For the characteristic polynomialp(s) =s5+ 2s4+ 24s3+ 48s2−25s−50(a) Use the Routh-Hurwitz Criterion to determine the number of roots ofp(s) in the right-half plane, in the left-half plane, and on thejω-axis.(b) Use Matlab to determine the roots ofp(s), and verify your results in part 2a.

Answers

Answer 1

Answer:

Answer:

1 root in the right half-plane
1 conjugate pair on the imaginary axis
2 roots in the left half-plane

Step-by-step explanation:

Without using the Routh-Hurwitz criterion at all, you know there is one positive real root. Descartes' rule of signs tells you the number of positive real roots is equal to the number of sign changes in the coefficients (perhaps less a multiple of 2). There is one sign change in + + + + - - , so there is one positive real root.

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(a) The Routh array starts as two rows of the polynomial's coefficients, alternate coefficients on each row. For this odd-degree polynomial, the number of coefficients is even, so no zero-padding is necessary at the right end of the second row. That is, we start with ...

$\begin{array}{cccc}s^5&1&24&-25\ns^4&2&48&-50\end{array}$

The next row is formed from combinations of coefficients in the two rows above. The computation is similar to that of a determinant. By matching the numbers to those in the array, you can see the pattern of the computation.

The next row values are ...

$\begin{array}{ccc}s^3&((2)(24)-(1)(48))/(2)&((2)(-25)-(1)(-50))/(2)\end{array}$

Simplifying, we find this row to be ...

$\begin{array}{ccc}s^3&0&0\end{array}$

The zero row is a special case that requires we proceed as follows. The row above (identified with s⁴) represents an "auxiliary polynomial":

$2s^4 +48s^2 -50$

To continue the process, we replace the zero row by the coefficients of the derivative of this auxiliary polynomial. Proceeding as before, the array now becomes ...

$\begin{array}{cccc}s^5&1&24&-25\ns^4&2&48&-50\ns^3&8&96\ns^2&24&-50\ns^1&112(2)/(3)&0\ns^0&-50\end{array}$

The number of sign changes in the first column (1) tells the number of roots in the right half-plane. The auxiliary polynomial will give us the remaining two pairs of roots:

$2s^4+48s^2-50=0\n\n2(s^2+25)(s^2-1)=0\n\ns=\pm 5i,\ s=\pm 1$

So, we have determined there to be ...

1 root in the right half-plane
2 roots on the jω axis
2 roots in the left half-plane

(b) The original polynomial can be factored as ...

p(s) = (s +2)(s² +25)(s +1)(s -1)

p(s) = (s +2)(s +1)(s -5i)(s +5i)(s -1)

This verifies our result from part (a).

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Additional comments

Any row can be multiplied by a convenient factor to simplify the arithmetic. Here, it would be convenient to divide the second row by 2 and the third row by 8.

A zero element (not row) in the first column is replaced by "epsilon" (a small positive number) and the rest of the arithmetic is continued as normal. That row is not counted (it is ignored) when counting sign changes in the first column.

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consider a population of voters. suppose that that there are n=1000 voters in the population, 30% of whom favor jones. identify the event favors jones as a success s. it is evident that the probability of s on trial 1 is 0.30. consider the event b that s occurs on the second trial. then b can occur two ways: the first two trials are both successes or the first trial is a failure and the second is a success. show that p(b) = 0.3

Answers

Answer:

P(B)=0.30

Step-by-step explanation:

Out of 1000 Voters, 30% favor Jones.

Event S=Favors Jones on First Trial

Event B=S occurs on Second Trial

P(S)=0.30

P(S')=1-0.30=0.70

Event B could occur in two ways

The first two trials are a success
The first trial is a failure and the second trial is a success.

Therefore,

P(B)=P(SS)+P(S'S)

=(0.3X0.3)+(0.7X0.3)

=0.09+0.21

=0.3

Therefore, the probability of event B(that event S occurs on the second trial), P(B)=0.30.

A certain firm has plants a, b, and c producing respectively 35\%, 15\%, and 50\% of the total output. The probabilities of a nondefective product are, respectively, 0.75, 0.95, and 0.85. A customer receives a defective product. What is the probability that it came from plant c?

Answers

The proportion of production that is defective and from plant A is

... 0.35·0.25 = 0.0875

The proportion of production that is defective and from plant B is

... 0.15·0.05 = 0.0075

The proportion of production that is defective and from plant C is

... 0.50·0.15 = 0.075

Thus, the proportion of defective product that is from plant C is

... 0.075/(0.0875 +0.0075 +0.075) = 75/170 = 15/34 ≈ 44.12%

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P(C | defective) = P(C&defective)/P(defective)

Final answer:

The question required the use of Bayes' theorem to determine the probability of a defective product coming from plant c. Given the probabilities of defectiveness for each plant, the calculation indicated that there is approximately a 54.55% chance that a defective product came from plant c.

Explanation:

The problem described can be solved using Bayes' theorem, which is a principle in Probability that is used when we need to revise/or update the probabilities of events given new data. Since a defective product is received, and we need to determine the probability of it coming from plant c, we apply Bayes' theorem for the probability of events a, b, and c (representative of the products from the respective plants).

The Bayesian formula we will use, given the probabilities of a, b and c respectively and the probability of receiving a nondefective product from these plants, is: P(c|defective) = [P(defective|c) * P(c)] / [P(defective|a) * P(a) + P(defective|b) * P(b) + P(defective|c) * P(c)].

First, calculate the probability of a defective product from each plant (1 minus the probability of a nondefective product): these are 0.25 for plant a, 0.05 for plant b, and 0.15 for plant c.

Then substitute the values: P(c|defective) = [0.15 * 0.50] / [(0.25 * 0.35) + (0.05 * 0.15) + (0.15 * 0.5)] = 0.075 / 0.1375 = 0.5454545.

So, given a defective product, there is approximately a 54.55% chance that it was produced by plant c.

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The expression 3a + 2b + c / d gives the average number of points a team earns in d games by winning a games in regulation, b games in overtime and losing c games in overtime. What is the average number of points earned for a team that wins 10 games in regulation, wins 8 in overtime, loses 6 in overtime, and loses 4 in regulation?

Answers

Answer:

Step-by-step explanation:

10+8-6-4=8

A bottler of drinking water fills plastic bottles with a mean volume of 999 milliliters (ml) and standard deviation 7 ml. The fillvolumes are normally distributed. What is the probability that a bottle has a volume greater than 992 mL?
1.0000
0.8810
0.8413
0.9987

Answers

The required probability that a bottle has a volume greater than 992 mL is 0.84134. Option C is correct

Given that,
A bottler of drinking water fills plastic bottles with a mean volume of 999 milliliters (ml) and a standard deviation of 7 ml. The fill volumes are normally distributed. What is the probability that a bottle has a volume greater than 992 mL, is to be determined

What is probability?

Probability can be defined as the ratio of favorable outcomes to the total number of events.

We use Z-statistic to find out the probability,

z = (x − μ) / σ

x = raw score = 992 mL
μ = population mean = 999 mL
σ = standard deviation
z = [992 − 999]/7
z = -1

P-value from Z-Table:

P(x<992) = 0.15866

P(x>992) = 1 - P(x<992) = 0.84134

Thus, the required probability that a bottle has a volume greater than 992 mL is 0.84134

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Answer:

0.8413

Step-by-step explanation:

Find the z score.

z = (x − μ) / σ

z = (992 − 999) / 7

z = -1

Use a chart or calculator to find the probability.

P(Z > -1)

= 1 − P(Z < -1)

= 1 − 0.1587

= 0.8413

PLZ IM ON THE CLOCK!!!!! A sports memorabilia store makes $6 profit on each football it sells and $5.50 profit on each baseball it sells. In a typical month, it sells between 35 and 45 footballs and between 40 and 55 baseballs. The store can stock no more than 80 balls total during a single month. What is the maximum profit the store can make from selling footballs and baseballs in a typical month? $457.50 $460.00 $462.50 $572.50