State the horizontal asymptote is the rational function. F(x)=x+9/x^2+8x+8
None
Y=x
Y=9
Y=0

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: 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.

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:

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:

He needs to walk 41 mins or 2,76 miles to burn 150 calories.

If he increases the speed of walking or eats snack with less calories, he will spend less time for walking.

Step-by-step explanation:

The person is burning

[tex]5,1*60=306[/tex]

calories per hour.

He needs to burn 150 calories plus 60 calories that comes from the snack. In total 210 calories to burn.

[tex]210/306=0,69[/tex]

He needs to walk 0,69 hours (aprrox. 41 mins) to burn 210 calories

[tex]0,69*4=2,76[/tex]

In total he need to walk 2,76 miles to burn 210 calories.

Final answer:

A 150-pound person must walk approximately 17.65 minutes at 4 mph to burn at least 150 calories, after accounting for a 60-calorie snack. To decrease walking time, they can increase activity intensity or choose lower-calorie snacks.

Explanation:

To calculate the time required for a 150-pound person to burn at least 150 calories at a rate of 5.1 calories per minute while walking at 4 miles per hour, we must account for the 60-calorie snack they consumed. We first subtract the 60 calories from the 150-calorie goal, which leaves 90 calories to be burned through exercise. Dividing 90 calories by the 5.1 calories burned per minute gives us the time needed in minutes.

Calculation:
Total calories to burn = 150 - 60 (from snack) = 90 calories
Calories burned per minute = 5.1
Time (in minutes) = Total calories to burn ÷ Calories burned per minute = 90 ÷ 5.1 = 17.65 minutes

So, the person must walk approximately 17.65 minutes to burn at least 150 calories, minus the calories from the snack. To reduce the time spent walking, the person could increase their walking speed, engage in a more vigorous exercise, or consume a lower-calorie snack.

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:

-45

Step-by-step explanation:

2/3x - 11 = x + 4

2/3x = x + 15

-1/3x = 15

x = -45

Answer:

-45.

Step-by-step explanation:

2/3 x - 11 = x + 4

2/3 x - x = 4 + 11

-1/3x = 15

x = 15 * -3

x = -45.

Answer:

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

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

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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:

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: [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]

Kira runs a total of 144 miles in 9 weeks.

Step-by-step explanation:

Given,

Distance covered on Monday = 2.5 miles

Distance covered on Wednesday = 5.75 miles

Distance covered on Friday = 7.75 miles

Total distance covered per week = 2.5+5.75+7.75 = 16 miles

Total distance in 9 weeks = Distance per week * 9

Total distance in 9 weeks = 16*9 = 144 miles

Kira runs a total of 144 miles in 9 weeks.

Keywords: addition, multiplication

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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:

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: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:

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)  

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

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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.

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Answer:

a) If we design the experiment on this way we can check if we have an improvement with the method used.

We assume that we have the same individual and we take a value before with the normal impaired condition and the final condition is the normal case.  

b) [tex]-0.96-2.306\frac{0.359}{\sqrt{9}}=-1.24[/tex]  

[tex]-0.96+2.306\frac{0.359}{\sqrt{9}}=-0.69[/tex]

The 95% confidence interval would be given by (-1.24;-0.69)

Step-by-step explanation:

Part a

If we design the experiment on this way we can check if we have an improvement with the method used.

We assume that we have the same individual and we take a value before with the normal impaired condition and the final condition is the normal case.

Part b

For this case first we need to find the differences like this :

Normal, Xi 4.47 4.24 4.58 4.65 4.31 4.80 4.55 5.00 4.79

Impaired, Yi 5.77 5.67 5.51 5.32 5.83 5.49 5.23 5.61 5.6

Let [tex]d_i = Normal -Impaired[/tex]

[tex] d_i : -1.3, -1.43, -0.93, -0.67,-1.52, -0.69, -0.68, -0.61, -0.81[/tex]

The second step is calculate the mean difference  

[tex]\bar d= \frac{\sum_{i=1}^n d_i}{n}=-0.96[/tex]

The third step would be calculate the standard deviation for the differences, and we got:

[tex]s_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n-1} =0.359[/tex]

The confidence interval for the mean is given by the following formula:  

[tex]\bar d \pm t_{\alpha/2}\frac{s_d}{\sqrt{n}}[/tex] (1)  

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:  

[tex]df=n-1=9-1=8[/tex]  

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,8)".And we see that [tex]t_{\alpha/2}=2.306[/tex]  

Now we have everything in order to replace into formula (1):  

[tex]-0.96-2.306\frac{0.359}{\sqrt{9}}=-1.24[/tex]  

[tex]-0.96+2.306\frac{0.359}{\sqrt{9}}=-0.69[/tex]  

So on this case the 95% confidence interval would be given by (-1.24;-0.69)

Random selection for first testing condition (impaired or unimpaired) was used to avoid order effects. The confidence interval on whether braking times under impaired and unimpaired conditions were significantly different can be determined using a paired t-test and if the interval includes zero, we can say that there is no significant difference.

(a) Random selection of whether the student had unimpaired or impaired vision was a good idea because it helps to prevent any order effects. An order effect occurs if the order in which the tests are performed can influence the results. For example, if the unimpaired test was always done first, the driver might be more cautious in the second test as they have learned from the first test.

(b) The confidence interval for a difference between two means (in this case the braking times) can be calculated with a paired t-test. We will compare the average of differences (impaired vision braking time - normal vision braking time) to zero, assuming that they follow a normal distribution.

The formula to calculate the confidence interval for paired data is:

(Avg(D) - (t * StdDev(D) / sqrt(n)), Avg(D) + (t * StdDev(D) / sqrt(n)))

Where Avg(D) is the average of the differences, StdDev(D) is the standard deviation of the differences, n is the sample size (9 in this case), and t is the t-value from the t-distribution table (which will be 2.306 considering 95% confidence for 8 degrees of freedom).

After calculating you'll get the confidence interval for the differences. If this interval includes zero, we can say there is no significant difference for the braking time under impaired and unimpressed conditions using the 95% confidence interval.

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Answer:

With the given zeros x=-2,1,4 the polynomial function is [tex]x^3-3x^2-6x-8[/tex]

Step-by-step explanation:

Given zeros are x=-2,1,4

Now to find the polynomial function:

With the given zeros we can write it as below:

x=-2  implies that x+2=0

x=-1  implies that  x-1=0

x=4  implies that  x-4=0

Then we can the zeros or factors by (x+2)(x-1)(x-4)

Now expanding the zeros or factors:

[tex](x+2)(x-1)(x-4)[/tex]

[tex](x+2)(x-1)(x-4)=(x^2-x+2x-2)(x-4)[/tex]  ( multiply each term with each term of another factor)

[tex]=(x^2+x-2)(x-4)[/tex] ( adding the like terms)

[tex]=x^3-4x^2+x^2-4x-2x+8[/tex]   ( multiply each term with each term of another factor)

[tex]=x^3-3x^2-6x+8[/tex]  ( adding the like terms)

[tex](x+2)(x-1)(x-4)=x^3-3x^2-6x+8[/tex]

Therefore the polynomial function is [tex](x+2)(x-1)(x-4)=x^3-3x^2-6x+8[/tex]

With the given zeros x=-2,1,4 the polynomial function is [tex]x^3-3x^2-6x-8[/tex]

Answer:

Option D.

Step-by-step explanation:

Form the given line plot, first we need to find the data set. So, our data set is

24, 26, 26, 26, 27, 27, 27, 28, 28, 28, 28, 28, 30, 30, 30, 32, 32, 32, 35

Divide the data in two equal parts.

(24, 26, 26, 26, 27, 27, 27, 28, 28), 28, (28, 28, 30, 30, 30, 32, 32, 32, 35)

Divide each of the parenthesis in two equal parts.

(24, 26, 26, 26), 27, (27, 27, 28, 28), 28, (28, 28, 30, 30), 30, (32, 32, 32, 35)

Now, we get

Minimum value = 24

First quartile = 27

Median = 28

Third quartile = 30

Maximum value = 35

It means the box lies between 27 and 30. The line inside the box at 28. Left point of the line isi 24 and right point of the line 35.

This description represented by the box plot in option D.

Hence, the correct option is D.

Answer:

Outer perimeter of the resulting 2-dimensional shape will be 18.84 units

Step-by-step explanation:

From the figure, we could see

AE =  FB = 1

And  minor arc EF  

=> [tex]\frac{2 \pi (1) }{6}[/tex]  

=>[tex]= \frac{2 \pi}{ 6}[/tex]

=>[tex]= \frac{ \pi}{ 3}[/tex]

So by symmetry the perimeter  is

=>[tex]3(2 + \frac { \pi}{3} )[/tex]

=>[tex]3(\frac {6 \pi}{3} )[/tex]

=>[tex]6 \pi[/tex]

=> [tex]6 \times 3.14[/tex]

=> 18.84 units

Answer:

6+pi or 9.14

Step-by-step explanation:

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:

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:

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:

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: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]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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Answer:

a. x = primes.difference(odds)

Step-by-step explanation:

Given the list of numbers primes and odds, the list x is made subtracting odds to primes. To do that in a object oriented language you use: x = primes.difference(odds) which is equivalent to x = primes  - odds

The correct line of code that will result in x containing {1} is option b: x = odds.difference(primes). This provides the set of elements in the 'odds' set that are not in the 'primes' set, which is {1}.

The question revolves around the concept of set operations in mathematics. Specifically, it focuses on the difference and intersection operations between two sets named primes and odds. To find the set containing only the number {1}, we should look for the difference between the odds and primes sets because 1 is in the odds but not in the primes.

Answer option a, x = primes.difference(odds), would result in a set that contains elements present in primes but not in odds, which would be {2}. However, that's not what we are looking for.

Answer option b, x = odds.difference(primes), is correct. It would result in a set containing elements that are in odds but not in primes, which is exactly {1}.

Answer options c and d, which refer to the intersection of the two sets, would result in the set {3, 5}, which are elements common to both primes and odds, and therefore not the correct answer.

For visual representation, you could draw a Venn diagram with two circles, one for primes containing {2, 3, 5, 7} and another for odds containing {1, 3, 5, 7}. The number 1 would be in the part of the odds circle that does not overlap with the primes circle.

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