10 pages are provided by server 1 and let \(B\) denote the event that at least 20 pages are provided by server 2 (a) Describe the sample space for the numbers of pages for the two servers graphically in an \(x-y\) plot. Show each of the following events on the sample space graph: (b) \(A\) (c) \(B\) (d) \(A \cap B\) (e) \(A \cup B\)

Each of the possible five outcomes of a random experiment is equally likely. The sample space is \(\\{a, b, c, d, e\\} .\) Let \(A\) denote the event \(\\{a, b\\},\) and let \(B\) denote the event \(\\{c, d, e\\} .\) Determine the following (a) \(P(A)\) (b) \(P(B)\) (c) \(P\left(A^{\prime}\right)\) (d) \(P(A \cup B)\) (e) \(P(A \cap B)\)
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