Question 5: What is the radius of the circle with general form x2+y2-4y=21?
Question 5 options:

√21

4.58

5

25

Answer:

[tex]x+\frac{-1}{3} \ln|x|+\frac{-8}{3}\ln|x+3|+C[/tex]

Step-by-step explanation:

To use partial fractions, I'm going to first do long division because the degree of the top is more than or equal to that of the bottom.

After I have that the degree of the bottom is more than the degree of the top, I will factor my bottom to figure out what kinds of partial fractions I'm going to have.

Let's begin with the long division:

The bottom goes outside.

The top goes inside.

                                      1

                                  -----------------

                    x^2+3x|    x^2          -1

                                  -(x^2+3x)

                              -----------------------------

                                          -3x       -1

We can not going any further since the divisor is more in degree than the left over part.

So we have so for that the integrand given equals:

[tex]\frac{x^2-1}{x^2+3x}=1+\frac{-3x-1}{x^2+3x}[/tex]

The 1 will of course not need partial fraction.

So we know our answer is x +  something   + C

Since the derivative of (x+c)=(1+0)=1.

Let's focus now on:

[tex]\frac{-3x-1}{x^2+3x}[/tex]

The bottom is not too bad too factor because it is binomial quadratic containing terms with a common factor of x:

[tex]x^2+3x=x(x+3)[/tex]

Since both factors our linear and there are two factors, then we will have two partial fractions where the numerators are both constants.

So we are looking to make this true:

[tex]\frac{-3x-1}{x^2+3x}=\frac{A}{x}+\frac{B}{x+3}[/tex]

Some people like to combine the fractions on the left and then regroup the terms and then compare coefficients.

Some people also prefer a method called heaviside method.

So I'm actually going to do this last way and I will explain it as I got.

We are going to clear the fractions by multiplying both sides by [tex]x(x+3)[/tex] giving me:

[tex]-3x-1=A(x+3)+Bx[/tex]

I know x+3 will be 0 when x=-3  so entering in -3 for x gives:

[tex]-3(-3)-1=A(-3+3)+B(-3)[/tex]

[tex]9-1=A(0)-3B[/tex]

[tex]8=-3B[/tex]

Divide both sides by -3:

[tex]\frac{8}{-3}=B[/tex]

[tex]B=\frac{-8}{3}[/tex]/

Now let's find A. If I replace x with 0 then Bx becomes 0 giving me:

[tex]-3(0)-1=A(0+3)+B(0)[/tex]

[tex]-1=A(3)+0[/tex]

[tex]-1=3A[/tex]

Divide both sides by 3:

[tex]\frac{1}{-3}=A[/tex]

[tex]\frac{-1}{3}=A[/tex]

Okay let me also who you other method of just comparing coefficients.

[tex]-3x-1=A(x+3)+Bx[/tex]

Distribute on the right:

[tex]-3x-1=Ax+3A+Bx[/tex]

Regroup terms on right so like terms are together:

[tex]-3x-1=(A+B)x+3A[/tex]

Now if this is to be true then we need:

-3=A+B        and         -1=3A

The second equation can be solved by dividing both sides by 3 giving us:

-3=A+B        and        -1/3=A

Now we are going to plug that second equation into the first:

-3=A+B  with A=-1/3

-3=(-1/3)+B

Add 1/3 on both sides:

-3+(-1/3)=B

-8/3=B

So either way you should get the same A and B if no mistake is made of course.

So this is the integral we are looking at now (I'm going to go ahead and include the 1 from earlier):

[tex]\int (1+\frac{\frac{-1}{3}}{x}+\frac{\frac{-8}{3}}{x+3})dx[/tex]

[tex]x+\frac{-1}{3} \ln|x|+\frac{-8}{3}\ln|x+3|+C[/tex]

Why I did choose natural log for both of those 2 terms' antiderivatives?

Because they are a constant over a linear expression. Luckily both of those linear expressions had a leading coefficient of 1.

Also recall the derivative of ln(x) is (x)'/x=1/x and

the derivative of ln(x+3) is (x+3)'/(x+3)=(1+0)/(x+3)=1/(x+3).

Let's check our answer:

To do that we need to differentiate what we have for the integral and see if we wind up with the integrand.

[tex]\frac{d}{dx}(x+\frac{-1}{3} \ln|x|+\frac{-8}{3}\ln|x+3|+C)[/tex]

[tex]1+\frac{-1}{3} \frac{1}{x}+\frac{-8}{3}\frac{1}{x+3}+0[/tex]

We are going to find a common denominator which would be the least common multiple of the denominators which is x(x+3):

[tex]\frac{x(x+3)+\frac{-1}{3}(x+3)+\frac{-8}{3}(x)}{x(x+3)}[/tex]

Now distribute property:

[tex]\frac{x^2+3x+\frac{-1}{3}x-1+\frac{-8}{3}x}{x^2+3x}[/tex]

Combine like terms:

[tex]\frac{x^2+3x+\frac{-9}{3}x-1}{x^2+3x}[/tex]

[tex]\frac{x^2+3x-3x-1}{x^2+3x}[/tex]

[tex]\frac{x^2-1}{x^2+3x}[/tex]

So that is the same integrand we started with so our answer has been confirmed.

Answer:

Option D. is the answer.

Step-by-step explanation:

The given statement is [tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]

Now we have to find a counter example from the given options that the statement is False.

A. a = 3, b = 5, c = -3, d = 5

[tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]=[tex]\frac{3}{5}+\frac{-3}{5}=\frac{3-3}{5+5}[/tex]

0 = 0

So the given statement is true.

B. a = 0, b = 4, c = 0, d= 9

[tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]

0 + 0 = 0

So for this example the given statement is true.

C. a = -2, b = 1, c = 2, d = 1

[tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]

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

0 = 0

Statement is true for these values.

D. a = 1, b = 2, c = 1, d = 2

[tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]

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

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

Therefore, for these values of a, b, c and d, the given statement is False.

Option D. is the answer.

To find a counter example for the given statement, we need to test the options provided. However, all the options satisfy the equation, indicating that the statement holds true for all nonzero real numbers.

To find a counter example for the given statement, we need to find values for a, b, c, and d that make the equation a/b + c/d = a+c/b+d false. Let's consider the options given:

We can see that for all the options, the equation holds true. Therefore, there is no counter example and the statement is true for all nonzero real numbers.

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Answer:

We have a normal distribution with a mean of 267 days and a standard deviation of 15 days. To solve this proble we're going to need the help of a calculator.

a. The probability of a pregnancy lasting 309 days or longer is:

P(z>309) = 0.0026 or 0.26%

b. The lowest 4% is separeted by the 240.74 days. The probability of pregnancy lasting 240.74 days is 4%.

Answer:

a) 0.26% probability of a pregnancy lasting 309 days or longer.

b) A pregnancy length of 241 days separates premature babies from those who are not premature.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 267, \sigma = 15[/tex]

a. Find the probability of a pregnancy lasting 309 days or longer.

This is 1 subtracted by the pvalue of Z when X = 309. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{309 - 267}{15}[/tex]

[tex]Z = 2.8[/tex]

[tex]Z = 2.8[/tex] has a pvalue of 0.9974

So there is a 1-0.9974 = 0.0026 = 0.26% probability of a pregnancy lasting 309 days or longer.

b. If the length of pregnancy is in the lowest 4​%, then the baby is premature. Find the length that separates premature babies from those who are not premature.

This is the value of X when Z has a pvalue of 0.04. So X when Z = -1.75

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-1.75 = \frac{X - 267}{15}[/tex]

[tex]X - 267 = -1.75*15[/tex]

[tex]X = 240.75[/tex]

A pregnancy length of 241 days separates premature babies from those who are not premature.

Answer:

at the end of 1st year we pay $ 11756

at the end of 2nd year we pay $ 24217

at the end of 3rd year we pay $ 37426

at the end of 4th year we pay $ 51427

at the end of 5th year we pay $ 66269

at the end of 6th year we pay $ 82001

Step-by-step explanation:

Given data

rate ( i ) = 6%

Future payment  = $82000

no of time period ( n ) = 6

to find out

size of all of 6 payments

solution

we know future payment formula i.e.

future payment = payment per period ( [tex](1 + rate)^{n}[/tex] - 1 )   / rate

put all these value and get payment per period

payment per period = future payment × rate  /  ( [tex](1 + rate)^{n}[/tex] - 1 )   / rate

payment per period = 82000 × 0.06  /  ( [tex](1 + 0.06)^{6}[/tex] - 1 )   / rate

payment per period = 82000 × 0.06 / 0.4185

payment per period = $ 11756.27

at the end of 1st year we pay $ 11756

and at the end of 2nd year we pay $ 11756 × ( 1  + 0.06) + 11756

and at the end of 2nd year we pay $ 24217

and at the end of 3rd year we pay $ 24217 × ( 1  + 0.06) + 11756

and at the end of 3rd year we pay $ 37426

and at the end of 4th year we pay $ 37426 × ( 1  + 0.06) + 11756

and at the end of 4th year we pay $ 51427

and at the end of 5th year we pay $ 51427 × ( 1  + 0.06) + 11756

and at the end of 5th year we pay $ 66269

and at the end of 6th year we pay $ 66269 × ( 1  + 0.06) + 11756

and at the end of 6th year we pay $ 82001

Answer:  

The converse and contrapositive of the given conditional statement are :

Converse : "If x is an even number, then 2 divides x".

Contrapositive : "If x is not an even number, then 2 does not divide x."

Step-by-step explanation:  We are given to write the converse and contrapositive of the following conditional statement :

"If 2 divides x, then x is an even number."

Let us consider that

p : 2 divides x

and

q : x is an even number.

We know that

the CONVERSE of a conditional statement p ⇒ q is written as q ⇒ p.

So, the converse of the given statement is

"If x is an even number, then 2 divides x".

The CONTRAPOSITIVE of the conditional statement  p ⇒ q is written as ~q ⇒ ~p (where ~p stands for the negation of p).

So, the contrapositive of the given statement is

"If x is not an even number, then 2 does not divide x."

Thus, the converse and contrapositive of the given statement are :

Converse : "If x is an even number, then 2 divides x".

Contrapositive : "If x is not an even number, then 2 does not divide x."

Answer: 1.8 servings of soybean and 1.2 servings rice will be needed.

Step-by-step explanation:

Let x be the number of soybeans and y be the number of rice.

Then , According to the question , we have

[tex]14x+4y=30\\\Rightarrow\ 7x+2y=15..................................(1)\\\\ 8x+\frac{1}{2}y = 15\\\Rightarrow\ 16x+y=30.............................(2)[/tex]

Multiply 2 on both the sides of equation (2), we get

[tex]32x+2y=60................................(3)[/tex]

Subtract equation (2) from equation (3), we get

[tex]25x=45\\\\\Rightarrow\ x=1.8[/tex]

Put x = 1.8 in equation (2), we get

[tex]16x+y=30\\\\\Rightarrow\ y=30-16x\\=30-16(1.8)=1.2[/tex]

Hence, 1.8 servings of soybean and 1.2 servings rice will be needed.

Answer:

$29,721

Step-by-step explanation:

We have been given that Wyatt is investing money into a savings account that pays 2% interest compounded annually, and plans to leave it there for 15 years. We are asked to find the amount deposited by Wyatt in order to have a balance of $40,000 in his savings account after 15 years.

We will use compound interest formula to solve our given problem.

[tex]A=P(1+\frac{r}{n})^{nT}[/tex], where,

A = Final amount after T years,

P = Principal amount,

r = Annual interest rate in decimal form,

n = Number of times interest is compounded per year,

T = Time in years.

Let us convert our given interest rate in decimal form.

[tex]2\%=\frac{2}{100}=0.02[/tex]

Upon substituting our given values in compound interest formula, we will get:

[tex]\$40,000=P(1+\frac{0.02}{1})^{1*15}[/tex]

[tex]\$40,000=P(1+0.02)^{15}[/tex]

[tex]\$40,000=P(1.02)^{15}[/tex]

[tex]\$40,000=P\times 1.3458683383241296[/tex]

Switch sides:

[tex]P\times 1.3458683383241296=\$40,000[/tex]

[tex]\frac{P\times 1.3458683383241296}{ 1.3458683383241296}=\frac{\$40,000}{1.3458683383241296}[/tex]

[tex]P=\$29,720.5891995[/tex]

Upon rounding our answer to nearest dollar, we will get:

[tex]P\approx \$29,721[/tex]

Therefore, Wyatt will have to invest $29,721 now in order to have a balance of $40,000 in his savings account after 15 years.

reflexive

NO -  e.g. 1+1=2 which is not a multiple of 4

irreflexive

NO - e.g. 2+2=4 which is multiple of 4

symmetric

YES - If [tex]a+b[/tex] is a multiple of 4, then [tex]b+a[/tex] is all multiple of 4, because addition is commutative.

antysymmetric

NO, because it's symmetric.

transitive

NO - e.g. 1+3=4 , 3+5=8 , 1+5=6 and 6 is not a multiple of 4.

Answer: a) 8359  b) 384

Step-by-step explanation:

Given : Significance level : [tex]\alpha=1-0.95=0.05[/tex]

Critical value : [tex]z_{\alpha/2}}=\pm1.96[/tex]

Margin of error : [tex]E=0.01[/tex]

a) If previous estimate of proportion : [tex]p=0.32[/tex]

Formula to calculate the sample size needed for interval estimate of population proportion :-

[tex]n=p(1-p)(\frac{z_{\alpha/2}}{E})^2[/tex]

[tex]\Rightarrow\ n=0.32(1-0.32)(\frac{1.96}{0.01})^2=8359.3216\approx 8359[/tex]

Hence, the required sample size would be 8359 .

b) If she does not use any prior estimate , then the formula to calculate sample size will be :-

[tex]n=0.25\times(\frac{z_{\alpha/2}}{E})^2\\\\\Rightarrow\ n=0.25\times(\frac{1.96}{0.05})^2=384.16\approx384[/tex]

Hence, the required sample size would be 384 .

Answer:

[tex]\frac{x-8}{x+3}[/tex]

Step-by-step explanation:

Given expression,

[tex]\frac{x^2-9x+8}{x^2+2x-3}[/tex]

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

[tex]=\frac{x^2-8x-x+8}{x^2+3x-x-3}[/tex]

[tex]=\frac{x(x-8)-1(x-8)}{x(x+3)-1(x+3)}[/tex]

[tex]=\frac{(x-1)(x-8)}{(x-1)(x+3)}[/tex]

[tex]=\frac{x-8}{x+3}[/tex]

Since, further simplification is not possible,

Hence, the given rational expression in lowest terms is,

[tex]\frac{x-8}{x+3}[/tex]

1197 hours, 48 minutes

is equal to

1197 + 48/60 hours

Dividing by 53 gives

1197/53 + 48/3180 hours

Since

1197 = 22*53 + 31

we get

22 + 31/53 + 48/3180 hours

22 + (1860 + 48)/3180 hours

22 + 3/5 hours

22 + 36/60 hours

22 hours, 36 minutes

Final answer:

The statement 'The chair is broken or the stove is not hot' translates to 'p v ~q' in symbolic form, where 'p' represents 'The chair is broken,' and 'q' represents 'The stove is hot.' The 'v' symbolizes 'or,' and '~' indicates negation.

Explanation:

The question requests the compound statement 'The chair is broken or the stove is not hot' be written in symbolic form using propositional logic. Given the statements, p is 'The chair is broken,' and q is 'The stove is hot,' we can translate the request into symbolic form. The use of 'or' in propositional logic is represented by the symbol 'v' (wedge), and the negation of a statement is represented by the symbol '~' (tilde). Therefore, 'The stove is not hot' can be represented as '~q'.

To combine these statements into the requested compound statement, we use:
p v ~q
This reads as 'p or not q', which translates back into the original statement 'The chair is broken or the stove is not hot.' This allows us to see how logical operators like disjunction ('or') and negation ('not') work together to construct compound statements in propositional logic.

Final answer:

The rate constant, k, can be found using the formula k = ln(1 + r), where r is the growth rate. In this case, the growth rate is 2.5%. The rate constant is approximately 0.0253. The exponential growth function is P(t) = 66,000 * e^(0.0253t). To find in what year the population will reach 176,000, we solve the equation 176,000 = 66,000 * e^(0.0253t) and find that it will take approximately 42 years.

Explanation:

To find the rate constant, we can use the formula:

k = ln(1 + r)

where k is the rate constant and r is the growth rate as a decimal.

In this case, the growth rate is 2.5%, which is equivalent to 0.025 as a decimal.

Using the formula, we have:

k = ln(1 + 0.025) = ln(1.025) ≈ 0.0253

Therefore, the rate constant k is approximately 0.0253.

To devise an exponential growth function, we can use the formula:

P(t) = P0 * ekt

where P(t) is the population at time t, P0 is the initial population, k is the rate constant, and t is the time in years.

In this case, the initial population P0 is 66,000 and we already found that the rate constant k is 0.0253.

So, the exponential growth function is:

P(t) = 66,000 * e0.0253t

To find in what year the population will reach 176,000, we can set up the following equation:

176,000 = 66,000 * e0.0253t

Divide both sides by 66,000:

176,000 / 66,000 = e0.0253t

Simplify:

2.6667 = e0.0253t

To solve for t, we can take the natural logarithm of both sides:

ln(2.6667) = 0.0253t

Divide both sides by 0.0253:

t = ln(2.6667) / 0.0253 ≈ 41.71

Therefore, the population will reach 176,000 in approximately 41.71 years, which can be rounded to 42 years.

Parameterize this surface (call it [tex]S[/tex]) by

[tex]\vec r(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(9-u^2)\,\vec k[/tex]

with [tex]0\le u\le3[/tex] and [tex]0\le v\le2\pi[/tex]. Take the normal vector to [tex]S[/tex] to be

[tex]\vec r_u\times\vec r_v=2u^2\cos v\,\vec\imath+2u^2\sin v\,\vec\jmath+u\,\vec k[/tex]

Then the area of [tex]S[/tex] is

[tex]\displaystyle\iint_S\mathrm dA=\iint_S\|\vec r_u\times\vec r_v\|\,\mathrm du\,\mathrm dv[/tex]

[tex]=\displaystyle\int_0^{2\pi}\int_0^3u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv[/tex]

[tex]=\displaystyle2\pi\int_0^3u\sqrt{1+4u^2}\,\mathrm du=\boxed{\frac{37\sqrt{37}-1}6\pi}[/tex]

Answer:

80

Step-by-step explanation:

Answer:

50% of students has a score less than 85

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 85, \sigma = 20[/tex]

What percentage of students has a score less than 85?

This is the pvalue of Z when X = 85. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{85 - 85}{20}[/tex]

[tex]Z = 0[/tex]

[tex]Z = 0[/tex] has a pvalue of 0.5

50% of students has a score less than 85

There is a loss of $4,400 on retirement. So option (b) is correct.

To calculate the gain or loss on retirement of the bonds, we need to compare the carrying value of the bonds with the price at which they are being called.

Given:

Par value of the bonds = [tex]$121,000$[/tex]

Carrying value of the bonds = [tex]$109,900$[/tex]

Call price = [tex]$105,500$[/tex]

The gain or loss on retirement is calculated as the difference between the carrying value and the call price.

Loss on retirement = Carrying value - Call price

Substitute the given values:

Loss on retirement = $109,900 - $105,500

Loss on retirement = $4,400

However, since the call price is lower than the carrying value, the loss is incurred by the company. Thus, the correct answer is a [tex]$\$4,400$[/tex] loss.

So, the correct option is: (b) $4,400 loss.

Clabber Company has bonds outstanding with a par value of [tex]$\$ 121,000$[/tex] and a carrying value of [tex]$\$ 109,900$[/tex]. If the company calls these bonds at a price of [tex]$\$ 105,500$[/tex], the gain or loss on retirement is:

Multiple Choice

(a) [tex]$\$ 15,500$[/tex] loss.

(b) [tex]$\$ 4,400$[/tex] loss.

(c) [tex]$\$ 11,100$[/tex] loss.

(d) [tex]$\$ 4,400$[/tex] gain.

The gain on retirement is $15,500.

To calculate the gain or loss on retirement of the bonds, we first need to understand the definitions involved:

1. Par value: This is the face value of the bonds, which is $121,000 in this case.

2. Carrying value: This is the book value of the bonds on the company's balance sheet, which is $109,900.

3. Call price: This is the price at which the company is redeeming (calling) the bonds, which is $105,500.

Now, let's calculate the gain or loss:

- Gain/Loss = Par Value - Call Price

If the call price is less than the carrying value, it results in a gain. If the call price is greater than the carrying value, it results in a loss.

Given:

- Par value = $121,000

- Carrying value = $109,900

- Call price = $105,500

[tex]\[ \text{Gain/Loss} = \text{Par Value} - \text{Call Price} \]\[ \text{Gain/Loss} = \$121,000 - \$105,500 \]\[ \text{Gain/Loss} = \$15,500 \][/tex]

Since the call price is less than the carrying value, it results in a gain.

912 cards in all

If there are 114 packs of cards and 8 in each you multiply 114 by 8 and get 912.

To find the total number of baseball cards, multiply the number of packs (144) by the number of cards in each pack (8), resulting in a total of 1,152 cards.

If a gross of baseball cards contains 144 packs of cards and each pack contains 8 baseball cards, the total number of cards would be calculated by multiplying the number of packs by the number of cards in each pack.

To get the total number of baseball cards.

144 packs x 8 cards/pack = 1,152 cards

Hence, a gross of baseball cards contains 1,152 cards in total.

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3(-3 + 5x) -1 (4 - 4x)

Simplify each term by first using the distributive property with each set of parenthesis:

3*-3 + 3*5x - 1*4 -1* -4x

Now do the multiplications:

-9 + 15x -4 + 4x

Combine like terms

15x +4x - 9 -4

19x - 13

The summand (R?) is missing, but we can always come up with another one.

Divide the interval [0, 1] into [tex]n[/tex] subintervals of equal length [tex]\dfrac{1-0}n=\dfrac1n[/tex]:

[tex][0,1]=\left[0,\dfrac1n\right]\cup\left[\dfrac1n,\dfrac2n\right]\cup\cdots\cup\left[1-\dfrac1n,1\right][/tex]

Let's consider a left-endpoint sum, so that we take values of [tex]f(\ell_i)={\ell_i}^3[/tex] where [tex]\ell_i[/tex] is given by the sequence

[tex]\ell_i=\dfrac{i-1}n[/tex]

with [tex]1\le i\le n[/tex]. Then the definite integral is equal to the Riemann sum

[tex]\displaystyle\int_0^1x^3\,\mathrm dx=\lim_{n\to\infty}\sum_{i=1}^n\left(\frac{i-1}n\right)^3\frac{1-0}n[/tex]

[tex]=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=1}^n(i-1)^3[/tex]

[tex]=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=0}^{n-1}i^3[/tex]

[tex]=\displaystyle\lim_{n\to\infty}\frac{n^2(n-1)^2}{4n^4}=\boxed{\frac14}[/tex]

The limit  expression for the area under the curve y = x³ as n approaches infinity is; ¹/₄

The given definition is area A of the region S that lies under the graph of the continuous function which is the limit of the sum of the areas of approximating rectangles.

The expression for the area under the curve y = x³ from 0 to 1 as a limit is;

[tex]\lim_{n \to \infty} \Sigma^{n} _{i = 1} (\frac{i}{n})^{3} * \frac{1}{n} }[/tex]

From the expression above, when we factor out 1/n⁴, we will get;

[tex]\lim_{n \to \infty} \frac{1}{n^{4} } \Sigma^{n} _{i = 1} \frac{i}{n} }[/tex]

This is further broken down to get;

[tex]\lim_{n \to \infty} \frac{1}{n^{4} } (\frac{n(n + 1)}{2} )^{2} } }[/tex]

This will be simplified to;

[tex]\lim_{n \to \infty} \frac{1}{n^{4} } \frac{(n^{4} + 2n^{3} + n^{2}) }{2}[/tex]

This would be simplified to;

[tex]\lim_{n \to \infty} \frac{1}{4} + \frac{1}{2n} + \frac{1}{4n^{2} }[/tex]

At limit of n approaches ∞, we have;

Limit = ¹/₄

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Answer:

static value come under the rejection value because it is less than critical value

Step-by-step explanation:

Given data

test = 300

wrong test = 30

significance level = 0.01

claim for wrong  = 10 %

to find out

test the claim that less than 10 percent of the test results are wrong

solution

we take test claim null hypo thesis  = 10 % = 0.10

and and alternate hypo thesis < 10% i.e. <0.10

and we know proportion of sample is = result/ test

sample proportion = 30/300 = 0.10

so the statistics of this test will  be = sample proportion - hypothesis / [tex]\sqrt{hyro(1-hypo)/test}[/tex]

so statistics of this test  = 0.10 - 0.10 / [tex]\sqrt{0.10(1-0.10)/300}[/tex]

so statistics of this test  =  0

and α = tail area critical value for  Z (0.01)  = 2.33

so here static value come under the rejection value because it is less than critical value

To test the claim that less than 10 percent of the test results are wrong, we'll use a hypothesis test with a 0.01 significance level. The calculated test statistic z will determine whether to reject or fail to reject the null hypothesis.

To test the claim that less than 10 percent of the test results are wrong, we'll use a hypothesis test with a 0.01 significance level. Let p represent the proportion of wrong test results. The null hypothesis is that p is greater than or equal to 0.10, and the alternative hypothesis is that p is less than 0.10.

We'll calculate the test statistic z = (x - np) / sqrt(np(1-p)), where x is the number of wrong test results and n is the total number of tested subjects. Using the given information, x = 30 and n = 300.

Using a z-table or calculator, we can find the z-score corresponding to a significance level of 0.01. If the calculated test statistic z is less than the z-score from the table, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

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Answer: The description are as follows:

Step-by-step explanation:

Correlation coefficients is a statistical measure that measures the relationship between the two variables.

(a) r = 1, it means that there is a Perfect positive relationship between the two variables. If there is positive increase in one variable then other variable also increases with a fixed proportion.

(b) r = -1, it means that there is a perfect negative relationship between the two variables. If there is positive increase in one variable then other variable decreases with a fixed proportion.

(c) r = 0, this is a situation which shows that there is no relationship between the two variables.

(d) r = 0.86, this is a situation  which shows that there is a fairly strong positive relationship between the two variables.

(e) r = 0.06, it is nearly zero which represents that either there is a very minor positive relationship between the two variables or there is no relationship between them.

(f) r = -0.89,  this is a situation  which shows that there is a fairly strong negative relationship between the two variables.

Final answer:

The correlation coefficient, r, measures the strength and direction of the linear relationship between x and y values in a sample.

Explanation:

The correlation coefficient, r, measures the strength and direction of the linear relationship between x and y values in a sample. Let's analyze each correlation coefficient:

a. r=1: This indicates a perfect positive correlation between x and y values, meaning that as x increases, y also increases at a constant rate. All the data points fall on a straight line with a positive slope.

b. r=-1: This indicates a perfect negative correlation between x and y values, meaning that as x increases, y decreases at a constant rate. All the data points fall on a straight line with a negative slope.

c. r=0: This indicates no linear relationship between x and y values. The data points are scattered randomly, and there is no consistent pattern or trend between the variables.

d. r=0.86: This indicates a strong positive correlation between x and y values. As x increases, y also increases, but not at a perfect constant rate. The data points approximately fall on a line with a positive slope.

e. r=0.06: This indicates a weak positive correlation between x and y values. As x increases, y also increases, but the relationship is not very strong. The data points have a scattered pattern around a slightly positive sloped line.

f. r=-0.89: This indicates a strong negative correlation between x and y values. As x increases, y decreases, but not at a perfect constant rate. The data points approximately fall on a line with a negative slope.

Answer:

The relator recived $8,186.4

Step-by-step explanation:

What you have to do is find the 60% of the commission. Commission is 4% of $341,100 (100%)

First do a cross multiplication to find 4% of $341,100

100% ___ $341,100

4%______x:

[tex]x=(4*341,100)/100=13,644[/tex]

So, the 4% of $341,100 is $13,466

Now you have to find the 60% of $13,466

100% ___ $13,466

60%______x:

[tex]x=(60*13,466)/100=8,186.4[/tex]

The answer is: $8,186.4

Answer:

The percentage of customer returns is 7%.

Step-by-step explanation:

Given,

Total gross sales = $ 7,500

Returns are,

$95 on Monday, $19 on Tuesday, $50 on Wednesday, $140 on Thursday, $160 on Friday, and $80 on Saturday,

So, the total returns = $ 95 + $ 50 + $ 140 + $ 160 + $ 80 = $ 525,

Hence, the percent of customer returns for the week = [tex]\frac{\text{Total returns}}{\text{Total gross sales}}\times 100[/tex]

[tex]=\frac{525}{7500}\times 100[/tex]

[tex]=\frac{52500}{7500}[/tex]

[tex]=7\%[/tex]

Answer:

[tex]\frac{208}{1091}[/tex]

Step-by-step explanation:

To find probability of x, we need to find the number of x divided by total number.

Here,

Total = 208 + 552 + 331 = 1091

Number of help dash wanted ad = 208

Hence, the probability that the ad for a specific week is a help dash wanted ad = 208/1091

Answer:

The correct option is A.

Step-by-step explanation:

The formula of probability is

[tex]P=\frac{a}{b}[/tex]

Where, a≤b, a is total favorable outcomes and b is total possible outcomes.

If an event is certain to occur, then a=b and the probability of an event that is certain to occur is

[tex]P=\frac{a}{a}=1[/tex]

If an event is impossible, then a=0 and the probability of an impossible event is

[tex]P=\frac{0}{a}=0[/tex]

Since total favorable outcomes a and total possible outcomes b can not be negative, a is always less than of equal to b. So,

[tex]0\leq \frac{a}{b}\leq 1[/tex]

[tex]0\leq P\leq 1[/tex]

Therefore the probability of any event is between 0 and 1 inclusive.

All events are not equally likely in any probability procedure. So, the statement "All events are equally likely in any probability procedure" is not true.

Therefore the correct option is A.

The statement 'All events are equally likely in any probability procedure' is NOT a fundamental principle of probability. Events in a probability procedure can have different probabilities based on the situation.

The question is asking to identify which of the given options is NOT a principle of probability. Here, the principles of probability suggest that the probability of an event that is certain to occur is 1 (option B), the probability of an impossible event is 0 (option C), and that the probability of any event is between 0 and 1 inclusive (option D). These are well-established principles of probability and hold true in most situations.

However, option A, 'All events are equally likely in any probability procedure', is NOT a fundamental principle of probability. This is not always true as the likelihood of events can vary greatly depending on the scenario. For instance, if you roll a fair six-sided dice, the probability of landing a 1 is 1/6, but in sampling with or without replacement, probabilities of different outcomes can differ. Thus, it is not a rule that all outcomes are always equally likely in any probabilistic process.

A simple real-life application of this can be seen in card games. If you draw one card from a standard deck of 52, the probability of drawing a heart is 1/4, not equal to the probability of drawing a specific number card like the 7 of clubs, which is 1/52. Therefore, the statement that 'all events are equally likely in any probability procedure' is not always true.

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The solutions of the given equation are:

              x=0,1,4 and 9

We are asked to find the solution of the equation:

[tex]x^2-x=0\ \text{mod}\ 12[/tex]

i.e. we have to find the possible values of x such that the equation is true.

then

[tex]x^2-x=0-0\\\\i.e.\\\\x^2-x=0[/tex]

Hence, x=0 is the solution of the equation.

then

[tex]x^2=1\\\\Hence,\\\\x^2-x=1-1\\\\i.e.\\\\x^2-x=0[/tex]

Hence, x=1 is  a solution.

then

[tex]x^2=4[/tex]

i.e.

[tex]x^2-x=4-2\\\\i.e.\\\\x^2-x=2\neq 0[/tex]

Hence, x=2 is not  a solution.

then

[tex]x^2=9[/tex]

i.e.

[tex]x^2-x=9-3\\\\i.e.\\\\x^2-x=6\neq 0[/tex]

Hence, x=3 is not  a solution.

then

[tex]x^2=16=4\ \text{mod}\ 12[/tex]

i.e.

[tex]x^2-x=4-4\\\\i.e.\\\\x^2-x=0[/tex]

Hence, x=4 is a solution to the equation.

then

[tex]x^2=25=1\ \text{mod}\ 12[/tex]

i.e.

[tex]x^2-x=1-4\\\\i.e.\\\\x^2-x=-3=9\ \text{mod}\ 12[/tex]

i.e.

[tex]x^2-x=9\neq 0[/tex]

Hence, x=5 is not a solution.

then

[tex]x^2=36\\\\i.e.\\\\x^2=0\ \text{mod}\ 12\\\\i.e.\\\\x^2=0[/tex]

Hence,

[tex]x^2-x=0-6\\\\i.e.\\\\x^2-x=-6=6 \text{mod}\ 12\\\\i.e.\\\\x^2-x=6\neq 0[/tex]

Hence, x=6 is not a solution

then,

[tex]x^2=49=1\ \text{mod}\ 12\\\\i.e.\\\\x^2=1[/tex]

Hence,

[tex]x^2-x=1-7\\\\i.e.\\\\x^2-x=-6=6\ \text{mod}\ 12\\\\i.e.\\\\x^2-x=6\neq 0[/tex]

Hence, x=7 is not a solution.

then,

[tex]x^2=64=4\ \text{mod}\ 12[/tex]

i.e.

[tex]x^2-x=4-8\\\\i.e.\\\\x^2-x=-4=8\ \text{mod}\ 12[/tex]

i.e.

[tex]x^2-x=8\neq 0[/tex]

Hence, x=8 is not a solution.

then,

[tex]x^2=81=9\ \text{mod}\ 12[/tex]

i.e.

[tex]x^2=9[/tex]

Hence,

[tex]x^2-x=9-9\\\\i.e.\\\\x^2-x=0[/tex]

Hence, x=9 is a solution.

then,

[tex]x^2=100=4\ \text{mod}\ 12[/tex]

i.e.

[tex]x^2-x=4-10\\\\i.e.\\\\x^2-x=-6=6\ \text{mod}\ 12\\\\i.e.\\\\x^2-x=6\neq 0[/tex]

Hence, x=10 is not a solution.

then,

[tex]x^2=121=1\ \text{mod}\ 12[/tex]

i.e.

[tex]x^2-x=1-11\\\\x^2-x=-10=2\ \text{mod}\ 12\\\\i.e.\\\\x^2-x=2\neq 0[/tex]

Hence, x=11 is not a solution.

Answer:

Construct validity

Step-by-step explanation:

Construct validity is the most important and outmost validity that is used in scientific methods. Construct validity tells us how an experiment or a test is performed well and how well is the outcome of the experiment, How can the experiments can be measured upto its claims. Construct validity is not concerned about the simple question or the factual question that if an experiment measures an attribute. Construct validity is thus an evaluation of the quality of the experiment.

Answer:

a. construct validity

Step-by-step explanation:

Construct validity refers to whether a particular test actually measures what it claims to be measuring. This is one of the main types of validity evidence. It has become the overriding objective in validity, subsuming all other types of validity evidence. Construct validity answers whether a particular measure behaves in the way that the theory says a measure of that construct should behave.

4 pizzas with 3 toppings each

only 12 toppings and 3 toppings per pizza

You have to do 12/3=4

Answer:

A, B, and E

if I read your functions right.

Step-by-step explanation:

It's zeros are x=-6,-2, and 2.

This means we want the factors (x+6) and (x+2) and (x-2) in the numerator.

It has a y-intercept of 4.  This means we want to get 4 when we plug in 0 for x.

And it's long-run behavior is y approaches - infinity as x approaches either infinity.  This means the degree will be even and the coefficient of the leading term needs to be negative.

So let's see which functions qualify:

A) The degree is 4 because when you do x^2*x*x you get x^4.

The leading coefficient is -4/144 which is negative.

We do have the factors (x+6), (x+2), and (x-2).

What do we get when plug in 0 for x:

[tex]\frac{-4}{144}(0+6)^2(0+2)(0-2)[/tex]

Put into calculator:  4

A works!

B) The degree is 6 because when you do x*x^4*x=x^6.

The leading coefficient is -4/192 which is negative.

We do have factors (x+6), (x+2), and (x-2).

What do we get when we plug in 0 for x:

[tex]\frac{-4}{192}(0+6)(0+2)^4(x-2)[/tex]

Put into calculator: 4

B works!

C) The degree is 4 because when you do x*x*x*x=x^4.

The leading coefficient is -4 which is negative.

Oops! It has a zero at 0 because of that factor of (x) between -4 and (x+6).

So C doesn't work.

D) The degree is 3 because x*x*x=x^3.

We needed an even degree.

D doesn't work.

E) The degree is 4 because x*x^2*x=x^4.

The leading coefficient is -4/48 which is negative.

It does have the factors (x+6), (x+2), and (x-2).

What do we get when we plug in 0 for x:

[tex]\frac{-4}{48}(0+6)(0+2)^2(0-2)[/tex]

Put into calculator: 4

So E does work.

F) The degree is 4 because x*x*x^2=x^4.

The leading coefficient is -4/48.

It does have factors (x+6), (x+2), and (x-2).

What do we get when we plug in 0 for x:

[tex]\frac{-4}{48}(0+6)(0+2)(0-2)^2[/tex]

Put into calculator: -4

So F doesn't work.

G. I'm not going to go any further. The leading coefficient is 4/48 and that is not negative.

So G doesn't work.

Answer:

Step-by-step explanation:

Notice that we have 3 zeros, which means there are only 3 roots, which are -6, -2 and 2, this indicates that our expression must be cubic with the binomials (x+6), (x+2) and (x-2).

We this analysis, possible choices are C and D.

Now, according to the problem, it has y-intercept at y = 4, so let's evaluate each expression for x = 0.

[tex]y=-4x(x+6)(x+2)(x-2)\\y=-4(0)(0+6)(0+2)(0-2)\\y=0[/tex]

[tex]y=-\frac{4}{24} (x+6)(x+2)(x-2)\\y=-\frac{4}{24}(0+6)(0+2)(0-2)\\y=-\frac{4}{24}(-24)\\ y=4[/tex]

Therefore, choice D is the right expression because it has all given characteristics.