Which system of equations can be used to find the roots of the equation 4x5-12x4+6x=5x3-2x?

Answer:

143

Step-by-step explanation:

Denote by x and y such integers. The hypotheses given can be written as:

[tex]x+y=24, x^2-y^2=48[/tex]

Use the difference of squares factorization to solve for x-y

[tex]48=x^2-y^2=(x-y)(x+y)=24(x-y)\text{ then }x-y=2[/tex]

Remember that

[tex](x+y)^2=x^2+2xy+y^2[/tex]

[tex](x-y)^2=x^2-2xy+y^2[/tex]

Substract the second equation from the first to obtain

[tex](x+y)^2-(x-y)^2=4xy[/tex]

Substituting the known values, we get

[tex]4xy=24^2-2^2=572\text{ then }xy=\frac{572}{4}=143[/tex]

The sum of the two integers is 24, and the difference of their squares is 48. By setting up a system of equations, we find the integers are 13 and 11. The product of these integers is 143.

We are given the sum of two positive integers is 24 and the difference of their squares is 48. Let's denote the integers as x and y, with x being the larger integer. So, we have:

We can factor Equation 2, which is a difference of squares, into (x + y)(x - y) = 48. Using the fact that x + y = 24 (from Equation 1), we can substitute into this to get 24(x - y) = 48, which simplifies to x - y = 2. Now we have a system of equations:

Adding these two equations, we get 2x = 26, so x = 13. Subtracting the second equation from the first, we get 2y = 22, so y = 11. Now to find the product of the two integers, we multiply x and y together: 13 * 11 = 143.

Therefore, the product of the two integers is 143.

Answer:

d) Y - 6 = x²; f) Y = 2x² + 5 - 3x²  

Step-by-step explanation:

Functions in which the exponent of x is not equal to one are nonlinear.

Functions in which the exponent of x is equal to one are linear.

Answer:option 3 is the correct answer.

Step-by-step explanation:

Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis

change in the value of y = y2 - y1

Change in value of x = x2 -x1

y2 = final value of y

y 1 = initial value of y

x2 = final value of x

x1 = initial value of x

From the graph given,

y2 = 3

y1 = 1

x2 = 0

x1 = - 1

Slope = (3 - 1)/(0 - - 1)

Slope = 2/1 = 2

Answer:

[tex]P(\mu -1< \bar X <\mu +1)=0.6826[/tex]

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Let X the random variable that represent the Shear strength of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(\mu,10)[/tex]  

Where [tex]\mu[/tex] and [tex]\sigma=10[/tex]

And let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:

[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]

On this case  [tex]\bar X \sim N(\mu,\frac{10}{\sqrt{100}})[/tex]

We are interested on this probability

[tex]P(\mu -1<\bar X<\mu +1)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

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

If we apply this formula to our probability we got this:

[tex]P(\mu -1<\bar X<\mu +1)=P(\frac{\mu- 1-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{X-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{\mu +1-\mu}{\frac{\sigma}{\sqrt{n}}})[/tex]

[tex]=P(\frac{\mu -1-\mu}{\frac{10}{\sqrt{100}}}<Z<\frac{\mu +1-\mu}{\frac{10}{\sqrt{100}}})=P(-1<Z<1)[/tex]

And we can find this probability on this way:

[tex]P(-1<Z<1)=P(Z<1)-P(Z<-1)[/tex]

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

[tex]P(-1<Z<1)=P(Z<1)-P(Z<-1)=0.8413-0.1587=0.6826[/tex]

The probability that the sample mean will be within 1 psi of the true population mean is approximately 68.2%, according to the properties of a normal distribution and the central limit theorem.

This is a problem of standard deviation and probability in relation to the sample mean. This type of problem can be solved by knowing the properties of a normal distribution.

The central limit theorem states that if we have a large enough sample, the distribution of the sample mean will approximate a normal distribution regardless of the distribution of the population.

For this scenario, where the true population mean is unknown, the standard deviation of the sampling distribution (also known as the standard error) can be calculated as the original standard deviation (10 psi) divided by the square root of the sample size (100 test welds in this case), hence 10 ÷ √100 = 1 psi.

Then, to find the probability that the sample mean is within 1 psi of the true population mean, we can refer to the Z-table (a standard normal distribution table) to find the corresponding probability for z = ±1 (because the z-score for ±1 standard error from the mean is ±1). This value is approximately 68.2%

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Blaire has 20 tomato plants altogether.

Step-by-step explanation:

Given,

Tomatoes plants ready to pick = 18

This represents 90% of total tomato plants.

Let,

x be the original number of tomato plants.

90% of x = 18

[tex]\frac{90}{100}*x=18[/tex]

[tex]0.9x=18[/tex]

Dividing both sides by 0.9

[tex]\frac{0.9x}{0.9}=\frac{18}{0.9}[/tex]

[tex]x=20[/tex]

Blaire has 20 tomato plants altogether.

Keywords: percentage, division

Learn more about division at:

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Answer:

Goal amount of $10,000

Deadline of next year

Step-by-step explanation:

Tyrone can make his financial goal ‘specific’ by deciding on a target amount for his emergency fund. He can make it 'timely' by assigning a deadline by which to save that amount.

To make Tyrone's financial goal specific, he could give himself a target amount to save for the emergency fund. This could be a fixed sum, like $1000, or a figure based on monthly expenses, like saving for 6 months' worth of living expenses. This clarity can help him to plan and track his progress.

To make his goal timely, he could give himself a deadline by which he wants to achieve this goal. For example, he might aim to save his specified amount within a year or two. The timetable can provide added motivation to adhere to a budget and save consistently.

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Answer:Yuan received $50 last year

Step-by-step explanation:

Yuan receives money from his relatives every year on his birthday.

Let x represent the amount of money that Yuan received last year on his birthday.

This year, Yuan received a total of $56. The amount that he received this year is 12% more than he received last year. This means that

the increment on the amount that he received last year is would be

12/100×x = 0.12x. Therefore,

x + 0.12x = 56

1.12x = 56

x = 56/1.12 = $50

Final answer:

The experimental probability of a string of lights being defective is calculated by dividing the number of defective strings found during the inspection by the total number of strings inspected, leading to a probability of 6/25.

Explanation:

The experimental probability that a string of lights is defective is determined by dividing the number of defective strings of lights by the total number of strings inspected. This probability can be calculated as follows:

Number of defective strings = 6

Total number of strings inspected = 25

Experimental Probability = Number of defective strings / Total number of strings

So, the experimental probability of finding a defective string of lights is 6/25.

Answer:

The ratio of the incumbent”s votes in the total number of votes = 7:8

Step-by-step explanation:

Given:

Number of students who cast vote for the incumbent = 7,000

Number of students who cast vote for the opponent = 900

Number of protest votes = 100

To find ratio of the incumbent”s votes in the total number of votes.

Solution:

Total number of votes cast = [tex]7000+900+100=8000[/tex]

Number of votes for incumbent = [tex]7000[/tex]

Ratio of incumbent”s votes in the total number of votes can be calculated as:

⇒ [tex]\frac{\textrm{The incumbent's votes}}{\textrm{Total number of votes}}[/tex]

⇒ [tex]\frac{7000}{8000}[/tex]

Simplifying to simplest fraction by dividing numerator and denominator by 1000.

⇒ [tex]\frac{7000\div1000}{8000\div1000}[/tex]

⇒ [tex]\frac{7}{8}[/tex]

Thus, ratio of the incumbent”s votes in the total number of votes = 7:8

Answer:

See explanation below

Step-by-step explanation:

Data given and notation  

First we need to find the sample mean and deviation from the data with the following formulas:

[tex]\bar X =\frac{\sum_{i=1}^n X_i}{n}[/tex]

[tex]s=\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]

[tex]\bar X[/tex] represent the sample mean  

[tex]s[/tex] represent the sample standard deviation

[tex]n[/tex] sample size  

[tex]\mu_o [/tex] represent the value that we want to test  

[tex]\alpha[/tex] represent the significance level for the hypothesis test.  

z would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We have three possible options for the null and the alternative hypothesis:

Case Bilateral  

Null hypothesis:[tex]\mu = \mu_o[/tex]  

Alternative hypothesis:[tex]\mu \neq \mu_o[/tex]

Case Right tailed

Null hypothesis:[tex]\mu \leq \mu_o[/tex]  

Alternative hypothesis:[tex]\mu > \mu_o[/tex]

Case Left tailed

Null hypothesis:[tex]\mu \geq \mu_o[/tex]  

Alternative hypothesis:[tex]\mu < \mu_o[/tex]

We assume that w don't know the population deviation, so for this case is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic  

We can replace in formula (1) and the value obtained is assumed as [tex]t_o[/tex]

Calculate the P-value  

First we need to find the degrees of freedom:

[tex] df=n-1[/tex]

Case two tailed

Since is a two-sided tailed test the p value would be:  

[tex]p_v =2*P(t_{df}>|t_o|)[/tex]  

Case Right tailed

Since is a one-side right tailed test the p value would be:  

[tex]p_v =P(t_{df}>t_o)[/tex]  

Case Left tailed

Since is a one-side left tailed test the p value would be:  

[tex]p_v =P(t_{df}<t_o)[/tex]  

Conclusion  

The rule of decision is this one:

[tex]p_v >\alpha[/tex] We fail to reject the null hypothesis at the significance level [tex]\alpha[/tex] assumed

[tex]p_v <\alpha[/tex] We reject the null hypothesis at the significance level [tex]\alpha[/tex] assumed

Answer:F=A-(A/2+A/4)

=> F=1/4

Step-by-step explanation:

Let A represent the initial quality of rectangular fabric.

Half of A was used for the sewing project

Quarter of the left over was used for project B

Hence a quarter of unused fabric(F) will be left.

Answer:

[tex]\displaystyle 5(4 - 2) = 20 - 10[/tex]

This is a genuine statement of you look real closely at it.

I am joyous to assist you anytime.

The distributive property is demonstrated in the equation 5(4 - 2) = 20 - 10, where multiplication outside the parentheses is distributed to each term within the parentheses.

The distributive property in mathematics is an algebraic property used to multiply a single term and two or more terms inside a set of parentheses. The correct statement that shows the distributive property among the given options is: 5(4 - 2) = 20 - 10.

Applying the distributive property, we would multiply the 5 by each term inside the parentheses: 5 * 4 = 20 and 5 * (-2) = -10. Hence, we have 5 * 4 - 5 * 2 = 20 - 10, which is a correct demonstration of this property.

To better understand, let me explain it step-by-step:

Multiply the term outside the parenthesis (5) by each of the terms inside the parenthesis (4 and -2).

Perform the multiplication: 5 * 4 = 20 and 5 * (-2) = -10

Combine the results to show that 5(4 - 2) is indeed equal to 20 - 10.

Answer: total Distance = 60km

Time = 4.5hrs

Speed = 60/4.5

13⅓km/hr

Step-by-step explanation:

Answer:

73 pencils

Step-by-step explanation:

There are 81 pencils in a box.

Abigail removes 5 pencils, thus we have 81-5 = 76 left

Barry removes 2 pencils,  it becomes 76-2 = 74

Cathy removes 6 pencils, now it is 74-6= 68

and David adds 5 pencils to the box,

Now we have 68+5=73 pencils left in the box.

Answer:

a) 31c + 19m ≤ 706

b) 7500c + 4300m > 84000

Step-by-step explanation:

To build a car, we need 31 employees and $7500.

To build a motorcycle, we need 19 employees and $4300.

Let C denote the number of cars they build.

Let M denote the number of motorbikes they build.

Recall that ;

To build a career, we need 31 employees. To build "c" cars, we will need 31*c = 31c employees

To build a motorcycle, we need 19 employees. To build "m" motorcycle, we will need 19*m = 19m

Since the maximum number of employees used to build the car and motorcycle is at most 706, we have

31c + 19m ≤ 706

It takes $7500 to build car. To build "c" cars, we need 7500*c = $7500c

It also takes $4300 to build "m" motorcycles. We need 4300*m = $4300m

Since Genghis motors wont to spend more than $84000 on both cars and motorcycles, we have

7500c + 4300m > 84000

For the condition based on the number of employees, we have

31c + 19m ≤ 706

For the condition based on the number of dollars, we have

7500c + 4300m > 84000

Answer:

31c + 19m ≤ 706 and 7500c + 4300m > 84000

Step-by-step explanation:

Answer:

There are a total of  [tex] { 6 \choose 3} = 20 [/tex] functions.

Step-by-step explanation:

In order to define an injective monotone function from [3] to [6] we need to select 3 different values fromm {1,2,3,4,5,6} and assign the smallest one of them to 1, asign the intermediate value to 2 and the largest value to 3. That way the function is monotone and it satisfies what the problem asks.

The total way of selecting injective monotone functions is, therefore, the total amount of ways to pick 3 elements from a set of 6. That number is the combinatorial number of 6 with 3, in other words

[tex] {6 \choose 3} = \frac{6!}{3!(6-3)!} = \frac{720}{6*6} = \frac{720}{36} = 20 [/tex]  

Answer:

Yes, the average (arithmetic mean) of the seven numbers would be negative.

Step-by-step explanation:

We have been given that list K consists of seven numbers. We have been given two cases about list K. We are asked to determine whether the average (arithmetic mean) of the seven numbers negative.

1st case: Four of the seven numbers in list K are negative.

For 1st case, if the sum of 3 positive numbers is greater than sum of four negative numbers, then the average would be positive.

2nd case: The sum of the seven numbers in list K is negative.

We know that average of a data set is sum of all data points of data set divided by number of data points.

Since we have been given that sum of the seven numbers in list K is negative, so a negative number divided by any positive number (in this case 7) would be negative.

Therefore, the average (arithmetic mean) of the seven numbers would be negative.

Answer:

The two watches will beep together for 20th time at 3:25 pm.  

Step-by-step explanation:

My watch beeps every 15 seconds and mom's watch beeps every 25 seconds.

Thus both the watches beep at the same time at an interval of 75 seconds.

75 is the smallest multiple of 15 and 25.

They beep together at 3 pm and when they beep together for the 20th time , we have to add 20 times the time taken for both the watches to beep together.

This time interval = 20 [tex]\times[/tex] 75 = 1500 seconds = 25 minutes.

The two watches will beep together for 20th time at 3:25 pm.  

Answer:

The answer to your question is (4, 6)

Step-by-step explanation:

Data

E ( 9 , 7 )

F ( - 1, 5)

Formula

[tex]Xm = \frac{x1 + x2}{2}[/tex]

[tex]Ym = \frac{y1 + y2}{2}[/tex]

Substitution and simplification

[tex]Xm = \frac{9 -1}{2}[/tex]

[tex]Xm = \frac{8}{2}[/tex]

      Xm = 4

[tex]Ym = \frac{7 + 5}{2}[/tex]

[tex]Ym = \frac{12}{2}[/tex]

      Ym = 6

Result

            (4 , 6)  

Answer:

Average rate of change [tex]=-35.7084[/tex]

Step-by-step explanation:

Given function is [tex]f(x)=480(0.3)^x[/tex] and we need to find average rate of change of the function from [tex]x=1\ to\ x=5[/tex].

Average rate of change [tex]=\frac{f(b)-f(a)}{b-a}[/tex]

So,

[tex]here\ b=5\ and\ a=1\\f(5)=480(0.3)^5\\=480\times0.00243=1.1664\\and\\f(1)=480(0.3)^1\\=480\times0.3=144[/tex]

Average rate of change

[tex]=\frac{f(b)-f(a)}{b-a}\\\\=\frac{f(5)-f(1)}{5-1}\\\\=\frac{1.1664-144}{5-1}\\\\=\frac{-142.8336}{4}= -35.7084[/tex]

Hence, average rate of change of the function [tex]f(x)=480(0.3)^x[/tex] over the intervel [tex]x=1\ to\ x=5[/tex] is [tex]=-35.7084[/tex].  

Answer:

-35.8074 is the correct answer

Step-by-step explanation:

In the context of renting a vacuum cleaner for an hourly rate plus a flat fee from a hardware store, the total rental cost can be represented as a linear function. If we consider the flat fee to be $31.50 and the hourly rate to be $32, the function would be f(x) = 31.50 + 32x, where x is the rental duration in hours.

The question pertains to a linear function, which is a fundamental concept in algebra and represents a straight line when graphed. Such a function is typically expressed in the form y = mx + b, where m and b are constants, y is the dependent variable, and x is the independent variable.

In the context of the question, the total rental cost for a vacuum cleaner from the hardware store can be a linear function if it involves both a fixed cost (the flat fee) and an hourly rate charge. Specifically, the flat fee can be represented as the constant b, which will be added to regardless of the number of hours the vacuum cleaner has been rented.

On the other hand, the hourly rate charge is the variable cost that alters in relation to the rental duration and can be shown as m times x. Thus, if we consider the flat fee to be $31.50 and the hourly rate to be $32 (as in the reference), the total rental cost function, f, can be formulated as follows: f(x) = 31.50 + 32x

In this equation, x stands for the number of hours the vacuum cleaner is rented. Consequently, by substituting the rental duration into the equation, it would be feasible to compute the total rental cost.

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Answer:

Therefore, HL theorem we will prove for Triangles Congruent.

Step-by-step explanation:

Given:

Label the Figure first, Such that

Angle ADB = 90 degrees,  

angle ADC = 90 degrees, and

AB ≅ AC

To Prove:

ΔABD ≅ ΔACD    by   Hypotenuse Leg theorem

Proof:

In  Δ ABD and Δ ACD

AB ≅ AC     ……….{Hypotenuse are equal Given}

∠ADB ≅ ∠ADC     ……….{Each angle measure is 90° given}

AD ≅ AD     ……….{Reflexive Property or Common side}

Δ ABD ≅ Δ ACD ….{By Hypotenuse Leg test} ......Proved

Therefore, HL theorem we will prove for Triangles Congruent.

Answer:

2 is the extraneous solution

Step-by-step explanation:

Given equation is

[tex]\frac{2x}{x-2} -\frac{11}{x} =\frac{8}{x^2-2x}[/tex]

Factor the denominator

[tex]\frac{2x}{x-2} -\frac{11}{x} =\frac{8}{x(x-2)}[/tex]

LCD is x(x-2), multiply all the fractions by LCD

[tex]2x \cdot x-11(x-2)=8[/tex]

[tex]2x^2-11x+22= 8[/tex], subtract 8 from both sides

[tex]2x^2-11x+14=0[/tex]

factor the left hand side

[tex]2x^2-7x-4x+14= 0[/tex]

[tex]x(2x-7)-2(2x-7)=0[/tex]

[tex](x-2)(2x-7)=0[/tex]

x-2=0, so x=2

2x-7=0,  [tex]x=\frac{7}{2}[/tex]

when x=2, then the denominator becomes 0 that is undefined

So 2 is the extraneous solution

The question is asking which statement is true regarding the potential extraneous solutions after solving an algebraic equation by multiplying both sides by the least common denominator. To determine if a solution is extraneous, it must be checked against the original equation. Without the specific manipulations made by Ren, we cannot assess the given options.

To solve the equation 2x x 2 11 x = 8 x 2 2x, Ren multiplied both sides by the least common denominator to eliminate the fractions and then used algebraic techniques to find the solutions for x. We know that when we have an equation of the form (ax + b)x = 0, there are two multiplicands, and we can set each equal to zero to solve for x. This leads to two solutions.

After solving, we need to check each solution by substituting it back into the original equation to confirm whether or not the solution is extraneous. An extraneous solution is one that does not satisfy the original equation after simplification. Checking is important as it ensures that the proposed solutions indeed make the original equation an identity, such as 6 = 6.

Without the specific equation after Ren's manipulations, we cannot evaluate the statements A, B, C, or D directly. However, we can understand that extraneous solutions arise when certain steps in solving an equation (like squaring both sides or multiplying by a variable expression) introduce results that are not true for the original equation.

Answer: [tex]24\ liters[/tex]

Step-by-step explanation:

Let be "x" the amount of gasoline in liters that the car's tank had at the beginning of the trip.

 1. In the first part of the trip the amount of gasoline the car used can be expressed as:

 [tex]\frac{2}{3}x[/tex]

2. After the first part of the trip, the remaining was:

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

3. In the second part of the trip the car used [tex]\frac{1}{2}[/tex] of the remaining. This is:

[tex](\frac{1}{3}x)(\frac{1}{2})=\frac{1}{6}x[/tex]

4. The total amount ot gasoline used in this trip was 20 liters.

5. Then, with this information, you can write the following equation:

[tex]\frac{2}{3}x+\frac{1}{6}x=20[/tex]

6. Finally, you must solve for "x" in order to find its value. This is:

[tex]\frac{2}{3}x+\frac{1}{6}x=20\\\\\frac{5}{6}x=20\\\\5x=120\\\\x=24[/tex]

Answer:

Step-by-step explanation:

The probability that you can turn your current $15,000 into $50,000. This means that the probability of success is 0.35. In terms of percentage, it is 0.35×100 = 35%

You have a 0.65 probability that fierce competition will drive you to ruin, losing all your money. This means that the probability of failure is 0.65. In terms if percentage, it is 0.65×100 = 65%

Looking at the percentage, entering the market will be too risky so I won't enter market since the chance of failing is very high compared to that of succeeding

Good evening ,

Answer:

(x-1) ; (x+3) and (x+5).

Step-by-step explanation:

Since  1 , -3 , -5 are roots of the polynomial function

then the factors of f are:

(x-1) ; (x+3) and (x+5).

:)

Answer: the measure of angle A is 12 degrees

Step-by-step explanation:

Let x represent the measure of angle A.

Let y represent the measure of angle B.

Let z represent the measure of angle C.

In triangle ABC, the measure of angle B is 60 more than A. This means that

y = x + 60

The measure of angle C is eight times the measure of A. This means that

z = 8x

Also, the sum of the angles in a triangle is 180 degrees. Therefore

x + y + z = 180 - - - - - - - - - 1

Substituting y = x + 60 and z = 8x into equation 1, it becomes

x + x + 60 + 8x = 180

10x + 60 = 180

10x = 180 - 60 = 120

x = 120/10 = 12

Answer:

Step-by-step explanation:

measure of A=x

∠C=8x

∠B=x+60

in a triangle sum of angles=180°

x+8x+x+60=180

10x=120

x=12

m∠A=12°

Answer:

[tex]\displaystyle 133\:square\:units[/tex]

Step-by-step explanation:

[tex]\displaystyle \frac{1}{2}hb = A, \frac{1}{2}bh = A, or\: \frac{hb}{2} = A[/tex]

For the two legs, no matter what you do, you can either take half of 19 [9½] and multiply it by 14, take half of 14 [7] and multiply it by 19, or you could multiply both 14 and 19 [266] then take of that.

I am joyous to assist you anytime.

Answer: 0.036756909

Step-by-step explanation:

Formula for Binomial probability distribution.

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]

, where x= number of success

n= total trials

p=probability of getting success in each trial.

According to the given information , we have

n= 10 , p= 0.30  and x= 6

Then, the required probability will be :

[tex]P(x=6)=^{10}C_6(0.3)^6(1-0.3)^{10-6}\\\\= \dfrac{10!}{6!(10-6)!}\times(0.3)^6(0.7)^4\\\\=\dfrac{10\times9\times8\times7\times6!}{6!4!}(0.3)^6(0.7)^4\\\\=(210)(0.000729)(0.2401)=0.036756909[/tex]

Hence, the required provability = 0.036756909

The probability of 6 successes given that a single success has a probability of 0.30 is given by the binomial distribution and P ( A ) = 0.03675 or 3.675 %

Given data ,

To find the probability of exactly 6 successes in 10 trials, with a probability of success (p) equal to 0.30, we can use the binomial probability formula:

P ( x ) = [ n! / ( n - x )! x! ] pˣqⁿ⁻ˣ

P(X = k) is the probability of getting exactly k successes,

n is the total number of trials,

k is the number of desired successes,

p is the probability of success for a single trial,

In this case, n = 10, k = 6, and p = 0.30. The binomial coefficient C(n, k) is calculated as:

P(n, k) = n! / (k! * (n - k)!)

Substituting the values into the formula, we have:

P(X = 6) = C(10, 6) x (0.30)⁶ * (1 - 0.30)⁽¹⁰⁻⁶⁾

Calculating the binomial coefficient:

C(10, 6) = 10! / (6! x (10 - 6)!)

= 10! / (6! x 4!)

= (10 x 9 x 8 x 7) / (4 x 3 x 2 x 1)

= 210

Substituting the values into the formula:

P(X = 6) = 210 x (0.30)⁶ (0.70)⁴

P ( X = 6 ) = P ( A ) = 210 ( 0.000729 ) ( 0.2401 )

P ( A ) = 0.036756909

Therefore, the probability of getting exactly 6 successes in 10 trials, with a probability of success of 0.30, is approximately 0.03675 or 3.675 %

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Answer:

Step-by-step explanation:

The standard form equation of a circle with radius r is expressed as

( x − h )^2 + ( y − k )^2 =r ^2 ,

where r represents the radius

h and k are the coordinates of the center of the circle C( h , k )

To determine the coordinates at the center of the circle, the midpoint formula would be used. It is expressed as

[(x1 + x2)/2 , (y1 + y2)/2]

Midpoint of the circle =

(6 - 4)/2 , (2 + 7)/2 = (1, 4.5)

h coordinate of the center = 1

k coordinate of the center = 4.5

r^2 = (x - h)^2 + (2 - k)^2

r^2 = (6 - 1)^2 + (2 - 4.5)^2

r^2 = 5^2 + (- 2.5)^2 = 25 + 6.25

r^2 = 31.25

Substituting r^2 = 31.25, h = 1 and k = 4.5 into (x − h )^2 + ( y − k )^2 = r^2, the standard equation of the circle becomes

(x − 1 )^2 + ( y − 4.5 )^2 = 31.25

Final answer:

The standard form of the equation is (x - 1)² + (y - 4.5)² = 31.25.

Explanation:

The student is asking for the standard form equation of a circle given the endpoints of a diameter. To find the center of the circle, we average the x-coordinates and the y-coordinates of the endpoints, resulting in the center coordinates (1, 4.5).

The radius can be calculated using the distance formula between the center and one of the endpoints, which gives us √((6-1)²+(2-4.5)²) = √(5²+2.5²) = √(25+6.25) = √31.25.

The radius in its simple form is √31.25.

The standard form of the equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Substituting the values we have, the equation becomes (x - 1)² + (y - 4.5)² = (√31.25)², which simplifies to

(x - 1)² + (y - 4.5)² = 31.25.