A standard deck of 52 cards contains four suits: clubs, spades, hearts, and diamonds. Each deck contains an equal number of cards in each suit. Rochelle chooses a card from the deck, records the suit, and replaces the card. Her results are shown in the table.



How does the experimental probability of choosing a heart compare with the theoretical probability of choosing a heart?

A.The theoretical probability of choosing a heart is 1/16 greater than the experimental probability of choosing a heart.

B.The experimental probability of choosing a heart is 1/16 greater than the theoretical probability of choosing a heart.

C.The theoretical probability of choosing a heart is 1/26 greater than the experimental probability of choosing a heart.

D.The experimental probability of choosing a heart is 1/26 greater than the theoretical probability of choosing a heart.

Answer:

3

Step-by-step explanation:

given: contribution margin=$450,000 and net profit= $150,000

To find operating leverage

Now, operating leverage is a cost accounting formula that measures the degree to which a firm or project can increase operating income by increasing revenue. A business that generates sales with a high gross margin and low variables costs has high leverage.

[tex]operating leverage=\frac{contribution margin}{net profit}[/tex]

therefore, operating leverage=[tex]\frac{450000}{150000}[/tex] =3

operating leverage is 3

Answer: (C) 0.9104

Step-by-step explanation:

Given : The hours studied follows the normal distribution

Mean : [tex]\mu=\text{20 hours}[/tex]

Standard deviation : [tex]\sigma=6\text{ hours}[/tex]

Sample size : [tex]n=144[/tex]

The formula to calculate the z-score is given by :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

Let x be the number hours taken by randomly selected undergraduated student.

Then for x = 19.25 , we have

[tex]z=\dfrac{19.25-20}{\dfrac{6}{\sqrt{144}}}=-1.5[/tex]

for x = 21.0  , we have

[tex]z=\dfrac{20-21}{\dfrac{6}{\sqrt{144}}}=2[/tex]

The p-value : [tex]P(19.25<x<21)=P(-1.5<z<2)[/tex]

[tex]P(2)-P(-1.5)= 0.9772498- 0.0668072=0.9104426\approx0.9104[/tex]

Thus, the probability that the mean of this sample is between 19.25 hours and 21.0 hours = 0.9104.

Answer:

[tex]f (-2) =-\frac{8}{3}[/tex]

[tex]f (4) =\frac{4}{3}[/tex]

[tex]f (1) = - 4[/tex]

Step-by-step explanation:

For this case it has a piecewise function composed of two functions.

To evaluate the piecewise function observe the condition.

[tex]f (x) = \frac{1}{3}x ^ 2 -4[/tex] when [tex]x \neq 1[/tex]

[tex]f (x) = -4[/tex] when [tex]x = 1[/tex]

We start by evaluating [tex]f(-2)[/tex], note that [tex]x = -2\neq 1[/tex]. Then we use the quadratic function:

[tex]f (-2) = \frac{1}{3}(-2) ^ 2 -4 = -\frac{8}{3}[/tex]

Now we evaluate [tex]f(4)[/tex] note that [tex]x = 4\neq 1[/tex]. Then we use the quadratic function:

[tex]f (4) = \frac{1}{3}(4) ^ 2 -4 = \frac{4}{3}[/tex]

Finally we evaluate [tex]f(1)[/tex] As [tex]x = 1[/tex]  then

[tex]f (1) = - 4[/tex]

Answer:The mean number of checks written per​ day = 0.3370

The standard deviation = 0.5805

The variance = 0.3370

Step-by-step explanation:

Let the random variable x represent the number of checks he wrote in one​ day.

Given : The number of checks written in last year = 123

Let the number of days in the year must be 365.

Now, the mean number of checks written per​ day will be  :-

[tex]\lambda=\dfrac{123}{365}=0.33698630137\approx0.3370[/tex]

We know that in Poisson distribution , the variance is equals to the mean value .

[tex]\text{Thus , Variance }=\sigma^2= 0.3370[/tex]

[tex]\Rightarrow\ \sigma=\sqrt{0.3370}=0.580517010948\approx0.5805[/tex]

Thus,  Standard deviation = 0.5805

Answer and Step-by-step explanation:

Since we have given that

Number of children in his state were eligible for Medicaid = 100,000

Number of children in his state were not eligible for Medicaid = 200,000

After a large expansion in Medicaid, we get that

Number of children in his state were eligible for Medicaid = 150,000

Number of children in his state were not eligible for Medicaid = 150,000

According to question, , “the number of children without access to health care fell by one-quarter".

So, we check whether it is correct or not.

Difference between previous and current data who were not eligible is given by

[tex]200000-150000\\\\=50000[/tex]

Percentage of decrement is given by

[tex]\dfrac{50000}{200000}=\dfrac{1}{4}[/tex]

Yes , it is fell by one quarter.

Answer:

The given function is

  f(x)=cos 3x-7 x²+ 4x

f'(x)=-3 sin 3 x-14 x+4

When you will draw the graph of the function , you will find that root of the function lie between (-1,0).

Consider initial root as,

 [tex]x_{0}=0[/tex]

Using Newton method to find the roots of the equation

 [tex]x_{n+1}=x_{n}-\frac{f{x_n}}{f'{x_{n}}}\\\\x_{1}=x_{0} - \frac{cos 3x_{0}-7 x_{0}^2+ 4x_{0}}{-3 sin 3 x_{0}-14 x_{0}+4}\\\\x_{1}=-\frac{\cos 0^{\circ}-0+0}{-3 \times 0-0+4}\\\\x_{1}=\frac{-1}{4}\\\\x_{1}= -0.25\\\\x_{2}=x_{1} - \frac{cos 3x_{1}-7 x_{1}^2+ 4x_{1}}{-3 sin 3 x_{1}-14 x_{1}+4}\\\\x_{2}=-0.25 -\frac{cos (-0.75)-7\times (0.0625)- 1}{-3 sin (-0.75)+3.50+4}\\\\x_{2}= -0.176054[/tex]

[tex]x_{3}=x_{2} - \frac{cos 3x_{2}-7 x_{2}^2+ 4x_{2}}{-3 sin 3 x_{2}-14 x_{2}+4}\\\\x_{3}=-0.176054 -\frac{cos (3\times -0.176054)-7\times (-0.176054)^2+4 \times -0.176054}{-3 sin (-0.176054)-14 \times (-0.176054)+4}\\\\x_{3}= -0.1689[/tex]

So, root of the equation is

         =0.1688878

       =0.1689(approx)

Answer:

M = ¼ k π R⁴

zG = 8/15 R

Step-by-step explanation:

Note: I'm using lower case r as the radius of each plate and upper case R as the radius of the hemisphere.

The mass of each plate is density times volume:

dm = ρ dV

Each plate has a radius r and a thickness dz.  So the volume of each plate is:

dV = π r² dz

Substituting:

dm = ρ π r² dz

We're told that ρ = kz.  Substituting:

dm = kz π r² dz

Next, we need to write the radius r in terms of the height z.  To do that, we need to look at the cross section (see image below).

The height z and the radius r form a right triangle, where the hypotenuse is the radius of the hemisphere R.

Using Pythagorean theorem:

z² + r² = R²

r² = R² − z²

Substituting:

dm = kπ z (R² − z²) dz

We now have the mass of each plate as a function of its height.  To find the total mass, we integrate between z=0 and z=R.

M = ∫ dm

M = ∫₀ᴿ  kπ z (R² − z²) dz

M = kπ ∫₀ᴿ (R² z − z³) dz

M = kπ (½ R² z² − ¼ z⁴) |₀ᴿ

M = kπ (½ R⁴ − ¼ R⁴)

M = ¼ k π R⁴

Next, to find the center of gravity, we use the weighted average:

zG = (∫ z dm) / (∫ dm)

zG = (∫ z dm) / M

We already found M, we just have to evaluate the other integral:

∫ z dm

∫₀ᴿ kπ z² (R² − z²) dz

kπ ∫₀ᴿ (R² z² − z⁴) dz

kπ (⅓ R² z³ − ⅕ z⁵) |₀ᴿ

kπ (⅓ R⁵ − ⅕ R⁵)

²/₁₅ k π R⁵

Plugging in:

zG = (²/₁₅ k π R⁵) / (¼ k π R⁴)

zG = ⁸/₁₅ R

Answer: [tex]\$\ 3600[/tex]

Step-by-step explanation:

Given : A bank charges a 1% fee to process a credit card cash advance which is taken out of the cash advance amount.

The rate of fee can be also written as 0.01.

The amount of money a customer wants as a cash advance = $4,000

The amount of bank fee = [tex]0.01\times4000=40[/tex]

The amount which customer receive after cash advance has been processed = [tex]4000-40=\$3600[/tex]

Hence, the amount of money will he receive from the bank after the cash advance has been processed = [tex]\$\ 3600[/tex]

Answer:

TT means pi and e means Euler

Step-by-step explanation:

Answer: 0.001

Step-by-step explanation:

Given : Number of true–false questions  =3

Choices of answer for true–false questions =2

Then , probability that the answer is correct for a true–false question =[tex]\dfrac{1}{2}=0.5[/tex]

Also, Number of multiple choice questions  =3

Choices of answer for multiple choice questions = 5

Then , probability that the answer is correct for a multiple choice question =[tex]\dfrac{1}{5}=0.2[/tex]

Now, the probability that the student will correctly answer all six questions :-

[tex](0.2)^3\times(0.5)^3=0.001[/tex]

Hence, the probability that the student will correctly answer all six questions = 0.001

The probability that a student blindly guessing will correctly answer all six questions on a pop quiz consisting of three true/false questions and three multiple choice questions is 0.001, or 0.1%.

The probability of correctly guessing the answer to a true/false question is 0.5, because there are two options, true or false. Therefore, the probability of correctly guessing all three true/false answers is 0.5^3, which is 0.125.

For a multiple choice question with five answers, the probability of guessing correctly is 0.2 (or 1/5). Therefore, the probability of correctly answering all three multiple choice questions is 0.2^3, which is 0.008.

To find the total probability of correctly answering all six questions, we multiply these two probabilities together: 0.125 * 0.008 = 0.001. So, the probability that a student guessing blindly will correctly answer all six questions is 0.001, or 0.1% when expressed as a percentage.

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Answer:

given y=2.69+0.0138x the slope is the coefficient of x slope=0.0138   this means that the dive duration is expected to increase by about 0.0138 per minute

Step-by-step explanation:

Answer:

Hi!

A) The intercept of the regression line is 2.56.

B)The slope of the regression line is 0.0135.

C)For every increase of 1-meter depth, the mean dive duration increases by exactly 0.0135 minutes.

Explanation:

The definition of the regression line of a function is the value of Y(x) when x=0.

The slope of a function is the value that configures the ratio of change vertically on the regression.

In this case is represented by 0.0135 multiplied by the x.

If we are talking about depth, and the values of x are all positives values.

Y(50)=2.56+0.0135(50)=3.235

So, when the penguins go deeper, they remain more time diving.

Answer:

The quadratic function is:

[tex]f (x) = x ^ 2 + 11x +30[/tex]

Step-by-step explanation:

The zeros of a function are all values of x for which the function is equal to zero.

If a function has two zeros then it is a quadratic function.

If the zeros are -6 and -5. Then the function will have the following form:

[tex]f (x) = (x + 6) (x + 5)[/tex]

We can expand the expression by applying the distributive property, and we obtain

[tex]f (x) = x ^ 2 + 5x + 6x +30[/tex]

[tex]f (x) = x ^ 2 + 11x +30[/tex]

The probability of all three dice landing on 4 is 1/216, calculated by multiplying the individual probabilities of each die landing on 4.

The probability of all three dice landing on 4 can be calculated using the concept of independent events. Each die has a 1/6 probability of landing on 4, so the probability of all three landing on 4 is (1/6) x (1/6) x (1/6) = 1/216.

Answer:

[tex]\boxed{\text{(a)}A = 100 - 99e^{-t/25}; \,\text{(b) 15 s}}[/tex]

Step-by-step explanation:

(a) Expression for mass of salt as function of time

[tex]\text{Let A = mass of salt after t min}\\\text{and }r_{i} = \text{rate of salt coming into talk}\\\text{and }r_{o}$ =\text{rate of salt going out of tank}[/tex]

i. Set up an expression for the rate of change of salt concentration.

[tex]\dfrac{\text{d}A}{\text{d}t} = r_{i} - r_{o}\\\\r_{i} = \dfrac{\text{4 gal}}{\text{1 min}} \times \dfrac{\text{1 lb}}{\text{1 gal}} = \text{4 lb/min}\\\\r_{o} = \dfrac{\text{4 gal}}{\text{1 min}} \times \dfrac {A\text{ lb}}{\text{100 gal}} =\dfrac{A}{25}\text{ lb/min}\\\\\dfrac{\text{d}A}{\text{d}t} = 4 - \dfrac{x}{25}[/tex]

ii. Integrate the expression

[tex]\dfrac{\text{d}A}{\text{d}t} = \dfrac{100 - A}{25}\\\\\dfrac{\text{d}A}{100 - A} = \dfrac{\text{d}t}{25}\\\\\int \frac{\text{d}A}{100 - A} = \int \frac{\text{d}t}{25}\\\\-\ln(100 - A) = \dfrac{t}{25} + C[/tex]

iii. Find the constant of integration

[tex]-\ln (100 - A) = \dfrac{t}{25} + C\\\\\text{At $t$ = 0, $A$ = 1, so}\\\\-\ln (100 - 1) = \dfrac{0}{25} + C\\\\C = -\ln 99[/tex]

iv. Solve for A as a function of time.

[tex]\text{The integrated rate expression is}\\\\-\ln (100-A) = \dfrac{t}{25} - \ln 99\\\\\text{Solve for } A\\\\\ln(100 - A) = \ln 99 - \dfrac{t}{25}\\\\100 - A = 99e^{-t/25}\\\\A = \boxed{\mathbf{100 - 99e^{-t/25}}}[/tex]

The diagram shows A as a function of time. The mass of salt in the tank starts at 1 lb and increases asymptotically to 100 lb.

(b) Time to 2 lb salt

[tex]A = 100 - 99e^{-t/25}\\\\2 = 100 - 99e^{-t/25}\\\\99e^{-t/25} = 98\\\\e^{-t/25} = 0.989899\\\\-t/25 = -0.01015\\\\t = 25\times 0.01015 =\text{0.25 min = 15 s}\\\\\text{The tank will contain 2 lb of salt after } \boxed{\textbf{15 s}}[/tex]

To find the amount of salt in the tank at any time t, we need to consider the rate at which the brine solution is being poured in and leaving the tank. Initially, there is 1 pound of salt in the tank. The rate at which salt is added is 1 pound per gallon, and the rate at which the mixture is being removed is 4 gallons per minute. Therefore, the amount of salt in the tank at time t can be calculated using the equation. To find the time at which the mixture in the tank contains 2 pounds of salt, we can set up the equation. Solving this equation for t will give us the time at which the mixture contains 2 pounds of salt.

To find the amount of salt in the tank at any time t, we need to consider the rate at which the brine solution is being poured in and leaving the tank. Initially, there is 1 pound of salt in the tank. The rate at which salt is added is 1 pound per gallon, and the rate at which the mixture is being removed is 4 gallons per minute. Therefore, the amount of salt in the tank at time t can be calculated using the equation:

Amount of salt at time t = 1 + (1 pound/gallon)(4 gallons/minute)(t minutes)

To find the time at which the mixture in the tank contains 2 pounds of salt, we can set up the equation:

2 = 1 + (1 pound/gallon)(4 gallons/minute)(t minutes)

Solving this equation for t will give us the time at which the mixture contains 2 pounds of salt.

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Answer with explanation:

The given differential equation is

x²y" -7 x y' +1 6 y=0---------(1)

  Let, y'=z

y"=z'

[tex]\frac{dy}{dx}=z\\\\y=zx[/tex]

Substitution the value of y, y' and y" in equation (1)

→x²z' -7 x z+16 zx=0

→x² z' + 9 zx=0

→x (x z'+9 z)=0

→x=0 ∧ x z'+9 z=0

[tex]x \frac{dz}{dx}+9 z=0\\\\\frac{dz}{z}=-9 \frac{dx}{x}\\\\ \text{Integrating both sides}\\\\ \log z=-9 \log x+\log K\\\\ \log z+\log x^9=\log K\\\\\log zx^9=\log K\\\\K=zx^9\\\\K=y'x^9\\\\K x^{-9}d x=dy\\\\\text{Integrating both sides}\\\\y=\frac{-K}{8x^8}+m[/tex]

is another independent solution.where m and K are constant of integration.

Answer:

[tex]y_2=x^4lnx[/tex]

Step-by-step explanation:

We are given that a differential equation

[tex]x^2y''-7xy'+16y=0[/tex]

And one solution is [tex]y_1=x^4[/tex]

We  have to find the other independent solution by using reduction order method

[tex]y''-\frac{7}{x}y'+\frac{16}{x^2}y=0[/tex]

Compare with the equation

[tex]y''+P(x)y'+Q(x)y=0[/tex]

Then we get P(x)=[tex]-\frac{7}{x}['/tex] Q(x)=[tex]\frac{16}{x^2}[/tex]

[tex]y_2=y_1\int\frac{e^{-\intP(x)dx}}{y^2_1}dx[/tex]

[tex]y_2=x^4\int\frac{e^{\frac{7}{x}}dx}}{x^8}dx[/tex]

[tex]y_2=x^4\int\frac{e^{7ln x}}{x^8}dx[/tex]

[tex]y_2=x^4\int\frac{x^7}{x^8}dx[/tex]

[tex]e^{xlny}=y^x[/tex]

[tex]y_2=x^4\int frac{1}{x}dx[/tex]

[tex]y_2=x^4lnx[/tex]

Answer:

Epistemology can be defined as the branch of philosophy which focuses on the theory of knowledge. It is the study of the nature of knowledge, and  analyses its relation concepts like truth, justifications and belief. It deals with the rationality of belief(logic or rationale behind a belief).

Yes, it is important to critical thinking as critical thinking is based on beliefs and actions that have reasons, logic and rationale behind them. But these logic and reasons are philosophically problematic as how one evaluate 'reason, 'logic', etc. These are abstract and difficult but question like these are a central part to epistemology.

Answer:

[tex]L\left(te^{2t }sin3t\right)=\frac{6s-12}{(s^2-4s+13)^2}[/tex].

Step-by-step explanation:

If F(s)= L{f(t)}

Then [tex]L\left\{(t^nf(t)\right\}=(-1)^n\frac{\mathrm{d^n}F(s)}{\mathrm{d^n}s}[/tex]

[tex]L\left\{te^{2t}sin3t\right\}[/tex]

f(t)=[tex]e^{2t}sin3t[/tex]

[tex]L\left\{e^{at}sinbt\right\}=\frac{b}{(s-a)^2+b^2}[/tex]

Therefore,[tex] L\left\{e^{2t}sin3t\right\}=\frac{3}{(s-2)^2+(3)^2}[/tex]

[tex]L\left\{e^{2t}sin3t\right\}=\frac{3}{s^2-4s+13}[/tex]

[tex]L\left\{te^{2t}sin3t\right}=-\frac{\mathrm{d}F(s)}{\mathrm{d}s}[/tex]

=-[tex]\frac{\mathrm{d}e^{2t}sin3t}{\mathrm{d}s}[/tex]

[tex]L\left\{te^{2t}sin3t\right\}[/tex]

[tex]=\frac{3(2s-4)}{(s^2-4s+13)^2}[/tex]

[tex]L\left\{te^{2t}sin3t\right\}=\frac{6s-12}{(s^2-4s+13)^2}[/tex].

The Laplace transform of te2tsin(3t) is 3(-1)(1)/(s2+9)2.

According to Theorem 7.4.1, the Laplace transform of a function tnf(t) can be calculated as (−1)n * dn/dsn * F(s), where F(s) is the Laplace transform of f(t). In this case, we need to evaluate the Laplace transform of te2tsin(3t).

First, we need to find the Laplace transform of te2t. Applying the formula F(s) = ℒ{f(t)} = ℒ{te2t}:

F(s) = -d/ds[2/(s-2)2]

Next, we need to find the Laplace transform of sin(3t). Applying the formula F(s) = ℒ{f(t)} = ℒ{sin(3t)}:

F(s) = 3/(s2+9)

Finally, we can evaluate the Laplace transform of te2tsin(3t) as the product of the Laplace transforms obtained:

F(s) = (-1)*(d1/d1s)*(-d1/d1s)[3/(s2+9)]

F(s) = 3(-1)(1)/(s2+9)2

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Answer:

The probability that the committee will consist of 1 from each type of monster is 0.1318 and Suppose that the Pokemon refuse to be on the same committee as a the Trolls.So,the probability that this type of committee is formed is 0.5357        

Step-by-step explanation:

No. of Trolls = 3

No. of  Nazguls =4

No. of Ents = 4

No. of Pokemon = 5

Total Monsters = 16

We are given that A committee of 4 is to be picked to represent the group at the Monster Bash.

(a) Find the probability that the committee will consist of 1 from each type of monster.

So, the probability that the committee will consist of 1 from each type of monster:

= [tex]\frac{^3C_1 \times ^4C_1 \times ^4C_1 \times ^5C_1}{^{16}C_4}[/tex]

= [tex]\frac{\frac{3!}{1!(3-1)!} \times \frac{4!}{1!(4-1)!} \times \frac{4!}{1!(4-1)!}\times \frac{5!}{1!(5-1)!}}{\frac{16!}{4!(16-4)!}}[/tex]

= [tex]\frac{\frac{3!}{1!(2)!} \times \frac{4!}{1!(3)!} \times \frac{4!}{1!(3)!}\times \frac{5!}{1!(4)!}}{\frac{16!}{4!(12)!}}[/tex]

= [tex]0.1318[/tex]

b)Suppose that the Pokemon refuse to be on the same committee as a the Trolls.

So,  the probability that this type of committee is formed

= [tex]\frac{(^3C_3 \times ^8C_1)+(^3C_2 \times ^8C_2)+(^3C_1 \times ^8C_3)+(^3C_0 \times ^8C_4)+(^5C_4 \times ^8C_0)+(^5C_3 \times ^8C_1)+(^5C_2 \times ^8C_2)+(^5C_1 \times ^8C_3)}{^{16}C_4}[/tex]

= [tex]\frac{(\frac{3!}{3!(3-3)!} \times \frac{8!}{1!(8-1)!})+(\frac{3!}{2!(3-2)!} \times\frac{8!}{2!(8-2)!})+(\frac{3!}{1!(3-1)!} \times\frac{8!}{3!(8-3)!})+(\frac{3!}{0!(3-0)!} \times \frac{8!}{4!(8-4)!})+(\frac{5!}{4!(5-4)!} \times \frac{8!}{0!(8-0)!})+(\frac{5!}{3!(5-3)!} \times \frac{8!}{1!(8-1)!})+(\frac{5!}{2!(5-2)!} \times\frac{8!}{2!(8-2)!})+(\frac{5!}{1!(5-1)!} \times \frac{8!}{3!(8-3)!})}{\frac{16!}{4!(16-4)!}}[/tex]

= [tex]0.5357[/tex]

Hence The probability that the committee will consist of 1 from each type of monster is 0.1318 and Suppose that the Pokemon refuse to be on the same committee as a the Trolls.So,the probability that this type of committee is formed is 0.5357          

The probability that the committee will consist of 1 member from each type of monster is 1. If the Pokemon refuse to be on the same committee as a Troll, the probability that this type of committee is formed is 4/5 or 0.8.

(a) To find the probability that the committee will consist of 1 member from each type of monster, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. There are 3 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 5 choices for a Pokemon. So the total number of possible outcomes is 3 * 4 * 4 * 5 = 240. Now, we need to calculate the number of favorable outcomes. Since we want 1 member from each type of monster, we choose 1 Troll from 3, 1 Nazgul from 4, 1 Ent from 4, and 1 Pokemon from 5. So the number of favorable outcomes is 3 * 4 * 4 * 5 = 240. Therefore, the probability is 240/240, which simplifies to 1.

(b) If the Pokemon refuse to be on the same committee as a Troll, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. To do this, we need to consider two cases: one where a Troll is chosen and one where a Troll is not chosen. Case 1: If a Troll is chosen, then there are 3 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 4 choices for a Pokemon (since one Pokemon is excluded). So the total number of outcomes in this case is 3 * 4 * 4 * 4 = 192. Case 2: If a Troll is not chosen, then there are 0 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 5 choices for a Pokemon. So the total number of outcomes in this case is 0 * 4 * 4 * 5 = 0. Therefore, the total number of favorable outcomes is 192 + 0 = 192. The total number of possible outcomes is still 240. So the probability is 192/240, which simplifies to 4/5 or 0.8.

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Answer: a) i-1; b) 4+i/13

Step-by-step explanation:

The complex number [tex]i[/tex] is defined as the number such that [tex]i^{2}=-1[/tex];

We use the propperty to notice that [tex]i^1=i \quad i^2 = -1 \quad i^3=-i \quad i^4=1 \quad i^5=1 \quad i^6=-1 \quad i^7=-i \quad i^8=1 \quad i^9=i \quad i^10= -1 etc...[/tex].

a) We notice that [tex]i^{21}=i \quad \text{ and \text} \quad i^{32}=1[/tex]. Hence, [tex]i^{21}-i^{32}=i-1[/tex].

b) We multiply the expression by [tex]1=\frac{3+2\cdot i}{3 + 2 \cdot i}[/tex]. Then we get that

[tex]\frac{2-i}{3-2 \cdot i}=\frac{2-i}{3-2\cdot i}\cdot\frac{3+2 \cdot i}{3 + 2\cdot i } = \frac{(2-i)\cdot(3+2 \cdot i)}{3^2+2^2}= \frac{6+4i-3i+2i^2}{13}=\frac{6+i-2}{13} = \frac{4+i}{13}[/tex]

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:

[tex]y=A+Bx^{2}[/tex]

Step-by-step explanation:

The given Cauchy-Euler equation is: [tex]x^2y''-xy'=0[/tex]

Comparing to the general form: [tex]ax^2y''+bxy'+cy=0[/tex], we have a=1,b=-1 and c=0

The auxiliary solution is given by: [tex]am(m-1)+bm+c=0[/tex]

[tex]\implies m(m-1)-m=0[/tex]

[tex]\implies m(m-1-1)=0[/tex]

[tex]\implies m(m-2)=0[/tex]

[tex]\implies m=0\:\:or\:\:m=2[/tex]

The general solution to this is of the form [tex]y=Ax^{m_1}+Bx^{m_2}[/tex], where A and B are constants.

[tex]y=Ax^{0}+Bx^{2}[/tex]

Therefore the general solution is;

[tex]y=A+Bx^{2}[/tex]

Let [tex]y_1=A[/tex] and [tex]y_2=Bx^2[/tex]

Since we CANNOT express the two solutions as constant multiple of each other, we say the two solutions are linearly independent.

[tex]y_1\neCy_2[/tex], where C is a constant.

The solutions to the differential equation x^2y" - xy' = 0 are y = 1 and y = x, which are linearly independent. The general solution is y = C1 + C2x, where C1 and C2 are constants.

For the Cauchy-Euler equation x^2y" - xy' = 0, we find solutions by substituting y = x^r. Differentiating yields y' = rx^{r-1} and y" = r(r-1)x^{r-2}. Substituting these into the equation and simplifying gives us r(r-1)x^r - rx^r = 0, which simplifies to the characteristic equation r^2 - r = 0. Solving this equation, we get the roots r1 = 0 and r2 = 1. Therefore, the two solutions are y1 = x^0 = 1 and y2 = x^1 = x. To show that they are linearly independent, we evaluate the Wronskian determinant W(y1,y2) = |1  x| = x which is nonzero for x > 0. The general solution is a combination of the two: y = C1y1 + C2y2, where C1 and C2 are arbitrary constants.

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:

Step-by-step explanation:

Since we are talking about compounded annual interest, we can use the Exponential Growth Formula to calculate the answer for this question.

[tex]y = a* (1+r)^{t}[/tex]

Where:

First we need to calculate the total after 2 years with a 9% interest.

[tex]y = 1000* (1+0.09)^{2}[/tex]

[tex]y =  1000* (1.09)^{2}[/tex]

[tex]y = 1000* 1.1881[/tex]

[tex]y = 1188.1[/tex]

So after 2 years there will be £1,188.10 in the account. Now we can add £3000 to that and use the new value as the initial amount, and calculate the new total in 5 years.

[tex]y = (1188.1+3000)* (1+0.09)^{5}[/tex]

[tex]y = 4188.1* (1.09)^{5}[/tex]

[tex]y = 4188.1* 1.5386 [/tex]

[tex]y = 6443.91[/tex]

So now we can subtract the £4000 purchase from the amount currently in the account, and calculate one more year of interest with the new initial amount.

[tex]y = (6443.91-4000)* (1+0.09)^{1}[/tex]

[tex]y = (2443.91)* 1.09[/tex]

[tex]y = 2663.86[/tex]

So at the end you would have £2,662.86 in the account one year after the purchase.

To calculate the amount left in the account one year after the purchase, calculate the interest earned on the initial and subsequent deposits, add them to the account balance, and subtract the purchase amount.

To calculate the amount left in the account one year after the purchase, we need to consider the interest earned on the initial deposit and the subsequent deposit in two years. Let's break it down step by step:

Therefore, there is a deficit of P2010 in the account one year after the purchase.

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Given:

area of square, A = 16 [tex]cm^{2}[/tex]

error in area, dA = 0.03 cm^{2}

Step-by-Step Explanation:

Let 'a' be the side of the square

area of square, A = [tex]a^{2}[/tex]                 (1)

A = 16 =  [tex]a^{2}[/tex]

Therefore, a = 4 cm

for max tolerable error in length 'da', differentiate eqn (1) w.r.t 'a':

dA = 2a da

[tex]0.03 = 2\times 4\times da[/tex]

da = [tex]\frac{0.03}{8}[/tex]

da = 0.0375 cm

The side length of the square is 4 cm, the maximum error that can be tolerated in the length of each side to ensure that the area is within the specified tolerance is 0.0375 cm (or 0.0375 mm).

Given specifications:

Desired Area (A) = 16 cm²

Tolerance (ΔA) = 0.03 cm²

The formula for the area of a square is:

A = side length (L) * side length

Calculate the derivative of the area formula with respect to the side length (L):

dA/dL = 2L

Now, we want to find the maximum error in the side length (ΔL) that can be tolerated while keeping the area within the specified tolerance:

ΔA = (dA/dL) * ΔL

Plug in the values we have:

0.03 cm² = (2L) * ΔL

Solve for ΔL:

ΔL = 0.03 cm² / (2L)

To ensure that the area is within the tolerance, the error in the side length should be no more than ΔL.

Now, let's calculate ΔL using the formula above and for a given side length (L):

ΔL = 0.03 cm² / (2L)

If we assume a side length of L = 4 cm (to achieve the desired area of 16 cm²), we can calculate ΔL:

ΔL = 0.03 cm² / (2 * 4 cm) = 0.03 cm² / 8 cm = 0.00375 cm = 0.0375 mm

So, if the side length of the square is 4 cm, the maximum error that can be tolerated in the length of each side to ensure that the area is within the specified tolerance is 0.0375 cm (or 0.0375 mm).

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Answer:

the total number of fur seal pups in Rookery A (a) is  45524

Step-by-step explanation:

Given data

in early August (b) = 5498

late August, a sample (c) = 1300

previously tagged (d ) = 157

to find out

the total number of fur seal pups in Rookery A (a)

solution

we will apply here proportion method

that is

a:b :: c :d

a/b = c/d

put all value and find a

a = c/d × b

a = 1300/157 × 5498

a = 45524.84

the total number of fur seal pups in Rookery A (a) is  45524

Answer:

190° rotation = c

80° rotation = a

Step-by-step explanation:

b = 180° rotation

d = 360° rotation

Answer:

The correct option is c.

Step-by-step explanation:

If the direction of rotation is not mentioned, then it is considered as counterclockwise rotation.

It is given that the figure XYZ rotated 190° and a 80° rotation(composition), both about point y.

It means figure is rotated 80° counterclockwise about the point y after that the new figure is rotated 190° counterclockwise about the point y.

[tex]80^{\circ}+190^{\circ}=270^{\circ}[/tex]

It means the figure XYZ rotated 270° counterclockwise about the point y.

In figure (a), XYZ rotated 90° counterclockwise about the point y.

In figure (b), XYZ rotated 180° counterclockwise about the point y.

In figure (c), XYZ rotated 270° counterclockwise about the point y.

In figure (d), XYZ rotated 360° counterclockwise about the point y.

Therefore the correct option is c.

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.

Answer: The answer is 135 murders.

Step-by-step explanation: The report tells us that statistically 67.5% of murders are committed using a firearm. It follows therefore that in a sample of 200 randomly selected murders, one would expect that 67.5% of those would be by a firearm. [tex]\frac{67.5}{100}[/tex] * 200 = 135.

It would certainly be higher that the expected value based on previous data collected but it would not be unusual because one sample may have a higher than "normal" amount of murders by firearm. Statistics aren't going to be exact for every sample.

Using the binomial distribution, it is found that:

a) The expected value is of 135.

b) Unusual, as 153 is more than 2.5 standard deviations above the mean.

It is the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

In this problem, the parameters are p = 0.675 and n = 200.

Item a:

E(X) = np = 200 x 0.675 = 135.

135 would be expected to be committed with a​ firearm.

Item b:

The standard deviation is given by:

[tex]\sqrt{V(X)} = \sqrt{200(0.675)(0.325)} = 6.624[/tex]

Then:

[tex]E(X) + 2.5\sqrt{V(X)} = 135 + 2.5(6.624) = 151.56 < 153[/tex]

Since 153 is more than 2.5 standard deviations above the mean, it would be unusual observe 153 murders by firearm in a random sample of 200 ​murders.

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Answer:

10% of the school are blonde haired boys.

Step-by-step explanation:

Let x be the total number of students,

Number of students who have blonde hair = 40% of the total students

= 40% of x

= 0.4x

Also, the ratio of three times as many girls as boys have blond hair,

So, the ratio of blond hair boys and blond hair girls = 1 : 3

Let the number of boys who have blonde hair = y

And, the number of girls who have blonde hair = 3y

So, y + 3y = 0.4x

4y = 0.4x ⇒ y = 0.1x

Thus, the number of boys who have blonde hair = 0.1x

Hence, the percentage of the school are blonde haired boys

= [tex]\frac{0.1x}{x}\times 100=10\%[/tex]