The average height of students at UH from an SRS of 12 students gave a standard deviation of 3.1 feet. Construct a 95% confidence interval for the standard deviation of the height of students at UH. Assume normality for the data.

Given that

[tex]\vec G(x,y,z)=xy\,\vec\imath+z\,\vec\jmath+3y\,\vec k[/tex]

has a fairly simple curl,

[tex]\nabla\times\vec G(x,y,z)=2\,\vec\imath-x\,\vec k[/tex]

we can take advantage of Stokes' theorem by transforming the line integral of [tex]\vec G[/tex] along the boundary of the square (call it [tex]S[/tex]) to the integral of [tex]\nabla\times\vec G[/tex] over [tex]S[/tex] itself. Parameterize [tex]S[/tex] by

[tex]\vec s(u,v)=u\,\vec\jmath+v\,\vec k[/tex]

with [tex]-\dfrac92\le u\le\dfrac92[/tex] and [tex]-\dfrac92\le v\le\dfrac92[/tex]. Then take the normal vector to [tex]S[/tex] to be

[tex]\vec s_u\times\vec s_v=\vec\imath[/tex]

so that

[tex]\displaystyle\int_{\partial S}\vec G\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec G)\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv[/tex]

[tex]=\displaystyle\int_{-9/2}^{9/2}\int_{-9/2}^{9/2}(2\,\vec\imath)\cdot(\vec\imath)\,\mathrm du\,\mathrm dv=\boxed{162}[/tex]

We have,the circulation of  [tex]G=xyi+zi+3yk[/tex]  around a square of side 3, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis is

[tex]\int_{\theta} G.dr=162[/tex]

From the Question we have the equation to be

[tex]G=xyi+zi+3yk[/tex]

Therefore

[tex]\triangle *G=\begin{Bmatrix}i & j & k\\\frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz}\\xy&z&3y\end{Bmatrix}[/tex]

Where

For i

[tex]i(\frac{d}{dy}*3y-(\frac{d}{dz}*xy))\\\\2i[/tex]

For j

[tex]j(\frac{d}{dx}*3y-(\frac{d}{dz}*xy))[/tex]

[tex]0j[/tex]

For z

[tex]z(\frac{d}{dy}*xy-(\frac{d}{dx}*z))[/tex]

[tex]xk[/tex]

Therefore

[tex]\triangle *G=2i-xk[/tex]

Generally considering [tex]\theta[/tex] as the origin

We apply Stoke's Theorem

[tex]\int_{\theta} G.dr=\int_{\theta}(\triangle *G) i ds[/tex]

[tex]\int_{\theta} G.dr=\int_{\theta}(=2i-xk) i ds[/tex]

[tex]\int_{\theta} G.dr=\int2ds[/tex]

[tex]\int_{\theta} G.dr=2*9^2[/tex]

[tex]\int_{\theta} G.dr=162[/tex]

Therefore

The circulation of [tex]G=xyi+zi+3yk[/tex] around a square of side 3, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis is

[tex]\int_{\theta} G.dr=162[/tex]

Answer:

Mean=18.32

Median=20.7

Standard deviation=7.10

Step-by-step explanation:

Mean is the average of the sample size. It is calculated by dividing the sum of all the observations by the number of observations.

[tex]mean=\frac{7.3+11.7+21.7+18.8+23.2+20.7+21.2+10.6}{9}[/tex]

[tex]\Rightarrow mean=18.32[/tex]

Median is the middle value of the observations if the number of observations is odd. If the number of the observations is even then it is the middle value and the next observations average. The values need to be arranged in ascending order.

Therefore the observations become

[tex]7.3, 10.6, 11.7, 18.8, 20.7, 21.2, 21.7, 23.2, 29.7[/tex]

In this case the number of observations is 9 which is odd

Therefore, the median is 20.7 i.e., the fifth observation

[tex]Standard\ deviation=\sqrt \frac{\sum_{i=1}^{N}\left ( x_{i}-\bar{x} \right )^2}{N-1}[/tex]

[tex]where,\ N=Number\ of\ observations\\\bar x=mean\\x_{i}=x_{1}+x_{2}+x_{3}+x_{4}.........x_{N}[/tex]

[tex]\left ({x_{i}}-\bar {x}\right )}^2[/tex]

[tex]\\121.49\\43.85\\11.41\\0.23\\23.79\\5.65\\129.45\\8.28\\59.63\\\sum_{i=1}^{N}\left ({x_{i}}-\bar {x}\right )}^2=403.80\\N-1=9-1\\=8\\\therefore Standard\ deviation=\sqrt {\frac{403.80}{8}}=7.10[/tex]

Standard deviation=7.10

Final answer:

This detailed answer addresses how to find the mean, median, and standard deviation for a given data set in a High School Mathematics context.

Explanation:

Mean: To find the mean, add up all the values and then divide by the total number of values. Add all the values provided: 7.3 + 11.7 + 21.7 + 18.8 + 23.2 + 20.7 + 29.7 + 21.2 + 10.6 = 164.9. Then, divide by 9 (total values) to get a mean of 164.9/9 = 18.32.

Median: To find the median, arrange the data in numerical order and find the middle value. The data in order is: 7.3, 10.6, 11.7, 18.8, 20.7, 21.2, 21.7, 23.2, 29.7. The median is the middle value, which is 20.7.

Standard Deviation: To calculate the standard deviation, you need to find the variance first. Then, take the square root of the variance. Using the given data above, the standard deviation is approximately 6.94.

If

[tex]y=\displaystyle\sum_{n=0}^\infty a_nx^n[/tex]

then

[tex]y'=\displaystyle\sum_{n=1}^\infty na_nx^{n-1}=\sum_{n=0}^\infty(n+1)a_{n+1}x^n[/tex]

The ODE in terms of these series is

[tex]\displaystyle\sum_{n=0}^\infty(n+1)a_{n+1}x^n+2\sum_{n=0}^\infty a_nx^n=0[/tex]

[tex]\displaystyle\sum_{n=0}^\infty\bigg(a_{n+1}+2a_n\bigg)x^n=0[/tex]

[tex]\implies\begin{cases}a_0=y(0)\\(n+1)a_{n+1}=-2a_n&\text{for }n\ge0\end{cases}[/tex]

We can solve the recurrence exactly by substitution:

[tex]a_{n+1}=-\dfrac2{n+1}a_n=\dfrac{2^2}{(n+1)n}a_{n-1}=-\dfrac{2^3}{(n+1)n(n-1)}a_{n-2}=\cdots=\dfrac{(-2)^{n+1}}{(n+1)!}a_0[/tex]

[tex]\implies a_n=\dfrac{(-2)^n}{n!}a_0[/tex]

So the ODE has solution

[tex]y(x)=\displaystyle a_0\sum_{n=0}^\infty\frac{(-2x)^n}{n!}[/tex]

which you may recognize as the power series of the exponential function. Then

[tex]\boxed{y(x)=a_0e^{-2x}}[/tex]

Solution of differential equation is, [tex]y=e^{-2x}+c[/tex]

Given differential equation is,

                   [tex]y'+2y=0\\\\\frac{dy}{dx}+2y=0[/tex]

Using separation of variable.

             [tex]\frac{dy}{y}=-2dx[/tex]

Integrating both side.

       [tex]ln(y)=--2x\\\\y=e^{-2x}+c[/tex]

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Answer:8

Step-by-step explanation:

We have given test consists of 40 multiple choice questions having five options for each question.

Suppose there is one correct answer to each question

therefore probabilty of getting a correct answer on making a guess is [tex]\frac{1}{5}[/tex]

i.e. 1 out of 5 questions is correct

Using binomial distribution

where n=40 p=[tex]\frac{1}{5}[/tex]

mean of binomial distribution is np

therefore mean of no of correct answers=[tex]40\times \frac{1}{5}[/tex]

                                                                    =8

Answer:

(-21)+(-20)+(-19)+...+50 is equal to 1044

Step-by-step explanation:

Let's divide the number series and find its solution.

The original number series is:

(-21)+(-20)+(-19)+...+50 which is the same as:

{(-1)*(21+20+19+18+....+0)} + (1+2+3+...+50) which is

-A+B where:

A=(21+20+19+18+....+0)=(0+1+2+3+...+21)

B=(1+2+3+...+50)

For this problem, we can use the Gauss method, which establishes that for a continuos series of numbers starting in 1, we can find the sum by:

S=n*(n+1)/2 where n is the last value of the series, so:

Using the method for A we have:

S=n*(n+1)/2

S(A)=(21)*(21+1)/2

S(A)=231

Using the method for B we have:

S=n*(n+1)/2

S(B)=(50)*(50+1)/2

S(B)=1275

So finally,

-A+B=-231+1275=1044

In conclusion, (-21)+(-20)+(-19)+...+50 is equal to 1044.

The sum of the sequence from -21 to 50 is calculated using the arithmetic series formula, resulting in a total sum of 1044.

The question asks for the sum of a sequence of integers starting from -21 and ending at 50. To find this sum, you can either add each number consecutively or use the formula for the sum of an arithmetic series.

In this case, the series is arithmetic because each term increases by 1 from the previous term. The formula for the sum of an arithmetic series is S = n/2 * (a_1 + a_n), where S is the sum of the series, n is the number of terms, a_1 is the first term, and a_n is the last term.

First, determine the number of terms in the series. Since our first term is -21 and our last term is 50, the series has 50 - (-21) + 1 = 72 terms in total. Now, using the formula, we can calculate the sum of the series:

S = 72/2 * (-21 + 50)

S = 36 * 29

S = 1044

Therefore, the sum of the integers from -21 to 50 is 1044.

The p-value is obtained by measuring how extreme the observed test statistic is compared to what is expected under the null hypothesis. In this case, a p-value less than the significance level (0.05) leads to the rejection of the null hypothesis, indicating that the actual proportion of readers owning the car is different from the reported 55%.

Given that the publisher claims that 55% of their readers own a particular make of car, this would be our null hypothesis (H0: p = 0.55), with the actual percentage of their readers owning a car being the alternate hypothesis (Ha: p ≠ 0.55). The sample found that 46% of 200 readers owned the car, so our sample proportion (p') is 0.46.

Using these values, we can calculate the test statistic (Z), which measures the number of standard deviations p' is from p under the null hypothesis. Once we have our test statistic, we can then determine the p-value.

The observed value of the test statistic falls into the critical region. The p-value is the probability that a test statistic will take on a value as extreme (or more extreme) than the observed value of the test statistic calculated from your sample data. If the p-value is less than or equal to the level of significance, we reject the null hypothesis. From the available reference, it is found that the p-value is less than the significance level of 0.05, and therefore, we would reject the null hypothesis and conclude that the proportion is different from 55% based on this sample data.

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Answer: There is a chance of 5% of adult regularly drinks coffee but doesn't drink soda.

Step-by-step explanation:

Since we have given that

Probability of all adults consume coffee P(C) = 50%

Probability of all adults consume carbonated soda P(S) = 65%

Probability of all adults consumes both coffee and soda P(C∩S) = 45%

We need to find the probability that adult regularly drinks coffee but doesn't drink soda.

So, it is talking about difference of sets in which we consider only one set completely i.e. it contains all element of one set but never contains any element of another set.

Here, P( Coffee - Soda) = P(C)-P(C∩S)

[tex]P(C-S)=0.50-0.45=0.05=0.05\times 100\%=5\%[/tex]

Hence, there is a chance of 5% of adult regularly drinks coffee but doesn't drink soda.

The probability that a randomly selected adult regularly drinks coffee but doesn't drink soda, given the provided data, is calculated to be 5%.

To find the probability that a randomly selected adult regularly drinks coffee but doesn't drink soda, we start with the information given: 50% of all adults regularly consume coffee, 65% regularly consume carbonated soda, and 45% regularly consume both coffee and soda. To find the probability of adults who consume coffee but not soda, we subtract the percentage of adults who consume both from the percentage of those who consume coffee. This is because those who consume both are also counted in the total number of coffee drinkers.

So, the calculation is as follows:

Therefore, the chance that a randomly selected adult regularly drinks coffee but doesn't drink soda is 5%.

Answer:  The correct option is (B) 1625.2 miles.

Step-by-step explanation:  Given that at a given speed a car can travel 95.6 miles on 4 gallons of gasoline.

We are to find the distance that the car can travel on 68 gallons at the same speed.

We will be using the UNITARY method to solve the given problem.

Distance traveled by the car on 4 gallons of gasoline = 95.6 miles.

So, the distance traveled by the car on 1 gallon of gasoline will be

[tex]\dfrac{95.6}{4}=23.9~\textup{miles}.[/tex]

Therefore, the distance traveled by the car in 68 gallons of gasoline is given by

[tex]23.9\times68=1625.2~\textup{miles}.[/tex]

Thus, the required distance traveled by the car on 68 gallons of gasoline is 1625.2 miles.

Option (B) is CORRECT.

52 cards  / 4 suits  = 13 cards of each suit.

Theoretically picking a heart would be 13/52 = 1/4 probability.

Experimentally she picked 15 hearts out of 80 total tries. for a 15/80 = 3/16 probability, which is less than the theoretical probability.

1/4 - 3/16 = 1/16

The answer is A.

Answer:

The right answer is A - The theoretical probability of choosing a heart is StartFraction 1 over 16 EndFraction greater than the experimental probability of choosing a heart.

Answer:

( )

Step-by-step explanation:

( ) shorthand label indicates an embedded design in mixed method research. It indicates that one form of data collection is embedded within another.Mostly ( ) is used used when data collection is embedded into larger data. So we can say that ( ) shorthand label indicates an embedded design in mixed research method

Answer:

The center is (3,1) and the radius is 3.

Step-by-step explanation:

The goal is to write in [tex](x-h)^2+(y-k)^2=r^2[/tex] because this tells us the center (h,k) and the radius r.

So we are going to need to complete the square 2 times here, once for x and the other time for y.

I'm going to use this formula to help me to complete the square:

[tex]x^2+bx+(\frac{b}{2})^2=(x+\frac{b}{2})^2[/tex].

So first step:

I'm going to group my x's together and my y's.

[tex]x^2-6x+\text{ ___ }+y^2-2y+\text{ ___ }+1=0[/tex]

Second step:

I'm going to go ahead and subtract that one on both sides. Those blanks are there because I'm going to fill them in with a number so that I can write the x part and y part as a square.  Remember whatever you add on one side you must add on the other.  So I'm going to put 2 more blanks to fill in on the opposite side of the equation.

[tex]x^2-6x+\text{ ___ }+y^2-2y+\text{ ___ }=-1+\text{ ___ }+\text{ ___}[/tex]

Third step:

Alright first blank I'm putting (-6/2)^2 due to my completing the square formula.  That means this value will also go on the other side in on of those blanks.

In the second blank I'm going to put (-2/2)^2 due to the completing the square formula.  This must also go on one of the blanks on the other side.

So we have:

[tex]x^2-6x+(\frac{-6}{2})^2+y^2-2y+(\frac{-2}{2})^2=-1+(\frac{-6}{2})^2+(\frac{-2}{2})^2[/tex]

Fourth step:

Don't make this more hurtful than it already is. Just use the formula drag down the things inside the square. Remember this:

[tex]x^2+bx+(\frac{b}{2})^2[/tex]

equals

[tex](x+\frac{b}{2})^2[/tex].

We are applying that left hand side there (that bottom thing I just wrote).

Let's try it:

[tex](x+\frac{-6}{2})^2+(y+\frac{-2}{2})^2=-1+(\frac{-6}{2})^2+(\frac{-2}{2})^2[/tex]

Fifth step:

The hard part is out of the way.

This is just a bunch of simplifying now:

[tex](x-3)^2+(y-1)^2=-1+9+1[/tex]

[tex](x-3)^2+(y-1)^2=9[/tex]

The center is (3,1) and the radius is 3.

Answer:

cos3x

Step-by-step explanation:

y" - y' + 9y = 3 sin 3x

[tex]D^{2}y-Dy+9y=3 sin3x[/tex]

[tex]y=\frac{3 sin 3x}{(D^{2} -D+9}=3 sin 3x[/tex]

here [tex]D^2[/tex] will be replaced by  [tex]\alpha^2[/tex] where [tex]\alpha[/tex] is coefficient of x

[tex]y=\frac{3 sin 3x}{-3^{2} -D+9}[/tex]

[tex]y=-3\frac{sin 3x}{D}[/tex]

[tex]y=-3\int\ {sin 3x} \, dx[/tex]

[tex]y=-3\frac{cos3x}{-3}[/tex]

y=cos3x

hence Particular solution is cos3x

Final answer:

To construct a 99% confidence interval for the proportion of people who accept the theory, you need to calculate the point estimate, standard error, margin of error, lower bound, and upper bound.

Explanation:

The psychologist wants to construct a 99% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. Let's calculate the confidence interval:

Calculate the point estimate: 64 out of 708 people agreed with the theory, so the estimated proportion is 64/708 = 0.0904.

Calculate the standard error: SE = sqrt((0.0904 * (1 - 0.0904)) / 708) = 0.0091.

Calculate the margin of error: ME = 2.57 * 0.0091 ≈ 0.0234.

Calculate the lower bound: Lower bound = 0.0904 - 0.0234 ≈ 0.067.

Calculate the upper bound: Upper bound = 0.0904 + 0.0234 ≈ 0.113.

So, with 99% confidence, the proportion of all people who accept the theory is between 0.067 and 0.113.

Answer:

220

Step-by-step explanation:

Given,

The total number of people = 12,

Out of which, top 3 finishers will be recognized with the same award,

So, the total possible way = total combination of 3 people out of 12 people

[tex]=^{12}C_3[/tex]

[tex]=\frac{12!}{3!(12-3)!}[/tex]

[tex]=\frac{12\times 11\times 10\times 9!}{3\times 2\times 9!}[/tex]

[tex]=\frac{1320}{6}[/tex]

[tex]=220[/tex]

Hence, there are 220 ways the top 3 racers can be grouped from the 12 people.

Answer:

d)9

Step-by-step explanation:

first , you have to take 1125and divide it by 5 , then the answer you should get is 225 ,second you need to take 225and divide it by the 25 children , then you get 9 , when you divide 225÷25

1125/25= 45

45/ 5= 9

Answer is 9 days - d)

Answer:

Option 1) p ←→ q

Step-by-step explanation:

The given statement is a combination of two phrases with a connective word.

Phrase 1 - The Fratellis will be arrested.

We consider this phrase as phrase p.

Phrase 2 - If the police believe chunk.

Phrase q.

Now these statements p and q are connected with a connecting word "If only and only".

When two phrases are connected by the connecting words "Only and only if", this connecting word is called as biconditional operator.

p (connecting word) q

p ( If and only if) q

p ←→ q

Therefore, Option 1) p ←→ q is the answer.

Answer:

Approximately 59 stacked cups.

Step-by-step explanation:

Given,

Height of a cup = 12.5 cm,

Two stacked cups = 14 cm,

Three stacked cups = 15.5 cm,

........, so on,....

Thus, there is an AP that represents the given situation,

12.5, 14, 15.5,....

First term is, a = 12.5,

Common difference, d = 1.5 cm,

Thus, the height of x cups is,

[tex]h(x) = a+(x-1)d = 12.5 + (x-1)1.5 = 1.5x + 11[/tex]

According to the question,

h(x) = 200

⇒ 1.5x + 11 = 200

⇒ 1.5x = 189

⇒ x = 59.3333333333 ≈ 59,

Hence, approximately 59 stacked cups will need.

Answer:

hx = 1.5cm . x + 11 cm

126 cups

Step-by-step explanation:

We have the following ordered pairs (x, hx).

We are looking for a linear equation of the form:

hx = a.x + b

where,

a is the slope

b is the y-intercept

To find the slope, we take any pair of ordered values and replace their values in the following expression.

[tex]a=\frac{\Delta hx }{\Delta x} =\frac{h2-h1}{2-1} =\frac{14cm-12.5cm}{2-1} =1.5cm[/tex]

Now, the general form is:

hx = 1.5cm . x + b

We can take any ordered pair and replace it in this expression to find b. Let's use h1.

h1 = 1.5cm . x1 + b

12.5 cm = 1.5 cm . 1 + b

b = 11 cm

The final equation is:

hx = 1.5cm . x + 11 cm

If hx = 200 cm,

200 cm = 1.5cm . x + 11 cm

189 cm = 1.5cm . x

x = 126

Answer:

   WXYZ = Ro(180°, (2, -3))(ABCD)

Step-by-step explanation:

A reflection across two perpendicular lines (y=-3, x=2) is equivalent to reflection across their point of intersection. That, in turn, is equivalent to rotation 180° about that point of intersection.

Your double reflection is equivalent to rotation 180° about (2, -3).

A secant is a line that intersects the curve of a circle at two points.

The line in the picture that passes through two points would be Line EF.

Answer:

EF is the secant line

Step-by-step explanation:

A tangent line of a circle is a line that touches the circle at any point on the circumference of the circle.

A secant line is a line that lies inside the circles . secant line crosses the circumference of the circle at two points. The secant line has not end points.

In the given diagram, secant line is EF because it has no end points and it intersects the circle at two points

EF is the secant line

Substitute [tex]v(x)=x-y(x)+1[/tex], so that

[tex]\dfrac{\mathrm dv}{\mathrm dx}=1-\dfrac{\mathrm dy}{\mathrm dx}[/tex]

Then the resulting ODE in [tex]v(x)[/tex] is separable, with

[tex]1-\dfrac{\mathrm dv}{\mathrm dx}=v^2\implies\dfrac{\mathrm dv}{1-v^2}=\mathrm dx[/tex]

On the left, we can split into partial fractions:

[tex]\dfrac12\left(\dfrac1{1-v}+\dfrac1{1+v}\right)\mathrm dv=\mathrm dx[/tex]

Integrating both sides gives

[tex]\dfrac{\ln|1-v|+\ln|1+v|}2=x+C[/tex]

[tex]\dfrac12\ln|1-v^2|=x+C[/tex]

[tex]1-v^2=e^{2x+C}[/tex]

[tex]v=\pm\sqrt{1-Ce^{2x}}[/tex]

Now solve for [tex]y(x)[/tex]:

[tex]x-y+1=\pm\sqrt{1-Ce^{2x}}[/tex]

[tex]\boxed{y=x+1\pm\sqrt{1-Ce^{2x}}}[/tex]

Answer:

[tex]a+b-c[/tex]

*Note c could be written as a/b

Step-by-step explanation:

-sin(-t - 8 π) + cos(-t - 2 π) + tan(-t - 5 π)

The identities I'm about to apply:

[tex]\sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)[/tex]

[tex]\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)[/tex]

[tex]\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}[/tex]

Let's apply the difference identities to all three terms:

[tex]-[\sin(-t)\cos(8\pi)+\cos(-t)\sin(8\pi)]+[\cos(-t)\cos(2\pi)+\sin(-t)\sin(2\pi)]+\frac{\tan(-t)-\tan(5\pi)}{1+\tan(-t)\tan(5\pi)}[/tex]

We are about to use that cos(even*pi) is 1 and sin(even*pi) is 0 so tan(odd*pi)=0:

[tex]-[\sin(-t)(1)+\cos(-t)(0)]+[\cos(-t)(1)+\sin(-t)(0)]+\frac{\tan(-t)-0}{1+\tan(-t)(0)[/tex]

Cleaning up the algebra:

[tex]-[\sin(-t)]+[\cos(-t)]+\frac{\tan(-t)}{1}[/tex]

Cleaning up more algebra:

[tex]-\sin(-t)+\cos(-t)+\tan(-t)[/tex]

Applying that sine and tangent is odd while cosine is even.  That is,

sin(-x)=-sin(x) and tan(-x)=-tan(x) while cos(-x)=cos(x):

[tex]\sin(t)+\cos(t)-\tan(t)[/tex]

Making the substitution the problem wanted us to:

[tex]a+b-c[/tex]

Just for fun you could have wrote c as a/b too since tangent=sine/cosine.

I'm guessing you were originally told to find the inverse of

[tex]A=\begin{bmatrix}6&7\\8&9\end{bmatrix}[/tex]

and you've found the inverse to be

[tex]A^{-1}=\begin{bmatrix}-\frac92&\frac72\\4&-3\end{bmatrix}[/tex]

I'm also guessing that "product of elementary matrices" includes the decomposition of [tex]A^{-1}[/tex] into lower and upper triangular as well as diagonal matrices.

First thing I would do would be eliminate the fractions by multiplying the first row of [tex]A^{-1}[/tex] by 2. In matrix form, this is done by multiplying [tex]A^{-1}[/tex] by

[tex]\begin{bmatrix}2&0\\0&1\end{bmatrix}[/tex]

which you can interpret as "multiply the first row by 2 and leave the second row alone":

[tex]\begin{bmatrix}2&0\\0&1\end{bmatrix}\begin{bmatrix}-\frac92&\frac72\\4&-3\end{bmatrix}=\begin{bmatrix}-9&7\\4&-3\end{bmatrix}[/tex]

Next, we make the matrix on the right side upper-triangular by eliminating the entry in row 2, column 1. This is done via the product

[tex]\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\begin{bmatrix}-\frac92&\frac72\\4&-3\end{bmatrix}=\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

which you can interpret as "leave the first row alone, and replace row 2 by 4(row 1) + 9(row 2)".

Lastly, multiply both sides by the inverses of all matrices as needed to isolate [tex]A^{-1}[/tex] on the left side. That is,

[tex]\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)A^{-1}=\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

[tex]\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)^{-1}\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)A^{-1}=\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)^{-1}\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

[tex]A^{-1}=\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)^{-1}\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

For two invertible matrices [tex]X[/tex] and [tex]Y[/tex], we have [tex](XY)^{-1}=Y^{-1}X^{-1}[/tex], so that

[tex]A^{-1}=\begin{bmatrix}2&0\\0&1\end{bmatrix}^{-1}\begin{bmatrix}1&0\\4&9\end{bmatrix}^{-1}\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

Compute the remaining inverses:

[tex]\begin{bmatrix}2&0\\0&1\end{bmatrix}^{-1}=\begin{bmatrix}\frac12&0\\0&1\end{bmatrix}[/tex]

[tex]\begin{bmatrix}1&0\\4&9\end{bmatrix}^{-1}=\begin{bmatrix}1&0\\-\frac49&\frac19\end{bmatrix}[/tex]

So we have

[tex]\begin{bmatrix}-\frac92&\frac72\\4&-3\end{bmatrix}=\begin{bmatrix}\frac12&0\\0&1\end{bmatrix}\begin{bmatrix}1&0\\-\frac49&\frac19\end{bmatrix}\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

Answer:

The value of [tex]P(\frac{F}{E})[/tex] is [tex]\frac{1}{2}[/tex]

Step-by-step explanation:

Given,

[tex]P(E\cap F)=0.1[/tex]

[tex]P(E)=0.2[/tex]

Thus, by the conditional probability formula,

[tex]P(\frac{F}{E})=\frac{P(E\cap F)}{P(E)}[/tex]

By substituting the values,

[tex]P(\frac{F}{E})=\frac{0.1}{0.2}[/tex]

[tex]=\frac{1}{2}[/tex]

The probability of getting an order from Restaurant A is [tex]0.321[/tex]

The probability of getting an order accurate in all the Restaurants is [tex]0.873[/tex]

The Probability of getting an order accurate in the Restaurant A is [tex]0.909[/tex]

Selecting an order from Restaurant A and selecting an accurate order are disjoint​ events

Total number of orders [tex]=321+276+235+126+32+56+40+11=1097[/tex]

The probability of getting an order from Restaurant A (the accurate +the not accurate) is   [tex]\dfrac{321+32}{1097}=0.321[/tex]  

Total Order Accurate  [tex]=321+276+235+126=958[/tex]

Probability of getting an order accurate is  [tex]\dfrac{958}{1097}=0.873[/tex]

Probability of getting an order accurate in the Restaurant A is   [tex]\dfrac{321}{353}=0.909[/tex]

Learn more about probability.

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To find the probability of getting an order from Restaurant A or an accurate order, sum the values of Restaurant A and the accurate orders and divide it by the total number of orders. The events of selecting an order from Restaurant A and selecting an accurate order are not disjoint.

To find the probability of getting an order from Restaurant A or an accurate order, we need to sum the values of Restaurant A and the accurate orders and divide it by the total number of orders.

Probability of getting an order from Restaurant A = (321+32)/(321+276+235+126+32+56+40+11)

Probability of getting an accurate order = (321+276+235+126)/(321+276+235+126+32+56+40+11)

Since both events can occur simultaneously (an order can be from Restaurant A and accurate), they are not disjoint. The final probability is the sum minus the probability of their intersection.

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Answer: The total amount played on the slot machines is $100416.67.

Step-by-step explanation:

Since we have given that

Percentage of amount payback on slot machines = 95.2%

Percentage of amount retained by the casino = 100 - 95.2 =4.8%

Amount retained by the casino = $4820

Let the total amount played on the slot machines be 'x'.

According to question, we get that

[tex]\dfrac{4.8}{100}\times x=\$4820\\\\0.048x=\$4820\\\\x=\dfrac{4820}{0.048}\\\\x=\$100416.67[/tex]

Hence, the total amount played on the slot machines is $100416.67.

Final answer:

To find the total amount played on slot machines with a casino retention of $4820 and a payback percentage of 95.2%, you solve the equation 0.048 * T = $4820 to get T = $100,416.67.

Explanation:

The student asked how to find the total amount played on slot machines if the casino retains $4820, given a payback percentage of 95.2%. To solve this problem, we use the concept of averages and percentages. The amount retained by the casino represents 100% minus the payback percentage, which is 4.8% in this case.

Let's denote the total amount played on the slot machines as T. The equation to find T is:

4.8% of T = $4820

We convert the percentage to a decimal and solve for T:

0.048 * T = $4820

T = $4820 / 0.048

T = $100,416.67

Therefore, the total amount played on the slot machines is $100,416.67

Answer:

16.2

Step-by-step explanation:

Since, if the sum of two angles is 90° then they are called complementary angles.

Here, the given angle is 73.8°,

Let x be the complementary angle of 73.8°,

Thus by the above definition,

x + 73.8° = 90°,

By subtraction property of equality,

⇒ x = 16.2°,

Hence, the complementary angle of 73.8° is 16.2°.

Answer:

(a)  $1.48

(b)  $0.0051 . . . . slightly more than 1/2¢

Step-by-step explanation:

  cost = (kW)(hours)(cost/kWh)

(a) cost = (0.040 kW)(24 h/da)(14 da/wk)($0.11/kWh) ≈ $1.478

The cost of leaving the porch light on is about $1.48.

__

(b) cost = (0.925 kW)(3 min)/(60 min/h)($0.11/kWh) ≈ $0.00509

The cost of toasting bread is about 0.5 cents.

The cost of leaving a 40-W porch light on for two weeks at an average electrical cost is $1.4784. The cost of making a toast using a 925-W toaster for 3 minutes at the same price is $0.0050875.

(a) We start by converting the wattage of the porch light, which is 40 Watts, or 40 Joules per second, into kilowatts. So, 40 W = 0.04 kW. Now, to find how much energy is used over two weeks (14 days), we need to multiply the wattage in kilowatts by the number of hours in two weeks.

There are 24 hours in a day and 60 minutes in an hour, so there are 14 days * 24 hours/day = 336 hours in two weeks.

Energy used = Power * Time = 0.04 kW * 336 hours = 13.44 kWh.

The cost is then Energy * Cost per unit of energy = 13.44 kWh * $0.110/kWh = $1.4784.

(b) In the case of the toaster, we repeat the same procedure, but using its wattage and the number of minutes it’s used to make toast. Convert the power to kW: 925 W = 0.925 kW. Convert the time to hours: 3 minutes = 0.05 hours.

Energy used = Power * Time = 0.925 kW * 0.05 hours = 0.04625 kWh.

Cost = Energy * Cost per unit of energy = 0.04625 kWh * $0.110/kWh = $0.0050875, or just over half a cent to make a piece of toast.

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Answer:

[tex]y=-\frac{158}{579}[/tex]

Step-by-step explanation:

To find the matrix A, took all the numeric coefficient of the variables, the first column is for x, the second column for y, the third column for z and the last column for w:

[tex]A=\left[\begin{array}{cccc}1&1&2&2\\-7&-3&5&-8\\4&1&1&1\\3&7&-1&1\end{array}\right][/tex]

And the vector B is formed with the solution of each equation of the system:[tex]b=\left[\begin{array}{c}3\\-3\\6\\1\end{array}\right][/tex]

To apply the Cramer's rule, take the matrix A and replace the column assigned to the variable that you need to solve with the vector b, in this case, that would be the second column. This new matrix is going to be called [tex]A_{2}[/tex].

[tex]A_{2}=\left[\begin{array}{cccc}1&3&2&2\\-7&-3&5&-8\\4&6&1&1\\3&1&-1&1\end{array}\right][/tex]

The value of y using Cramer's rule is:

[tex]y=\frac{det(A_{2}) }{det(A)}[/tex]

Find the value of the determinant of each matrix, and divide:

[tex]y==\frac{\left|\begin{array}{cccc}1&3&2&2\\-7&-3&5&-8\\4&6&1&1\\3&1&-1&1\end{array}\right|}{\left|\begin{array}{cccc}1&1&2&2\\-7&-3&5&-8\\4&1&1&1\\3&7&-1&1\end{array}\right|} =\frac{158}{-579}[/tex]

[tex]y=-\frac{158}{579}[/tex]

Answer:

m∠1=80°, m∠2=35°, m∠3=33°

Step-by-step explanation:

we know that

The sum of the interior angles of a triangle must be equal to 180 degrees

step 1

Find the measure of angle 1

In the triangle that contain the interior angle 1

∠1+69°+31°=180°

∠1+100°=180°

∠1=180°-100°=80°

step 2

Find the measure of angle 2

In the small triangle that contain the interior angle 2

∠2+45°+(180°-∠1)=180°

substitute the value of angle 1

∠2+45°+(180°-80°)=180°

∠2+45°+(100°)=180°

∠2+145°=180°

∠2=180°-145°=35°

step 3

Find the measure of angle 3

In the larger triangle that contain the interior angle 3

(∠3+31°)+69°+47°=180°

∠3+147°=180°

∠3=180°-147°=33°

Answer:

The growth rate is 1.02; the value of the house in 2010 is $185,388

Step-by-step explanation:

This is an exponential growth equation, therefore, it follow the standard form:

[tex]y=a(b)^x[/tex]

where y is the value of the house after a certain number of years,

a is the initial value of the house,

b is the growth rate, and

x is the year number.

We are going to make this easy on ourselves and call year 1985 year 0.  Therefore, is year 1985 is year 0, then year 2005 is year 20, and year 2010 is year 25.  We will make these the x coordinates in our coordinate pairs.

(0, 113000) and (20, 155000)

Filling into our standard form using the first coordinate pair will give us the initial value of the house at the start of our problem:

[tex]113000=a(b)^0[/tex]

Anything raised to the 0 power is equal to 1, so

113000 = a(1) and

a = 113000

Now we will use that value of a along with the second pair of coordinates and solve for b, the growth rate you're looking for:

[tex]155000=113000(b)^{20}[/tex]

Start by dividing both sides by 113000 to get a decimal:

[tex]1.371681416=b^{20}[/tex]

To solve for b, we have to undo that power of 20 by taking the 20th root of b.  Because this is an equation, we have to take the 20th root of both sides:

[tex]\sqrt[20]{1.371681416}=\sqrt[20]{b^{20}}[/tex]

The 20th root and the power of 20 undo each other so all we have left on the right is a b, and taking the 20th root on your calculator of the decimal on the left gives you:

b = 1.0159 which rounds to

b = 1.02  This is our growth rate.

Now we can use this growth rate and the value of a we found to write the model for our situation:

[tex]y=113000(1.02)^x[/tex]

If we want to find the value of the house in the year 2010 (year 25 to us), we sub in a 25 for x and do the math:

[tex]y=113000(1.02)^{25}[/tex]

Raise 1.02 to the 25th power and get:

y = 113000(1.640605994) and multiply to get a final value of

y = $185,388

The annual exponential growth rate from 1985 to 2005 is 1.40%. Using this rate, the estimated value of the house in 2010 would be approximately $176,927.

In order to calculate the exponential growth rate, we use the formula: R = (final value/initial value)^(1/n) - 1, where R is the annual rate, n is the number of years. So, R = (155,000/113,000)^(1/20) - 1 = 0.0140 or 1.40%.

To find the value of the house in 2010, we use the exponential growth formula: future value = present value * (1 + annual rate)^n. The value in 2010 would be $155,000 * (1 + 0.0140)^5 = $176,927 (rounded to the nearest dollar).

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