A student bought a juice pouch forn$2.50 and 3 bags of chips. The total cost was $5.05.Write and solve an equation to determine the cost of a bag of chips,c

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: total Distance = 60km

Time = 4.5hrs

Speed = 60/4.5

13⅓km/hr

Step-by-step explanation:

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:

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:

The product of 'x' and 'y' is [tex]\boxed 8[/tex].

Step-by-step explanation:

Given:

[tex]\log_{5\sqrt5}125=x\\\\\log_{2\sqrt2}64=y[/tex]

We need to determine the product of 'x' and 'y'.

Using the following logarithmic property:

[tex]\log_ab=\frac{\log b}{\log a}[/tex]

Here, [tex]a=5\sqrt5\ and\ 2\sqrt2[/tex]

[tex]b=125\ and\ 64[/tex]

So, [tex]log_{5\sqrt5}125=\frac{\log 125}{\log 5\sqrt{5}}\\\\log_{5\sqrt5}125=\frac{\log 5^3}{\log 5\times5^{1/2}}.......[\sqrt5=5^{1/2}][/tex]

[tex]log_{2\sqrt2}64=\frac{\log 64}{\log 2\sqrt{2}}\\\\log_{2\sqrt2}64=\frac{\log 2^6}{\log 2\times2^{1/2}}.......[\sqrt2=2^{1/2}][/tex]

Now, we use another property of log and exponents.

[tex]\log a^m=m\log a\\a^m\times a^n=a^{m+n}[/tex]

[tex]log_{5\sqrt5}125=\frac{3\log 5}{\log 5^{1+{1/2}}}=\frac{3\log 5}{\log 5^{\frac{3}{2}}}=\frac{3\log 5}{\frac{3}{2}\log 5}=2\\\\\\\\log_{2\sqrt2}64=\frac{6\log 2}{\log 2^{1+{1/2}}}=\frac{6\log 2}{\log 2^{\frac{3}{2}}}=\frac{6\log 2}{\frac{3}{2}\log 2}=\frac{12}{3}=4[/tex]

So, [tex]x=2\ and\ y=4[/tex]

The product of 'x' and 'y' = [tex]2\times 4=8[/tex]

Therefore, the product of 'x' and 'y' is 8.

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:

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:

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) P({a, b, {a, b}})=[tex] 2*2*2=8[/tex]

b) P({∅, a, {a}, {{a}}})=[tex] 2*2*2*2=16[/tex]

c) P(P(∅))=P(P{∅})=2

Step-by-step explanation:

Previous concepts

For this case we need to remember the concpts of subset and the power set.

A is a subset of B if every element of A is an eelement if B and we write [tex]A \subseteq B[/tex]

The power set of a set R is th set of all the possible subsets of R and we writ this as P(R)

Solution to the problem

Part a

P({a, b, {a, b}}) as we can see this set contains 3 elements a,b and {a,b}

And since the power set contains all the possible subsets of the elements. For this case for each element we have 3-1 = 2 options in order to combine, then the number of possible subsets would be equal to the product of possible options and we got:

P({a, b, {a, b}})=[tex] 2*2*2=8[/tex]

Part b

P({∅, a, {a}, {{a}}}) as we can see this set contains 4 elements ∅,a,{a} and {{a}}

And since the power set contains all the possible subsets of the elements. For this case for each element we have 3-1 = 2 options in order to combine, then the number of possible subsets would be equal to the product of possible options and we got:

P({∅, a, {a}, {{a}}})=[tex] 2*2*2*2=16[/tex]

Part c

P(P(∅)) for this case we need to remember that P(∅) have all the possible subsets empty, but the power set for the empty set is also empty.

P(∅)={∅}

And we see that this last one set have just one element.

For this special case we have again 2 options since we can have the element in the subset or the element that is not on the subset, so then we have this:

P(P(∅))=P(P{∅})=2

The number of elements of the set are 8, 16, and 2.

It should be noted that the objects in a set are known as the elements or members of the set.

The number of elements in the set will be calculated thus:

P({a, b, {a, b} = 2³ = 2 × 2 × 2 = 8

P({∅, a, {a}, {{a}}) = 2⁴ = 16

P(P(∅)) = 2¹ = 2

In conclusion, the correct options are 8, 16, and 2.

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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:

40%

Step-by-step explanation:

135 of 225 is equal to 60%. Therefore, the remaining percent, which should equal to 100, did not wear spirit shirts. 100 - 60 = 40. Therefore, 40% of students did not wear spirit shirts on Friday.

The percentage of students who did NOT wear spirit shirts on Friday is 40%

To find the percentage of students who did NOT wear spirit shirts on Friday, we subtract the number of students who wore spirit shirts from the total number of students and divide by the total number of students.

Then, multiply by 100 to get the percentage.

Number of students who did NOT wear spirit shirts = Total number of students - Number of students who wore spirit shirts = 225 - 135 = 90.

Percentage of students who did NOT wear spirit shirts = (90 / 225) x 100 = 40%

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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: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]\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.

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:

-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:

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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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:

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:

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)  

The power developed in the first crane will be twice the power developed in the second crane.

Given information:

Two construction cranes are used to lift identical 1200-kilogram loads of bricks at the same vertical distance.

The first crane lifts the bricks in 20 seconds and the second crane lifts the bricks in 40 seconds.

Now, the load is the same and the vertical lift is also the same. So, the work done W by both the lifts will also be the same.

The power developed by first crane will be,

[tex]P_1=\dfrac{W}{20}[/tex]

The power developed by second crane will be,

[tex]P_2=\dfrac{W}{40}[/tex]

So, the ratio of power developed in two cranes will be,

[tex]P_1:P_2=\dfrac{W}{20}:\dfrac{W}{40}\\P_1:P_2=2:1[/tex]

Therefore, the power developed in the first crane will be twice the power developed in the second crane.

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Final answer:

The power developed by the first crane is twice as great as the power developed by the second crane because it lifts the bricks in half the time.

Explanation:

To compare the power developed by the first crane with the power developed by the second crane, we can use the formula:

Power = Work / time

Since both cranes are lifting identical 1200-kilogram loads of bricks the same vertical distance, the work done by both cranes is the same. Therefore, the power developed by the first crane is twice as great as the power developed by the second crane. This is because the first crane lifts the bricks in 20 seconds, which is half the time it takes the second crane to do the same work in 40 seconds.

Answer:

we have 448 ways of putting them.

Step-by-step explanation:

For the first piece we have no restrictions, so we have 8*8 = 64 ways of puting it. For the second piece we have 7 ways to put it in the same row and 7 ways of put it in the same column, so we have 14 ways.

This gives us a total of 14*64 = 896 ways.

However, since the pieces are indistinguishable, we need to divide the result by two, because we were counting each possibility twice, (if we swap the pieces, then it counts as the same way), thus we have 896/2 = 448 of putting two pieces on the board.

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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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 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:

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:

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:

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.