Steve has ​$25,000 to invest and wishes to earn an overall annual rate of return of 8​%. His financial advisor recommends that he invest some of the money in a​ 5-year CD paying 5​% per annum and the rest in a corporate bond paying 9​% per annum. How much should be placed in each investment in order for Steve to achieve his​ goal?

Answer:

P(T) = 1/20 = 0.05

The probability of randomly selecting an umbrella and a shaving kit in that order is 0.05

Step-by-step explanation:

The probability of randomly selecting an umbrella and a shaving kit in that order.

P(T) = Probability of selecting umbrella first P(U) × probability of selecting shaving kit second P(S)

P(U) = 1/5 (1 umbrella out of five possible gifts)

P(S) = 1/4 (1 shaving kit out of four remaining possible gifts)

P(T) = 1/5 × 1/4

P(T) = 1/20 = 0.05

The probability that a person will select an umbrella and a shaving kit in that order is 1/20.

The following can be deduced from the information given:

P(U) = 1/5 (1 umbrella out of five possible gifts)

P(S) = 1/4 (1 shaving kit out of four remaining possible gifts)

Therefore, the probability that a person will select an umbrella and a shaving kit in that order will be:

P(T) = 1/5 × 1/4

P(T) = 1/20 = 0.05

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Answer:

The Matrix of T is

[tex]\left[\begin{array}{cc}cos(\pi/6)&-sin(\pi/6)\\-sin(\pi/6)&-cos(\pi/6)\end{array}\right][/tex]

Step-by-step explanation:

Rotate -pi/6 Clockwise is the same as rotating pi/6 anticlockwise. The matrix of that rotation is

[tex]\left[\begin{array}{cc}cos(\pi/6)&-sin(\pi/6)\\sin(\pi/6)&cos(\pi/6)\end{array}\right][/tex]

The matrix of the reflection through the x1-axis is

[tex]\left[\begin{array}{cc}1&0\\0&-1\end{array}\right][/tex]

Therefore, the composition is the product of both matrices is the matrix of T

[tex]MT = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right] * \left[\begin{array}{cc}cos(\pi/6)&-sin(\pi/6)\\sin(\pi/6)&cos(\pi/6)\end{array}\right] = \left[\begin{array}{cc}cos(\pi/6)&-sin(\pi/6)\\-sin(\pi/6)&-cos(\pi/6)\end{array}\right][/tex]

I hope that works for you!

To find the standard matrix of the linear transformation [tex]\( T \)[/tex], we need to perform two operations in sequence: a rotation through [tex]\( -\frac{\pi}{6} \)[/tex] radians (clockwise) and a reflection through the horizontal [tex]\( x_1 \)-axis[/tex]. We will find the matrices for each of these transformations and then multiply them to get the standard matrix for [tex]\( T \)[/tex].

1. Rotation Matrix [tex]\( R \)[/tex]:

The matrix that represents a rotation through an angle [tex]\( \theta \)[/tex] in the counterclockwise direction is given by:

[tex]\[ R(\theta) = \begin{bmatrix} \cos(\theta) -\sin(\theta) \\ \sin(\theta) \cos(\theta) \end{bmatrix} \][/tex]

For a clockwise rotation, we use a negative angle, so for [tex]\( -\frac{\pi}{6} \)[/tex] radians, the rotation matrix is:

[tex]\[ R\left(-\frac{\pi}{6}\right) = \begin{bmatrix} \cos\left(-\frac{\pi}{6}\right) -\sin\left(-\frac{\pi}{6}\right) \\ \sin\left(-\frac{\pi}{6}\right) \cos\left(-\frac{\pi}{6}\right) \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{3}}{2} \frac{1}{2} \\ -\frac{1}{2} \frac{\sqrt{3}}{2} \end{bmatrix} \][/tex]

2. Reflection Matrix [tex]\( M \)[/tex]:

The matrix that represents a reflection through the horizontal[tex]\( x_1 \)[/tex]-axis is given by:

[tex]\[ M = \begin{bmatrix} 1 0 \\ 0 -1 \end{bmatrix} \][/tex]

3. Standard Matrix of [tex]\( T \)[/tex]

To find the standard matrix of[tex]\( T \)[/tex], we multiply the rotation matrix [tex]\( R \)[/tex] by the reflection matrix [tex]\( M \)[/tex]:

[tex]1 0 \\ 0 -1[/tex]

[tex]\end{bmatrix} \begin{bmatrix} \frac{\sqrt{3}}{2} \frac{1}{2} \\ -\frac{1}{2} \frac{\sqrt{3}}{2} \end{bmatrix} \] \[ T = \begin{bmatrix} \frac{\sqrt{3}}{2} \frac{1}{2} \\ \frac{1}{2} -\frac{\sqrt{3}}{2} \end{bmatrix} \][/tex]

Therefore, the standard matrix of the linear transformation [tex]\( T \)[/tex] is:

[tex]\[ \boxed{T = \begin{bmatrix} \frac{\sqrt{3}}{2} \frac{1}{2} \\ \frac{1}{2} -\frac{\sqrt{3}}{2} \end{bmatrix}} \][/tex]

This matrix represents the transformation that first rotates points through [tex]\( -\frac{\pi}{6} \)[/tex] radians (clockwise) and then reflects them through the horizontal [tex]\( x_1 \)-axis.[/tex]

Answer:

Step-by-step explanation:

This is a circle with radius 2 and z = y

All points on or within the circle x2 + y2 +4 and in the plane z = y

The value of B is determined by the equation 2π / 12.7, which corresponds to the tide's period. The variable C represents the time delay from midnight to the first high tide, which is a phase shift in the function.

The Bay of Fundy tidal pattern can be modeled using a cosine function. Since tides go through a complete cycle (360 degrees or 2π radians) every 12.7 hours, the value of B, the frequency, can be determined by dividing 2π by the period of the tide in hours.

Therefore, B = 2π / 12.7.

The variable C in the equation represents a phase shift. In this context, a phase shift refers to a horizontal shift of the cosine function, which corresponds to a time delay or advance of the tides. The meaning of C is the time delay between midnight and the first high tide of the day.

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Answer:

0.5882 or 58.82%

Step-by-step explanation:

The probability that both balls were red (A) is:

[tex]P(A)=\frac{6}{9}*\frac{5}{9}=0.3704[/tex]

The probability that the first ball was blue and the second ball was red (B) is:

[tex]P(B) = \frac{3}{9}*\frac{7}{9}=0.2593[/tex]

The conditional probability that the first ball selected is red, given that the second ball was red is:

[tex]P = \frac{P(A)}{P(A)+P(B)}=\frac{0.3704}{0.3704+0.2593} =0.5882[/tex]

Answer:

[tex] P(A \cap B) = P(X=9) =\frac{16}{78} \neq 0[/tex]

The correct answer would be:

NO

Step-by-step explanation:

For this case we have the following dataset given

Hours    Number of students (f)

_______________________________

   4                     15

   5                     11

    6                     19

   7                      6

   8                      9

    9                     16

    10                     2

______________________________

Total                  78

For this case we have defined the following events:

A = event the student took at most 9 hours

B = event the student took at least 9 hours

And we can find the empirical probability for both elements like this:

[tex] P(A) = \frac{78-2}{78}= \frac{76}{78}[/tex]

[tex] P(B) = \frac{16+2}{78}= \frac{18}{78}[/tex]

And for this case we want to see if A and B are disjoint

From definition two events X and Y are disjoint if the two sets not have a common elements, and we satisfy that:

[tex] P(X \cap Y) =0[/tex]

So this case the intersection for the events A and B is X=9, because at most 9 means [tex] X \leq 9[/tex] and at least 9 means [tex] X \geq 9[/tex] and the intersection between [tex] X \leq 9[/tex]  and [tex] X \geq 9[/tex]  is X=9

So then the probability:

[tex] P(A \cap B) = P(X=9) =\frac{16}{78} \neq 0[/tex]

So then we can conclude that the two events not are disjoint

The correct answer would be:

NO

No, the events A and B are not disjoint.

If two events have no outcomes in common, then they are called disjoint.

We have data of the number of hours sixth grade students took to complete a research project as:

For this case we have the following dataset given

Hours    Number of students (f)

 4                     15

5                     11

6                     19

7                      6

8                      9

9                     16

10                     2

Total                  78

Two events are:

A = event the student took at most 9 hours

B = event the student took at least 9 hours

Now, the number of students who took at most 9 hours

= 78 - 2

= 76

So, [tex]P(A)=\frac{76}{78}[/tex]

The number of students who took at least 9 hours

=16 +2

=18

So, [tex]P(B)=\frac{16}{78}[/tex]

Number of students who read exactly 9 hours

P(A n B)[tex]=\frac{16}{78}[/tex][tex]\neq 0[/tex]

Therefore the events A and B disjoint are not disjoint.

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Answer:

The answer to your question is 146 adult tickets

Step-by-step explanation:

Data

Total number of tickets = 584

student ticket = s

adult ticket = a

Condition

                      3a = s

Equation

                     s + a = 584

Substitution

                     3a + a = 584

Simplification

                            4a = 584

Solve for a

                              a = 584/4

                              a = 146

Find s

                             s = 3(146)

                             s = 438

Answer:146 adult tickets were sold.

Step-by-step explanation:

Let x represent the number of adult tickets that were sold.

Let y represent the number of student tickets that were sold.

A total of 584 tickets were sold for the school play. This means that

x + y = 584 - - - - - - - - - - - - - - 1

The number of student tickets sold was three times the number of adult tickets sold. This means that

y = 3x

Substituting y = 3x into equation 1, it becomes

x + 3x = 584

4x = 584

x = 584/4 = 146

y = 3x = 3 × 146

y = 438

Answer:

(a) x1 = 11/13, x2 = 50/13, x3 = -17/13

(b) x = 54/235, y = 6/47, z = 24/235

(c) x1 = 22/9, x2 =164/9, x3 = 139/9, x4 = -37/3

Step-by-step explanation:

Gaussian Elimination Method was the matrix method used in solving the system of equations.

It is done by writing the equations given in an augmented form, this is shown in the attachment. The coefficients of each variable is taken to form a matrix.

Row operations are then performed on the augmented matrix. This operation can be addition, subtraction, multiplication, or division.

For convenience, Row is written as R1, Row 2 as R2, and so on

R2 - R3 means Subtract Row 3 from Row 2, and so on.

The step by step operations for each question are shown in the attachment.

The matrix method involves representing the systems of equations as matrices, and then using matrix operations or the inverse matrix method to solve for the variables. This method can only be used when the system has a unique solution.

Using the matrix method to solve systems of equations involves first representing the system as a matrix. For example, the first system of equations can be represented as a matrix: [[3,2,4], [2,5,3], [7,2,2]][[x1],[x2],[x3]] = [[5], [17], [11]]. Similarly, the second and third systems can be written in matrix form. Then you can use various matrix operations or the inverse matrix method to solve for the variables. Note that this method is used only when the system has a unique solution - that is, when the coefficient matrix is invertible.

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Answer:

...the size tolerance controls both the measurements or dimensions of a piece, and the size.

Step-by-step explanation:

However, dimensional tolerance controls neither the shape, nor the position, nor the orientation of the elements to which said tolerance applies. In manufacturing, geometric irregularities occur that can affect the shape, position or orientation of the different elements of the pieces. An applied dimensional tolerance, for example, has an effect on the parallelism and flatness of that piece.

Answer:

Our answer is 0.8172

Step-by-step explanation:

P(doubles on a single roll of pair of dice) =(6/36) =1/6

therefore P(in 3 rolls of pair of dice at least one doubles)=1-P(none of roll shows a double)

=1-(1-1/6)3 =91/216

for 12 players this follows binomial distribution with parameter n=12 and p=91/216

probability that at least 4 of the players will get “doubles” at least once =P(X>=4)

=1-(P(X<=3)

=1-((₁₂ C0)×(91/216)⁰(125/216)¹²+(₁₂ C1)×(91/216)¹(125/216)¹¹+(₁₂ C2)×(91/216)²(125/216)¹⁰+(₁₂ C3)×(91/216)³(125/216)⁹)

=1-0.1828

=0.8172

Answer:

The probability is 0.27

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this problem, we have that:

Odds of a win are 27 to 73.

This means that for each 27 games that you are expcted to win, you are also expected to lose 73.

So

Desired outcomes:

27 wins

Total outcomes:

27 + 73 = 100 games

Probability

[tex]P = \frac{27}{100} = 0.27[/tex]

Answer:

The probability is 0.27

Focus on the sub-triangle on the left. It is a right triangle with legs 9 and 6, so its hypothenuse is

[tex]\sqrt{9^2+6^2}=\sqrt{81+36}=\sqrt{117}[/tex]

Now focus on the sub-triangle on the right. It is a right triangle with legs 6 and x, so its hypothenuse is

[tex]\sqrt{6^2+x^2}=\sqrt{x^2+36}[/tex]

Now, the entire triangle has legs [tex]\sqrt{117}[/tex] and [tex]\sqrt{x^2+36}[/tex], and its hypothenuse is [tex]9+x[/tex]. Write the Pytagorean theorem one last time to get

[tex]117+(x^2+36)=(9+x)^2\iff x^2+153=81+18x+x^2 \iff 18x+81=153[/tex]

Subtract 81 from both sides to get

[tex]18x=72 \iff x=\dfrac{72}{18}=4[/tex]

Answer: x = 4

Step-by-step explanation:

The attached photo shows a clearer illustration of the given triangle.

Looking at the photo, assuming ∆BCD is a right angle triangle. To determine BC, we would apply Pythagoras theorem which is expressed as

Hypotenuse² = opposite side² + adjacent side²

BC² = 9² + 6²

BC² = 81 + 36 = 117

BC = √117

To determine θ, we would apply the tangent trigonometric ratio.

Tan θ opposite side/adjacent side

Tan θ = 6/9 = 0.6667

θ = 33.6914

Considering ∆ABC,

Hypotenuse = x + 9

Adjacent = √117

Cos θ = adjacent side/ hypotenuse

Cos 33.6914 = √117/(x + 9)

Cross multiplying, it becomes

0.8320 = √117/(x + 9)

x + 9 = √117/0.8320

x + 9 = 13

x = 13 - 9 = 4

Answer:

11,713 bacteria

Step-by-step explanation:

Integrating the growth rate function gives us the population of bacteria at any given moment 't', in hours:

[tex]r(t) = (450.263)e^{1.12567t} \\\int\ {r(t)} \, dt =P(t) = \frac{450.263}{1.12567}*e^{1.12567t} +C[/tex]

Since at t=0, P(t) = 400, the value of C is:

[tex]P(0) = \frac{450.263}{1.12567}*e^{1.12567*0} +C\\400 = 400*1+C\\C=0[/tex]

The number of bacteria after 3 hours is:

[tex]P(3) = 400*e^{1.12567*3}\\P(3) =11,713[/tex]

Answer:

2.2 times

Step-by-step explanation:

Given that,

Sales = $4,400,000

Average total assets (excluding long-term investments) = 2,000,000

Therefore, it is as follows;

Asset turnover ratio:

= Sales ÷ Average total assets (excluding long-term investments)

= $4,400,000 ÷ 2,000,000

= 2.2 times

Hence, the asset turnover ratio of this company is 2.2 times.

The manufacturer's profit is P(p) = 120p^2 - 6000p. The average profit for producing 0 to 15 recorders is 15 * 120. At 15 units, the rate of profit change is -3000.

The function for the manufacturer's profit P is given by P(p) = pq - 50q where q=120p. Thus, P(p) = 120p * p - 50 * 120p, simplifying to P(p) = 120p^2 - 6000p.

To express the average rate of profit from q = 0 to q=15, we need to calculate P(15)-P(0) and divide it by 15 - 0. Thus, ((120 * 15^2 - 6000 * 15) - (120 * 0^2 - 6000 * 0)) / (15 - 0) = 15*120.

The rate at which profit is changing when q = 15 is the derivative of the profit function evaluated at q = 15. We obtain P'(q) = 240p - 6000, and by inserting 15 for p, we obtain -3000 (at dollars per unit).

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a) The manufacturer's profit can be expressed as P = (p - 50)q. b) The average rate of profit obtained as the level of production increases from q=0 to q=15 is p - 50 dollars per unit. c) When q=15 recorders are produced, the rate at which profit is changing is p - 50 dollars per unit.

a) To express the manufacturer's profit P as a function of q, we can use the equation: P = (p - 50)q. This equation represents the total revenue minus the total cost, where the total revenue is the selling price p multiplied by the quantity q, and the total cost is the cost of production per unit (50 dollars) multiplied by the quantity q.

b) The average rate of profit obtained as the level of production increases from q=0 to q=15 can be found by calculating the change in profit divided by the change in quantity. In this case, the change in profit is (p - 50)(15) - (p - 50)(0) = 15(p - 50) dollars, and the change in quantity is 15 units. Therefore, the average rate of profit is (15(p - 50))/15 = p - 50 dollars per unit.

c) To find the rate at which profit is changing when q=15 recorders are produced, we can calculate the derivative of the profit function with respect to q. Taking the derivative of P = (p - 50)q with respect to q gives us dP/dq = p - 50 dollars per unit. So, when q=15, the rate at which profit is changing is p - 50 dollars per unit.

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Answer:

4.23% probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill.

Step-by-step explanation:

For each installation, there are only two possible outcomes. Either it reduces the utility bill, or it does not. The probabilities for each installation reducing the utility bill are independent. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

In this problem we have that:

Solar-heat installations successfully reduce the utility bill 60% of the time, which means that [tex]p = 0.6[/tex]

What is the probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill?

This is [tex]P(X \geq 9)[/tex] when [tex]n = 10[/tex]. So

[tex]P(X \geq 9) = P(X = 9) + P(X = 10)[/tex]

In which

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 9) = C_{10,9}.(0.6)^{9}.(0.4)^{1} = 0.0363[/tex]

[tex]P(X = 10) = C_{10,10}.(0.6)^{10}.(0.4)^{0} = 0.0060[/tex]

So

[tex]P(X \geq 9) = P(X = 9) + P(X = 10) = 0.0363 + 0.0060 = 0.0423[/tex]

4.23% probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill.

The probability of at least 90% success in solar-heat installations is found by using the binomial probability formula to calculate and add together the probabilities of exactly 9 and 10 successful installations out of 10.

The problem in question is a classic scenario of binomial probability. Here, each solar-heat installation attempt is independent and each attempt is a success (reduces the utility bill) 60% of the time. We are interested in the probability of having 90% or more success in ten attempts.

In a binomial distribution, the formula for calculating the probability of k successes in n attempts is:

P(X=k) = C(n, k) * (p^k) * (1-p)^(n-k)

where C(n, k) is the binomial coefficient ('n choose k'), p is the probability of success on an individual trial, n is the number of trials, and k is the number of successes.

To calculate the probability that at least 9 out of 10 solar-heat installations are successful, we need to calculate P(X=9) and P(X=10) and add these probabilities together.

Calculations like these help inform decisions in a range of fields - from individual choices about energy saving at homes to policy and planning decisions at the level of energy utilization for entire nations.

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Answer:

Step-by-step explanation:

Since each trial is independent of the other

no of mistakes he does is binomial with p = 1/3

a) the probability that he makes no mistakes on his first 10 orders but the 11th order is a mistake

= [tex](\frac{2}{3}) ^{10} *\frac{1}{3}\\=\frac{2^{10} }{3^{11} }[/tex]

b) Prob that shanker quits = P(Shankar does I one mistake and Fran does not do the first one)+Prob (Shanker does mistake in the II one while Fran does both right)

= [tex]\frac{1*5}{3*6} +\frac{2}{3} \frac{1}{3}(\frac{5}{6})^2 =\frac{5}{18} +\frac{50}{216} \\=\frac{55}{108}[/tex]

Final answer:

Calculating probabilities in scenarios involving making mistakes in orders in a coffee shop setting. he probability that Shankar quits and goes to grad school before Fran is approximately 0.651.

Explanation:

a. For Shankar to make no mistakes on his first 10 orders but the 11th order is a mistake, the probability of making no mistakes on the first 10 orders and then making a mistake on the 11th order can be calculated as follows:

[tex]\[ P(\text{No mistakes on first 10 orders}) \times P(\text{Mistake on 11th order}) \][/tex]

[tex]\[ = \left(\frac{2}{3}\right)^{10} \times \frac{1}{3} \][/tex]

[tex]\[ = \left(\frac{1024}{59049}\right) \times \frac{1}{3} \][/tex]

[tex]\[ \approx 0.0173 \][/tex]

b. For Shankar to quit and go to grad school before Fran, Shankar must make a mistake before Fran does. The probability of Shankar quitting and going to grad school can be calculated as follows:

[tex]\[ P(\text{Shankar quits}) = 1 - P(\text{Fran quits first}) \][/tex]

Since Fran's probability of making a mistake on an order is [tex]\( \frac{1}{6} \)[/tex], the probability of Fran making no mistakes on an order is [tex]\( \frac{5}{6} \).[/tex] Thus, the probability of Fran not making a mistake before Shankar is:

[tex]\[ P(\text{Fran makes no mistakes before Shankar}) = \left(\frac{5}{6}\right)^{10} \][/tex]

Therefore,

[tex]\[ P(\text{Shankar quits}) = 1 - \left(\frac{5}{6}\right)^{10} \][/tex]

[tex]\[ \approx 0.651 \][/tex]

So, the probability that Shankar quits and goes to grad school before Fran is approximately \(0.651\).

Answer: QINGHAOSU

Step-by-step explanation:Artemisinin is an antimalarial (Drug used to cure Malaria) drug whose active ingredient QINGHAOSU was isolated in the 1970s from the Plant called Artemisia annua by a Chinese Physician.

Artemisinin has been in use in ancient Chinese communities since the 4th century to cure Diseases. Artemisinin is potent in killing several forms of the Plasmodium specie and has been used to derive other antimalarial Drugs in use today like Artemeter and Artesunate.

Answer:

24, 204

Step-by-step explanation:

to find the height

we would use the heron formula  when all the sides are given

S which is equal to half of the perimeter of the triangle which is (a+b+c)/2.

S = (17+ 25+ 26)/2 = 68/2 = 34

Area = √S(S-A)(S-B)(S-C)

Area= (1/2)bh

we equate it back to the formula Area = √S(S-A)(S-B)(S-C)

it becomes

(1/2)bh  = √S(S-A)(S-B)(S-C)

A = IABI= 25

B= IBCIM = 26

C= IACI = 17

b = base = IACI = 17

S = 34

(1/2)bh  = √S(S-A)(S-B)(S-C)

(1/2)17h  = √34(34-25)(34-26)(34-17)

(17/2)h = √34(9)(8)(17) = √34 x 9 x 8 x 17

(17/2)h = √41616 = 204

17h/2 = 204

17h = 204 x 2 = 408

h = 408/17 = 24 inch

height = h = IBDI = 24 in

Area = (1/2)bh

      = (1/2) x 17 x 24

    =  12 x 17 = 204  or we use the heron formula just like the above which we get 204 before multiplication by 2.

Answer:

The distance Plane A traveled in 1 h is greater than the distance Plane B traveled in 1 h.

Step-by-step explanation:

The equation of the distance traveled by Plane A is

[tex]d(t) = 290t[/tex]

The plane B traveled 540 miles in 2 hours.

So in 1 hour, plane B traveled 540/2 = 270 miles:

How does the distance Plane A traveled in 1 hour compare to the distance Plane B traveled in 1 hour?

Plane A:

d(1) = 290*1 = 290

Plane A traveled 290 miles in 1 hour.

Plane B travaled 270 miles in 1 hour.

So the correct answer is:

The distance Plane A traveled in 1 h is greater than the distance Plane B traveled in 1 h.

Answer:

The distance Plane A traveled in 1 h is greater than the distance Plane B traveled in 1 h.

Step-by-step explanation:

Answer:

(a) 0.25

(b) 0.5

(c) 0.5256

(d) 0.025

Step-by-step explanation:

(a) There are 20 even numbers out of 40 the probability that both numbers are even is:

[tex]P=\frac{20}{40} *\frac{20}{40} =\frac{1}{4}=0.25[/tex]

(b) The events for which one number is even and one number is odd are:

- First is odd, second is even

- First is even, second is odd.

The probability is:

[tex]P = \frac{20}{40}*\frac{20}{40}+\frac{20}{40}*\frac{20}{40}=\frac{1}{2}=0.5[/tex]

(c) There are 29 numbers that are less than 30, the probability that both numbers are less than 30 is:

[tex]P=\frac{29}{40}*\frac{29}{40}=\frac{841}{1600}=0.5256[/tex]

(d) If any number from 1 to 40 is selected in the first pick, the probability that the same number is selected again is:

[tex]P=\frac{1}{40} =0.025[/tex]

Answer:

How many measurements must be made = 9

Step-by-step explanation:

The steps are as shown in the attachment.

To reduce the uncertainty in the diameter of a disk from 1.5mm to 0.5mm, three times more measurements of the circumference would need to be made. This is due to the relationship between the circumference and diameter, and how the uncertainty propagates through this relationship.

This question pertains to the areas of accuracy, precision, and uncertainty in measurements. Understanding these concepts is vital in the field of physics. The circumference of a disk and its diameter are related by the constant π (Pi): Diameter = Circumference / π.

The question states that there is an uncertainty of 1.5 mm in measuring the circumference. Given that the diameter and circumference are directly connected, when you reduce the uncertainty in the measurement of the circumference (e.g., by taking more measurements), you also reduce the uncertainty in the diameter. However, the relationship is not linear. Through propagation of uncertainty principles, the uncertainty in diameter would be the uncertainty in the circumference divided by π. To reduce this to 0.5 mm, you would require three times more measurements.

The precision of a measurement system is closely linked to the size of its measurement increments. The smaller the measurement increment, the more precise the tool. Various factors can contribute to the uncertainty of a measurement, including the smallest division on a given tool, the ability of the person making the measurement, irregularities in the object being measured, and unforeseen circumstances that affect the outcome.

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Answer:

[tex]\frac{3}{14}[/tex]

Step-by-step explanation:

There are 252 (=31+54+42+20+47+58) employees in total. 54 of those are in research. So the chances that one check that gets lost belongs to a research employee can be calculated as follows:

[tex]P=\frac{54}{252}= \frac{3}{14}[/tex]

Answer:

Mean ;

Median;

Step-by-step explanation:

The mean is used when the data under consideration is more of quantitative and in which the data is devoid of outliers as such the values are assumed to follow a normal distribution.

The median on the other hand is considered when the data are more of qualitative and usually contain outliers. Median on the other hand is best used when there is a skewed symmetry in the values given.

Mean ;

Median;

Whether mean or median provides a better description of a data set depends on the skewness and outliers in the data. Generally, the mean is more sensitive to outliers whereas the median can better represent the central tendency of skewed distributions.

In statistics, mean and median are two measures of central tendency. The mean is the average of the data points, while the median is the middle value. Whether the mean or median provides a better description depends on the distribution of the data.

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Answer:

a) [tex]\bar X = \frac{\sum_{i=1}^{27} X_i }{27}= \frac{809}{27}=29.96[/tex]

[tex] Median = 25[/tex]

b) [tex] Mode = 25, 35[/tex]

Since 25 and 35 are repeated 4 times, so then the distribution would be bimodal.

c) [tex] Midrange = \frac{70+13}{3}=41.5[/tex]

d) [tex] Q_1 = \frac{20+21}{2} =20.5[/tex]

[tex] Q_3 =\frac{35+35}{2}=35[/tex]

e) Min = 13 , Q1 = 20.5, Median=25, Q3= 35, Max = 70

f) Figura attached.

g) When we use a quantile plot is because we want to show the percentage or the fraction of values below or equal to an specified value for the distribution of the data.

By the other hand the quantile-quantile plot shows the quantiles of the distribution values against other selected distribution (specified, for example the normal distribution). If the points are on a straight line we assume that the data values fit very well to the hypothetical distribution selected.

Step-by-step explanation:

For this case w ehave the following dataset given:

13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30,33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70.

Part a

The mean is calculated with the following formula:

[tex]\bar X = \frac{\sum_{i=1}^{27} X_i }{27}= \frac{809}{27}=29.96[/tex]

The median on this case since we have 27 observations and that represent an even number would be the 14 position in the dataset ordered and we got:

[tex] Median = 25[/tex]

Part b

The mode is the most repeated value on the dataset on this case would be:

[tex] Mode = 25, 35[/tex]

Since 25 and 35 are repeated 4 times, so then the distribution would be bimodal.

Part c

The midrange is defined as:

[tex] Midrange = \frac{Max+Min}{2}[/tex]

And if we replace we got:

[tex] Midrange = \frac{70+13}{3}=41.5[/tex]

Part d

For the first quartile we need to work with the first 14 observations

13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25

And the Q1 would be the average between the position 7 and 8 from these values, and we got:

[tex] Q_1 = \frac{20+21}{2} =20.5[/tex]

And for the third quartile Q3 we need to use the last 14 observations:

25, 30,33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70

And the Q3 would be the average between the position 7 and 8 from these values, and we got:

[tex] Q_3 =\frac{35+35}{2}=35[/tex]

Part e

The five number summary for this case are:

Min = 13 , Q1 = 20.5, Median=25, Q3= 35, Max = 70

Part f

For this case we can use the following R code:

> x<-c(13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30,33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70)

> boxplot(x,main="boxplot for the Data")

And the result is on the figure attached. We see that the dsitribution seems to be assymetric. Right skewed with the Median<Mean

Part g

When we use a quantile plot is because we want to show the percentage or the fraction of values below or equal to an specified value for the distribution of the data.

By the other hand the quantile-quantile plot shows the quantiles of the distribution values against other selected distribution (specified, for example the normal distribution). If the points are on a straight line we assume that the data values fit very well to the hypothetical distribution selected.

Answer: The strength of an electric field is E = - 0,05.10⁹ N/C.

Step-by-step explanation: According to the question, the plastic sphere is in equilibrium in an electric field. This sugests that the forces acting on the sphere, which are Gravitational Force (Fg) and Electric Force (Fe) are also in equilibrium, denotating Fg=Fe.

As Fg = m . g, with m = 0,009kg and g= 9,8m/s², we have Fg = 0,0882N.

Knowing the value of Fe, the strength of the electric field can be calculated as

E = Fe/Q, in which Q is the electric charge.

E = (0,0882) / (-1,6·10⁻⁹)

E = - 0,05·10⁹N/C

Answer:

$7826.78

Step-by-step explanation:

Total income = 7950

With a service fee charge of 1.55%, the service fee charge = [tex]1.55\%\times7950 = \dfrac{1.55}{100}\times7950 = 123.225[/tex]

Net income = 7950 - 123.225 = 7826.775 = $7826.78

An alternative solution:

Service charge = 1.55%

Net income = (100 - 1.55)% × 7950

= 98.45% × 7950 = [tex]\dfrac{98.45}{100}\times7950 = 7826.775=7826.78[/tex]

Answer:

Step-by-step explanation:

Accepted credit= $ 7950

Charges for services =1.55% of the total  

7950 - 1.55% of 7950

7950 - (1.55/100)*7950

7950 - 123.225

7826.775

7826,78  two decimal places

This is the net income earned during the month of July  

Answer:

A

Step-by-step explanation:

The type I error arises when we wrongfully reject the null hypothesis. The probability of  occurrence of type I error is denoted as α. Thus, α=0.05 means that there is 5% probability that we reject the null hypothesis when it is true. So, we can say that the α=0.05, means that 5% of the time, we reject the null hypothesis when it is correct.

Answer:

A

Step-by-step explanation:

Answer:

B. 72

Step-by-step explanation:

The total number of different systems that can be bundled together is the product of the possible number of ways to select 1 out of 4 stereo receivers, by 1 out of 3 CD decks and by 1 out of 6 speakers. Assuming that the order at which products are picked does not matter, the number of different systems is:

[tex]n=\frac{4!}{(4-1)!1!}*\frac{3!}{(3-1)!1!}*\frac{6!}{(6-1)!1!}\\n=4*3*6\\n=72\ systems[/tex]

The electronics firm can offer 72 different systems.

The electronics firm can offer 72 different systems by combining 4 models of stereo receivers, 3 CD decks, and 6 speaker brands together.

To find the number of different systems the electronics firm can offer, we use the combinations formula. Since there are 4 models of stereo receivers, 3 CD decks, and 6 speaker brands, the total number of different systems can be calculated as:

4 models * 3 CD decks * 6 speaker brands = 72 different systems

Therefore, the answer to the question is 72 different systems (option B).

Answer:

5.70456 slug

Step-by-step explanation:

Data provided in the question:

Dimensions of the room = 20 ft long, 15 ft wide, and 8 ft high

Now,

Volume of the room = 20 × 15 × 8

or

Volume of the room = 2400 ft³

we know,          

Density of air = 0.0023769 slug/ft³

Therefore,

Mass of air in the room = Volume × Density

= 0.0023769 × 2400

= 5.70456 slug

The mass of air in the room is approximately 1268 slugs and the weight is approximately 40825.6 pounds.

To calculate the mass of air in a room, we first need to find the volume of the room. The volume of a rectangular room can be calculated by multiplying its length, width, and height. So, the volume of the room is 20 ft x 15 ft x 8 ft = 2400 ft³.

Next, we need to convert the volume from cubic feet to cubic meters. Since 1 ft³ is approximately equal to 0.0283 m³, we can multiply the volume in cubic feet by 0.0283 to get the volume in cubic meters. Therefore, the volume of the room is 2400 ft³ x 0.0283 m³/ft³ = 67.92 m³.

Lastly, we need to find the mass of air in the room. The average molar weight of air is approximately 28.8 g/mol. Since the mass of one cubic meter of air is 1.28 kg, the mass of air in the room is 67.92 m³ x 1.28 kg/m³ = 86.86 kg. To convert the mass from kg to slugs, we divide it by the conversion factor of 0.0685218 slugs/kg. Therefore, the mass of air in the room is 86.86 kg / 0.0685218 slugs/kg ≈ 1268 slugs.

To calculate the weight in pounds, we multiply the mass in slugs by the acceleration due to gravity. The acceleration due to gravity is approximately 32.2 ft/s². Therefore, the weight of the air in the room is 1268 slugs x 32.2 ft/s² = 40825.6 lb.