senior management of a consulting services firm is concerned about a growing decline in the firm's weekly number of billable hours. the firm expects each professional employee to spend at least 40 hours per week on work. in an effort to understand this problem better, management would like to estimate the standard deviation of the number of hours their employees spend on work-related activities in a typical week. rather than reviewing the records of all the firm's full-time employees, the management randomly selected a sample of size 51 from the available frame. the sample mean and sample standard deviations were 48.5 and 7.5 hours, respectively. construct a 88% confidence interval for the mean of the number of hours this firm's employees spend on work-related activities in a typical week. place your lower limit, in hours, rounded to 1 decimal place, in the first blank. for example, 6.7 would be a legitimate entry. place your upper limit, in hours, rounded to 1 decimal place, in the second blank. for example, 12.3 would be a legitimate entry.

Answer:

A) Yes, Rolle's Theorem can be applied!

Step-by-step explanation:

Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.

Here, for our continuous function [tex]f(x)=-x^2+2x[/tex] over the closed interval [tex][0,2][/tex], we can tell that the function is clearly differentiable over the interval [tex](0,2)[/tex] as [tex]f'(x)=-2x+2[/tex], so we'll need to check if [tex]f(0)=f(2)[/tex]:

[tex]f(0)=-0^2+2(0)=0\\f(2)=-(2)^2+2(2)=-4+4=0[/tex]

Next, we'll need to check if f'(x) = 0 for some x within the closed interval:

[tex]f'(x)=-2x+2=0\\-2x+2=0\\-2x=-2\\x=1[/tex]

As x=1 is contained in [0,2] and the previous conditions were met, Rolle's Theorem can be applied!

Answer:

20%

Step-by-step explanation:

The statement that every diagonal of a trapezoid divides it into two congruent triangles is false.

Two or more triangles are said to be congruent if and only if sides and angles of one triangle is equal to the corresponding sides and angles of another triangle.

A trapezoid is a quadrilateral which has at least one pair of parallel sides.

The sides may or may not be equal.

Consider an isosceles trapezium, ABCD, given below which has two parallel opposite sides and the other pair of sides are equal.

If we draw a diagonal, AC, we have two triangles ABC and ADC.

AC = AC (common side)

AD = BC (given)

But AB ≠ CD

Hence the triangles cannot be congruent.

Hence the given statement is false.

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Answer: the only y-intercept is 8

Step-by-step explanation:

One way I remembered equations when I was doing algebra with only an x or a y was HOY VUX.

H - horizontal
O = slope
Y = the variable in the line equation

V  =vertical
U = undefined slope
X = the variable in the line equation

As we can see this is a horizontal line so there will be no x-intercepts.

6y = 48

y = 8

the only y-intercept is 8

The cross product of two vectors v and w is explicitly defined as ijkv jwk and also transformed as a dual vector under rotation.

The cross product of two vectors v and w is defined as:

v × w = (v2w3 - v3w2)i - (v1w3 - v3w1)j + (v1w2 - v2w1)k

To show that this cross product transforms as a dual vector under rotation, consider a rotation matrix R that transforms a vector u to u'. The transformation of u can be written as:

u' = Ru

We can write the cross product of v and w in terms of their components as:

v × w = (v1, v2, v3) × (w1, w2, w3) = (v2w3 - v3w2, -(v1w3 - v3w1), v1w2 - v2w1)

To see how this changes under rotation, we can use the matrix-vector product to transform v × w:

(v × w)' = R(v × w) = (Rv × Rw)

Rotation matrix R in terms of its components:

R = (r11, r12, r13; r21, r22, r23; r31, r32, r33)

(v × w)' = (r11r12r13r21r22r23r31r32r33(v2w3 - v3w2) - (v1w3 - v3w1) + (v1w2 - v2w1))

Since the Levi-Civita symbol is a symbol and does not transform under rotation, we have:

(v × w)' = det(R)(v × w)

This shows that the cross product of two vectors v and w transforms as a dual vector under rotation, and that the transformation is proportional to the determinant of the rotation matrix R.

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The Emily soccer team has the higher ratio of wins to losses.

Ratio is defined as the relationship between two quantities where it tells how much one quantity is contained in the other.

The ratio of a and b is denoted as a : b.

Given that,

Ratio of win to loss of Emily soccer team = 9 : 12

Ratio of win to loss of Grant soccer team = 10 : 15

9 : 12 = 9 / 12 = (9 × 5) / (12 × 5) = 45 / 60

10 : 15 = 10 / 15 = (10 × 4) / (15 × 4) = 40 / 60

If 60 games are losses, then 45 games are wins for Emily soccer team.

If 60 games are losses, then 40 games are wins for Grant soccer team.

Higher ratio is for Emily soccer team.

Hence higher ratio of wins to losses is for Emily soccer team.  

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The side lengths of the given squares are 11 cm and 14 cm

A square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides.

Given that, the sides of two squares differ by 3cm and the sum of their areas is 317cm², we need to find the length of the sides of the two squares.

Let the sides of the squares be x and y, we know that the area of a square is their side's square.

Therefore, establishing the system of equations,

x-y = 3

x = y+3.......(i)

x²+y² = 317....(ii)

Put eq(i) in eq(ii)

(y+3)²+y² = 317

y²+9+6y+y² = 317

2y²+6y-308 = 0

On factorizing, we get,

y = 11 and y = -14

Since, the length can not be negative so, ignore -14

Put y = 11 in eq(i)

x = 11+3

x = 14

Therefore, the side lengths of the given squares are 11 cm and 14 cm

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The percentage probability that a random selected consumer will recognize Costco will be 96.3%.

Given, the total number of consumers who recognized the brand was 1036. And the total number of consumers who didn't recognize the brand was 40. Therefore, the total number of our sample will be :

⇒1036 + 40

⇒1076

Now, we know that probability(P) of an event = f/n

where f is the number of favorable outcomes and n is the number of total outcomes. Here, number of favorable outcome is 1036.

Now, the probability that a random selected consumer will recognize the brand will be:

⇒ P = f/n

⇒ P = 1036/1076

⇒P = 0.96282

Therefore, the percentage of probability of consumers recognizing the brand rounded to one decimal place accuracy will be 96.3%.

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The length of the curve defined by from the point (-3, 2136) and (3, 2238), by use of distance formula is approximately 102.17 units.

The formula for the distance between two points (x1, y1) and (x2, y2) is given by:

[tex]d = \sqrt{((x2 - x1)^2 + (y2 - y1)^2)[/tex]

In this case, we have:

x1 = -3, y1 = 2136

x2 = 3, y2 = 2238

Substituting these values into the formula, we get:

[tex]d = \sqrt{((3 - (-3))^2 + (2238 - 2136)^2)[/tex]

[tex]= \sqrt{(6^2 + 102^2)[/tex]

= [tex]\sqrt{(10440)[/tex]

≈ 102.17

Therefore, the length of the curve defined by the points (-3, 2136) and (3, 2238) is approximately 102.17 units.

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_____The given question is incorrect, the correct question is given below:

To find the length of the curve defined by from the point (-3,2136) to the point (3,2238), you'd have to compute by using distance formula.

You can solve this by using the addition and multiplication properties of equality

the original equation is:

a divided by 6 - 11 = 25

then you can add 11 to both sides to get rid of it (addition property of equality)

a divided by 6 = 36

then you use the multiplication property of equality

a = 216

To prove that ΔAED ≅ ΔCFD, we will use the two given angle equalities and the fact that D is the midpoint of AC:

Given: D is the midpoint of AC, ∠AED ≅ ∠CFD, and ∠EDA ≅ ∠FDC

To prove: ΔAED ≅ ΔCFD

Proof:

Thus, we have proved that ΔAED ≅ ΔCFD.

The probability of the complement of event E[tex](E^c)[/tex] is 7/12.

a. Outcomes in [tex]E^c[/tex] = {3, 9, 10, 11, 12, 13, 14}

b. The probability of the complement of event E[tex](E^c)[/tex] is the probability of all outcomes in S that are not in E. P[tex](E^c)[/tex] = 1 - P(E).

The probability of E is calculated by counting the number of elements in E and dividing by the total number of elements in the sample space S.

P(E) = 5/12

Therefore, P[tex](E^c)[/tex] = 1 - P(E) = 1 - (5/12) = 7/12

The probability of the complement of event E [tex](E^c)[/tex] is 7/12.

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The value of the expression (1/2)^3 include the following: D. 1/8.

In Mathematics, an exponent can be defined as a mathematical operation that is used in conjunction with an algebraic expression to raise a quantity to the power of another and it is generally written as;

bⁿ

Where:

the variables b and n represent numerical values or an algebraic expression.

From the information provided, we can logically deduce the following as the only true statement:

Expression = (1/2)^3

Expression = 1/2 × 1/2 × 1/2

Expression = 1/8

In conclusion, the numerical value 3 represents the power of the given expression.

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Complete Question:

What is the value of the expression (1/2)^3?

Responses

19

1/9

18

1/8

16

1/6

32

A and B are not independent since the result of one toss affects the second toss. Specifically, if the first toss result is a head (A), then the second toss will be more likely to result in a head (B).

A and B are conditionally independent given C since the probability of getting head(A) for the first toss does not depend on the type of coin selected (C).

P(A|C) = 1/2, P(B|C) = 1/2, P(A∩B|C) = 1/4, P(A) = 1/2, P(B) = 3/4, and P(A∩B) = 1/4. Therefore, given the information that a regular coin has been selected (C), the probability of A and B occurring independently is 1/4. This suggests that A and B are conditionally independent given C.

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The depreciated price of the car in the year 2022 is $148.47

Depreciation in economics is a measure of the amount of value an asset loses from influential factors affecting its market value.

Given that, in 1960 the average price of a new car was $2752. Between 1960 and 2022 car prices have decreased by an average of 4.6% per year.

We need to calculate the average price of a new car in 2022​

The question is depreciated price of the car,

Depreciated price =

A = P(1-r)ⁿ

A = final amount

P = initial amount

r = 4.6%

n = time

Here, the time = 2022 - 1960 = 62 years

Therefore,

A = 2752(1-0.046)⁶²

A = 2752(0.954)⁶²

A = 148.47

Hence, the depreciated price of the car in the year 2022 is $148.47

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The final volume of a system with initially of 2 liters if the pressure is tripled and the temperature is tripled, is 11.57 liters.

The combined gas law equation relates the initial and final conditions of a system of gas by considering the effect of changes in pressure, volume, and temperature:

P1 * V1 / T1 = P2 * V2 / T2

Given the initial conditions of the system:

V1 = 2 liters

P1 = 1 atm

T1 = 273 K (room temperature)

And the changes in pressure and temperature:

P2 = 3 * P1 = 3 atm

T2 = 3 * T1 = 819 K

We can calculate the final volume of the system:

V2 = (P1 * V1 / T1) * (T2 / P2) = (1 atm * 2 liters / 273 K) * (819 K / 3 atm) = approximately 11.57 liters

So the final volume of the system would be approximately 11.57 liters if the pressure is tripled and the temperature is tripled from the initial conditions.

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Answer:

£16.38

Step-by-step explanation:

To calculate the cost of 180 eggs during the sale, we first need to determine the cost of a single egg and then apply the 30% discount.

The initial cost of 20 eggs is £2.60. To find the cost of a single egg, we divide the total cost by the number of eggs:

Now, we need to apply the 30% discount to the cost of a single egg:

The cost of a single egg during the sale is £0.091. To find the cost of 180 eggs, we multiply the cost of a single egg by the number of eggs:

Therefore, the cost of 180 eggs during the sale is £16.38.

________________________________________________________

To solve this problem, we can use the formula for calculating the quarterly payment of a loan with compound interest:

Payment = (r * P) / (1 - (1 + r)^(-n))

where:

r = quarterly interest rate (12% / 4 = 0.03)

P = principal amount (remaining debt after down payment) = $150,000

n = total number of quarterly payments (10 years * 4 quarters per year = 40)

(1) Calculation of Quarterly Payment:

Payment = (0.03 * 150,000) / (1 - (1 + 0.03)^(-40)) = $2,365.87 (rounded to the nearest cent)

Therefore, the quarterly payment is $2,365.87.

(2) Calculation of Total Amount of Payment:

Total amount of payment = (number of payments) * (quarterly payment)

number of payments = 10 years * 4 quarters per year = 40

Total amount of payment = 40 * 2,365.87 = $94,634.80

Therefore, the total amount of payment is $94,634.80.

(3) Calculation of Total Amount of Interest Paid:

Total amount of interest paid = total amount of payment - principal amount

Total amount of interest paid = $94,634.80 - $150,000 = -$55,365.20

Note that the result is negative because the down payment ($50,000) exceeds the total amount of interest paid over the life of the loan. Therefore, in this case, the total amount of interest paid is $0, and the man ends up paying a total of $100,000 ($50,000 down payment + $50,000 in quarterly payments).

The number of units manufactured at the cost of $8200 will be 95.

A quadratic equation is a polynomial with a degree of 2 or the maximum power of the variable is 2 in quadratic equations. It has two solutions as its maximum power is 2.

Given that the cost, C(x), for manufacturing x units of a certain product is

C(x) = x² - 9x + 30.

The number of units will be calculated as:-

C(x) = x² - 9x + 30

8200 = x² - 9x + 30

x² - 9x - 8170 = 0

Solve the equation as below,

x² - 9x - 8170 = 0

x² - 95x + 86x - 8170 = 0

x ( x - 95 ) + 86 ( x - 95 ) = 0

( x - 95 ) ( x + 86 ) = 0

x = 95 units

Therefore, the number of units manufactured at the cost of $8200 will be 95.

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a) We cannot determine the probability

b) The mean of x - 55 is $28 - $55 = -$27, and the standard deviation of x - 55 is $0.45.

c) By the central limit theorem, the sampling distribution of x^-x will be approximately normal if the sample size is large enough.

d)  The probability that the average fee paid by the sample of households exceeds $55 is very close to 0.

a. We cannot determine the probability that the amount a randomly selected household pays for access to the internet exceeds $55 because we don't have information about the distribution of individual payments, only the mean and standard deviation of the population.

b. The mean of the sampling distribution of x is equal to the population mean, which is $28. The standard deviation of the sampling distribution of x is equal to the population standard deviation divided by the square root of the sample size, which is $10/sqrt(500) ≈ $0.45. Therefore, the mean of x - 55 is $28 - $55 = -$27, and the standard deviation of x - 55 is $0.45.

c. By the central limit theorem, the sampling distribution of x^-x will be approximately normal if the sample size is large enough. In this case, since the sample size is 500 and greater than 30, we can assume that the sampling distribution is approximately normal.

d. To find the probability that the average fee paid by the sample of households exceeds $55, we need to standardize the value of x using the sampling distribution of x^-x and then find the probability of a z-score greater than or equal to:

z = (55 - 28)/0.45 ≈ 60. The probability of a z-score greater than or equal to 60 is extremely small, as the normal distribution tails off quickly. Therefore, the probability that the average fee paid by the sample of households exceeds $55 is very close to 0.

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The shaded region represents the solution set of the given inequality.

An inequality is used to make comparisons between the numbers or expressions. For example -

2x + 4 > 5

4x + 6 > 2

Given is the linear inequality as given below -

y < - x + 1

The graph of the inequality is attached. The shaded region represents the solution set of the given inequality. This means that all the coordinate points on the shaded graph satisfy the inequality -

y < - x + 1

Therefore, the shaded region represents the solution set of the given inequality.

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The required one point and two non-parallel vectors are (0, -3, 0) and (1, 0, 0), (0, 0, 1).

A quantity or phenomena with independent qualities for both magnitude and direction is called a vector. The term can also refer to a quantity's mathematical or geometrical representation. Velocity, momentum, force, electromagnetic fields, and weight are a few examples of vectors in nature.

One point on the plane y=-3 is (0, -3, 0).

Two non-parallel vectors that are parallel to the plane y=-3 can be obtained by taking any two vectors whose y-coordinate is 0,

since any vector with a y-coordinate of 0 lies in the x-z plane, which is parallel to the y=-3 plane.

For example, the vectors (1, 0, 0) and (0, 0, 1) are non-parallel vectors that are parallel to the plane y=-3.

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The length of the diagonal of the rectangle is 11.70 in.

A diagonal of a polygon is a line segment that joins two vertices of the polygon which are not already joined by an edge of the polygon.

Since, there is no information about which polygon's diagonal length we are asked to find, so let us consider that the given side lengths are of a rectangle,

Now, please refer to the figure attached in the solution,

We know that, the diagonal of a rectangle divides it into two right triangles,

Here our rectangle is ABCD and the diagonal AC divides it into rt. Δ ABC and rt. Δ ADC,

Considering the rt. Δ ABC,

The diagonal AC is acting as the hypotenuse of rt. Δ ABC,

So, using Pythagoras theorem,

AC² = AB² + BC²

AC = √4²+11²

AC = √121+16

AC = 11.70

Since, the diagonals of a rectangle are congruent so, AC = BD = 11.70

Hence, the length of the diagonals of the rectangle is 11.70 in.

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The car could drive approximately 293.04 miles on a full tank of gas. Drag and drop the number "293.04" into the response area to complete the statement.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that contains variables, numbers, and mathematical operations such as addition, subtraction, multiplication, and division. Algebraic expressions are used to represent mathematical relationships and can be used to solve equations and perform calculations.

We can use the given information to find the car's fuel efficiency in miles per gallon (mpg) as follows:

mpg = miles driven / gallons of gas used

mpg = 249.48 miles / 12.6 gallons

mpg ≈ 19.8

This means that the car can travel approximately 19.8 miles on one gallon of gas. To find out how far the car could drive on a full tank of 14.8 gallons of gas, we can multiply the tank capacity by the car's fuel efficiency:

miles on full tank = mpg * gallons in full tank

miles on full tank = 19.8 mpg * 14.8 gallons

miles on full tank ≈ 293.04 miles

Therefore, the car could drive approximately 293.04 miles on a full tank of gas. Drag and drop the number "293.04" into the response area to complete the statement.

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The solution of the inequality is x > -2 and x < 7.

Inequality Notation: -2 < x < 7.

The graph is attached.

An inequality is a relationship that makes a non-equal comparison between two numbers or other mathematical expressions e.g 2x > 4.

-5x - 7 < 3

-5x < 3 + 7

-5x < 10

x > 10/(-5)

x > -2

-5x - 7 > -42

-5x > -42 + 7

-5x > -35

x < -35/(-5)

x < 7

We have x > -2 and x < 7.

x > -2 can be written as -2 < x.

Inequality Notation: -2 < x < 7

This implies that x is in the range of -2 and 7. The graph is attached.

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The system of linear equations x = 4 and y = -x -1 has a unique solution.

A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation.

Given is a graph of lines,

We need to find the number of solutions do the lines x = 4 and y = -x-1 have,

In the graph of the lines x = 4 and y = -x -1, intersecting at a point, we know that the lines which intersects at a single point will have only one solution.

Also,

The system of equation has the solution at the point where the line intersects,  

By observing the graph it can be concluded that the graph of x = 4 and y = - x - 1 intersect only at one point i.e (4, -5).

Therefore, the solution of the given system of equations is (4, -5)

Hence, the system of linear equations x = 4 and y = -x -1 has a unique solution.

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let x represent the number of milliliters she should add and the equation represents this situation is 0.18(100) + 0.25x = 0.2(100 + x).

Acid makes up number 0.18(100) = 18 cc of the 18% solution.

Let's suppose Sam needs to blend x millilitres of the 25% fluid.

0.25x times as much acid is present in x millilitres of the 25% solution.

Acid should make up 0.2(100 Plus x) of the entire mixture.

Therefore, 0.18(100) + 0.25x = 0.2(100 + x) represents this scenario mathematically.

Acid makes up 0.18(100) = 18 cc of the 18% solution. Let's suppose Sam needs to blend x millilitres of the 25% fluid. 0.25x times as much acid is present in x millilitres of the 25% solution. Acid should make up 0.2(100 Plus x) of the entire mixture. As a result, 0.18(100) + 0.25x = 0.2(100 + x) represents this scenario mathematically.

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