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1 month ago

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Rick surveyed 50 students at his school to see how many have after-school jobs and how many are getting the new video game conso

le that is coming out next month. From his data, 30% of the students are getting the new console. Also, of the 20 students who have after-school jobs, 45% are getting the new console. Using this information, complete the following table with the relative frequency of the columns. 0.30 0.70 1.00 0.20 0.80 0.45 0.55 Mathematics

1 answer:

tester [12.3K]1 month ago

7 0

I believe it was option d when I completed the test; I hope I’m right, sorry if I’m mistaken.

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Find c1 and c2 such that M2+c1M+c2I2=0, where I2 is the identity 2×2 matrix and 0 is the zero matrix of appropriate dimension.

Inessa [12570]

The question lacks details. Below is the complete version provided.

Let M = $\left[\begin{array}{cc}6&5\\-1&-4\end{array}\right]$ . Find $c_{1}$ and $c_{2}$ such that $M^{2}+c_{1}M+c_{2}I_{2}=0$ , where $I_{2}$ is the identity 2x2 matrix and 0 is the zero matrix of the appropriate dimension.

Answer: $c_{1} = \frac{-16}{10}$

$c_{2}=\frac{-214}{10}$

Step-by-step explanation: The identity matrix is a square matrix with 1's along its main diagonal and 0's elsewhere. For example, a 2x2 identity matrix is:

$\left[\begin{array}{cc}1&0\\0&1\end{array}\right]$

$M^{2} = \left[\begin{array}{cc}6&5\\-1&-4\end{array}\right]\left[\begin{array}{cc}6&5\\-1&-4\end{array}\right]$

$M^{2}=\left[\begin{array}{cc}31&10\\-2&15\end{array}\right]$

When solving the equation:

$\left[\begin{array}{cc}31&10\\-2&15\end{array}\right]+c_{1}\left[\begin{array}{cc}6&5\\-1&-4\end{array}\right] +c_{2}\left[\begin{array}{cc}1&0\\0&1\end{array}\right] =\left[\begin{array}{cc}0&0\\0&0\end{array}\right]$

The multiplication of a matrix by a scalar results in each term being scaled by that scalar. Matrices of different sizes cannot be combined.

So, we structure the equation as follows:

$\left[\begin{array}{cc}31&10\\-2&15\end{array}\right]+\left[\begin{array}{cc}6c_{1}&5c_{1}\\-1c_{1}&-4c_{1}\end{array}\right] +\left[\begin{array}{cc}c_{2}&0\\0&c_{2}\end{array}\right] =\left[\begin{array}{cc}0&0\\0&0\end{array}\right]$

And the system of equations can be written as:

$6c_{1}+c_{2} = -31\\-4c_{1}+c_{2} = -15$

Various methods are available to solve this system. One way is to multiply the second equation by -1 and then add the equations together:

$6c_{1}+c_{2} = -31\\(-1)*-4c_{1}+c_{2} = -15*(-1)$

$6c_{1}+c_{2} = -31\\4c_{1}-c_{2} = 15$

$10c_{1} = -16$

$c_{1} = \frac{-16}{10}$ Following this, substitute variables back into one of the equations to find $c_{2}$ :

$6c_{1}+c_{2}=-31$

$c_{2}=-31-6(\frac{-16}{10} )$

$c_{2}=-31+(\frac{96}{10} )$

$c_{2}=\frac{-310+96}{10}$

$c_{2}=\frac{-214}{10}$

For the equation, $c_{1} = \frac{-16}{10}$ and $c_{2}=\frac{-214}{10}$

6 0

1 month ago

Given A is between Y and Z and YA=22, AZ=16x, and YZ=166, find AZ

AnnZ [12381]

YA + AZ = YZ

22 + 16x = 166

16x = 166 - 22

16x = 144

x = 144/16

x = 9

AZ = 144

8 0

1 month ago

Read 2 more answers

Nadia wants to enclose a square garden with fencing. It has an area of 141 square feet. To the nearest foot, how much fencing w

Zina [12379]

Nadia requires 47 feet of fencing.

8 0

2 months ago

The Big River Casino is advertising a new digital lottery-style game called Instant Lotto. The player can win the following mone

Inessa [12570]

Answer:

(a) For a single play of Instant Lotto, the anticipated value of the prize amounts to $3.50.

(b) The likelihood that a visitor secures a prize at least two times during the 20 free plays is 0.2641.

(c) The chance of a randomly chosen day having 1000 or more players of Instant Lotto is 0.2579.

Step-by-step explanation:

(a)

The probability distribution for the monetary awards in Instant Lotto is defined as follows:

X P (X = x)

$10 0.05

$15 0.04

$30 0.03

$50 0.01

$1000 0.001

$0 0.869

___________

Total = 1.000

The expected value for one Instant Lotto play can be computed as follows:

$E(X)=\sum x\cdot P (X=x)$

$=(10\times 0.05)+(15\times 0.04)+(30\times 0.03) \\+ (50\times 0.01)+(1000\times 0.001)+(0\times 0.869)\\=0.5+0.6+0.9+0.5+1+0\\=3.5$

Consequently, the expected prize value for a single Instant Lotto play is $3.50.

(b)

Define X as the count of prizes a visitor wins.

A visitor receives n = 20 complimentary plays of Instant Lotto.

The probability of winning in any of the 20 games is p = 1/20 = 0.05.

The outcomes in the 20 plays are independent of one another.

The variable X adheres to a Binomial distribution characterized by parameters n = 20 and p = 0.05.

To find the probability that the visitor wins a prize at least twice in these 20 plays, perform the following calculation:

P (X ≥ 2) = 1 - P (X < 2)

= 1 - P (X = 0) - P (X = 1)

$=1-[{20\choose 0}0.05^{0}(1-0.05)^{20-0}]-[{20\choose 1}0.05^{1}(1-0.05)^{20-1}]\\=1-0.3585-0.3774\\=0.2641$

The resulting probability that the visitor wins a prize at least twice during the 20 plays is 0.2641.

(c)

Let X denote the number of individuals playing Instant Lotto each day.

The variable X is assumed to be normally distributed with a mean value of μ = 800 players and a standard deviation of σ = 310 players.

To ascertain the probability that on a random day, at least 1000 people participate in Instant Lotto, consider the following:

Implementing continuity correction:

P (X ≥ 1000) = P (X > 1000 + 0.50)

= P (X > 1000.50)

$=P(\frac{X-\mu}{\sigma}>\frac{1000.50-800}{310})$

$=P(Z>0.65)\\=1-P(Z$

The probability of having at least 1000 players on any random day is 0.2579.

5 0

3 months ago