A yogurt stand gave out 120 free samples of frozen yogurt, one free sample per person. The three sample choices were vanilla, chocolate, or chocolate & vanilla twist. 97 people tasted the vanilla and 72 people tasted the chocolate, some of those people tasted both because they chose the chocolate and vanilla twist. How many people chose chocolate and vanilla twist?

Answer:

a)

Reduced Row Echelon:

[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&0&1&-4\end{array}\right][/tex]

Solution to the system:

[tex]x_3=-4\\x_2=-\frac{7}{4}x_3=7\\x_1=-\frac{1}{2}x_2=-\frac{7}{2}[/tex]

b)

Reduced Row Echelon:

[tex]\left[\begin{array}{cccc}4&3&0&7\\0&0&2&-17\\0&0&2&-17\end{array}\right][/tex]

Solution to the system:  

[tex]x_3=-\frac{17}{2}\\x_1=\frac{7-3x_2}{4}[/tex]

x_2 is a free variable, meaning that it has infinite possibilities and therefore the system has infinite number of solutions.

Step-by-step explanation:

To find the reduced row echelon form of the matrices, let's use the Gaussian-Jordan elimination process, which consists of taking the matrix and performing a series of row operations. For notation, R_i will be the transformed column, and r_i the unchanged one.

a) [tex]\left[\begin{array}{cccc}0&4&7&0\\2&1&0&0\\0&3&1&-4\end{array}\right][/tex]

Step by step operations:

1. Reorder the rows, interchange Row 1 with Row 2, then apply the next operations on the new rows:

[tex]R_1=\frac{1}{2}r_1\\R_2=\frac{1}{4}r_2[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&3&1&-4\end{array}\right][/tex]

2. Set the first row to 1

[tex]R_3=-3r_2+r_3[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}1&1/2&0&0\\0&1&7/4&0\\0&0&1&-4\end{array}\right][/tex]

3. Write the system of equations:

[tex]x_1+\frac{1}{2}x_2=0\\x_2+\frac{7}{4}x_3=0\\x_3=-4[/tex]

Now you have the  reduced row echelon matrix and can solve the equations, bottom to top, x_1 is column 1, x_2 column 2 and x_3 column 3:

[tex]x_3=-4\\x_2=-\frac{7}{4}x_3=7\\x_1=-\frac{1}{2}x_2=-\frac{7}{2}[/tex]

b)

[tex]\left[\begin{array}{cccc}4&3&0&7\\8&6&2&-3\\4&3&2&-10\end{array}\right][/tex]

1. [tex]R_2=-2r_1+r_2\\R_3=-r_1+r_3[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}4&3&0&7\\0&0&2&-17\\0&0&2&-17\end{array}\right][/tex]

2. Write the system of equations:

[tex]4x_1+3x_2=7\\2x_3=-17[/tex]

Now you have the reduced row echelon matrix and can solve the equations, bottom to top, x_1 is column 1, x_2 column 2 and x_3 column 3:

[tex]x_3=-\frac{17}{2}\\x_1=\frac{7-3x_2}{4}[/tex]

x_2 is a free variable, meaning that it has infinite possibilities and therefore the system has infinite number of solutions.

Answer:

The required factors are: x, (x + 6) and (x - 3).

Step-by-step explanation:

As per the question,

The given polynomial is:

[tex]x^{3}+3x^{2}-18x[/tex]

Now,

BY factorization, we get

[tex]x^{3}+3x^{2}-18x[/tex]

[tex]=x(x^{2}+3x-18)[/tex]

By splitting the mid-term, that is split 3x like:

3x = 6x - 3x

Therefore,

[tex]x(x^{2}+6x-3x-18)[/tex]

Now on further solving by taking common factor out, we get

[tex]=x[x(x+6)-3(x+6)][/tex]

[tex]=x(x+6)(x-3)[/tex]

Therefore, the given second polynomial (x - 4), is not a factor of given polynomial [tex]x^{3}+3x^{2}-18x[/tex].

Hence, the given polynomial has three factor x, (x + 6) and (x - 3).

Answer:

1. Group A and B agree with each other.

2. Group A and C do not agree with each other.

Step-by-step explanation:

When we are analizing this problem, we will see what are the ranges of this measured times. Since we are taking into account the error we can see that :

Question 1 is about the overlapping response in Group A and Group B. And yes, we have an overlap between 7.35 to 7.39. Among this times both group A and B are in agree with each other within the experimental uncertainty.

Question 2 is now referring to Group A and Group C. And no, there isn't any common time where both groups agree with each other.

Answer:

[tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]

Step-by-step explanation:

Given : [tex]f(t)=16\cos (t-\frac{\pi}{6})[/tex]

To find : ℒ{f(t)} by first using a trigonometric identity ?

Solution :

First we solve the function,

[tex]f(t)=16\cos (t-\frac{\pi}{6})[/tex]

Applying trigonometric identity, [tex]\cos (A-B)=\cos A\cos B+\sin A\sin B[/tex]

[tex]f(t)=16(\cos t\cos (\frac{\pi}{6})+\sin t\sin(\frac{\pi}{6})[/tex]

[tex]f(t)=16(\frac{\sqrt3}{2}\cos t+\frac{1}{2}\sin t)[/tex]

[tex]f(t)=\frac{16}{2}(\sqrt3\cos t+\sin t)[/tex]

[tex]f(t)=8(\sqrt3\cos t+\sin t)[/tex]

We know, [tex]L(\cos at)=\frac{s}{s^2+a^2}[/tex] and [tex]L(\sin at)=\frac{a}{s^2+a^2}[/tex]

Applying Laplace in function,

[tex]L\{f(t)\}=8\sqrt3L(\cos t)+8L(\sin t)[/tex]

[tex]L\{f(t)\}=8\sqrt3(\frac{s}{s^2+1})+8(\frac{1}{s^2+1})[/tex]

[tex]L\{f(t)\}=\frac{8\sqrt3s+8}{s^2+1}[/tex]

[tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]

Therefore, The Laplace transformation is [tex]L\{f(t)\}=\frac{8(\sqrt3s+1)}{s^2+1}[/tex]

Final answer:

In this college-level mathematics question, the task is to find ℒ{f(t)} by employing a trigonometric identity. By rewriting the function f(t) = 16cos(t−π/6) in terms of sine and utilizing a trigonometric identity, we find ℒ{f(t)} = 16s / (s^2 + 1).

Explanation:

To find ℒ{f(t)} by utilizing a trigonometric identity, first rewrite the function f(t) = 16cos(t−π/6) in terms of sine. Use the trigonometric identity cos(a) = sin(a + π/2) to rewrite cos(t−π/6) as sin(t−π/6 + π/2). This simplifies to sin(t−π/3). Thus, ℒ{f(t)} = 16 * ℒ{cos(t−π/6)}

= 16 * ℒ{sin(t−π/3)} = 16 * (s / (s^2 + 1)).

Therefore, the answer is 16s / (s^2 + 1).

Answer:

The desviation is 8 4/7 or 8.571

Step-by-step explanation:

The conversion for any variable X to a standard z is

[tex]Z=\frac{X-\[Mu]}\\{\[Sigma]}[/tex]

where mu is the mean and sigma de desviation

You can find the value of Z whith the tables of the accumulated probability function. The accumulated probability for the mean is 0.5 .Remenber that the accumlated probability function represent the area at the left of an abscissa. Then

0.5+0.3531=0.8531

Acording to the accumulated probability function table, a Z=1.05 has an area of 0.8531 at its left.

Now it is only solving the equation

[tex]\[Sigma]=\frac{X-\[Mu]}{Z}[/tex]

σ=[tex]\frac{34-25}{1.05}[/tex]

σ=8.571

Answer:

Step-by-step explanation:

Sin t . Sin 3t . Sin 5t = 1/4 [ - Sin t + Sin 3t + Sin 7t - Sin 9t ]

Take Right hand side and use the following formula

[tex]sin C - sin D = 2 Cos\left ( \frac{C+D}{2} \right )Sin\left ( \frac{C-D}{2} \right )[/tex]

[tex]Cos C - Cos D = 2 Sin\left ( \frac{C+D}{2} \right )Sin\left ( \frac{D-C}{2} \right )[/tex]

Take right hand side

[tex]\frac{1}{4}\left (Sin 3t - Sin t + Sin 7t - Sin 9t  \right )[/tex]

[tex]\frac{1}{4}\left (2 Cos 2t Sin t +2 Sin (-t)Cos 8t \right )[/tex]

[tex]\frac{1}{4}\times 2 Sin t\left (Cos 2t-Cos8t \right )[/tex]

[tex]\frac{1}{4}\times 2 Sin t\ \times 2 \times Sin 5t\times 3t[/tex]

Sin t . Sin 3t . Sin 5t

So, LHS = RHS

Answer:

[tex]E(k)= \left[ \begin{array}{c} {6+0.10k\,\,if\,\,0\leq k \leq500} & 6+500\times0.10+0.11(k-500)\,\,if\,\,500<k\leq1000& 6+500\times0.10+500\times0.11+0.15(k-1000)\,\,if\,\,k>1000\end{array}[/tex]

Step-by-step explanation:

The cost function has 3 branches,

So the first branch the consumer pays 6 dollars plus .10cents for any additional kWh (k)

In the second, they pay the same as the first up to 500kWh, and after that they pay  0.11 for the additional kWh above 500: (k-500) but bellow 1000

In the third branch, for consumptions above 1000, they pay the fix amount, plus .10 for the first 500 ([tex]500\times0.10[/tex]) , .11 for the additional 500 ([tex]500\times0.11[/tex]) , and finally 0.15 for consumptions above 1000: [tex]0.15(k-1000)[/tex]

Final answer:

The monthly cost of electricity, E, as a function of the electricity used, x, is defined using a piecewise function with different rates for usage tiers, including a fixed base charge and variable rates for different consumption brackets.

Explanation:

To express the monthly cost E as a function of the amount x of electricity used by the customer, we consider the following rates:

Therefore, the function is a piecewise function defined as:

E(x) =

This function helps calculate the monthly electricity bill based on the usage levels. Remember that the first bracket is for the initial 500 kWh, the second bracket is for 501 to 1000 kWh, and the last bracket applies to usage above 1000 kWh.

Answer:

The solution of given equation are -1 and 5.

Step-by-step explanation:

The given equation is

[tex]|x-2|-3=0[/tex]

We need to solve the above equation by finding the zeros of

[tex]y=|x-2|-3[/tex]

The vertex form of an absolute function is

[tex]y=a|x-h|+k[/tex]

where, a is constant and (h,k) is vertex.

Here, h=2, k=-3. So vertex of the function is (2,-3).

The table of values is

   x           y

   0         -1

   2         -3

   4         -1

Plot these points on a coordinate plane and draw a V-shaped curve with vertex at (2,-3).

From the given graph it is clear that the graph intersect x-axis at -1 and 5. So, zeroes of the function y=|x-2|-3 are -1 and 5.

Therefore the solution of given equation are -1 and 5.

Now solve the given equation algebraically.

[tex]|x-2|-3=0[/tex]

Add 3 on both sides.

[tex]|x-2|=3[/tex]

[tex]x-2=\pm 3[/tex]

Add 2 on both sides.

[tex]x=\pm 3+2[/tex]

[tex]x=3+2[/tex] and [tex]x=-3+2[/tex]

[tex]x=5[/tex] and [tex]x=-1[/tex]

Therefore the solution of given equation are -1 and 5.

Answer: There are 133 chickens and 20 pigs.

Step-by-step explanation:

Let x be the number of chickens and y be the number of pigs.

Given : Number of heads of pigs and chicken = 153

Number of feet = 346

Since one chicken has 2 legs and one pig has 4 legs.

By considering the given information, we have the following system of equations:-

[tex]x+y=153------(1)\\\\ 2x+4y=346---------(2)[/tex]

Multiply 2 on both sides of (1), we get

[tex]2x+2y=306-----(3)[/tex]

Subtract (3) from (2), we get

[tex]2y=40\\\\\Rightarrow\ y=\dfrac{40}{2}=20[/tex]

Put value of y in (1), we get

[tex]x+20=153\\\\\Rightarrow\ x=153-20=133[/tex]

Hence, there are 133 chickens and 20 pigs.

Final answer:

The problem is a system of linear equations in Mathematics, where we find that there are 133 chickens and 20 pigs in the barnyard after setting up and solving the equations based on the given number of heads and feet.

Explanation:

Let's denote the number of chickens as C and the number of pigs as P. Therefore, we have two equations based on the given information:

C + P = 153 (since each animal has one head)

2C + 4P = 346 (since chickens have 2 feet and pigs have 4 feet)

By solving these equations, we can find the values for C and P. Multiplying the first equation by -2 and adding it to the second equation, we eliminate C and get:

-2C - 2P = -306
2C + 4P = 346
-----------------
2P = 40

Dividing both sides by 2, we find that P = 20. Substituting this value back into the first equation, we get C + 20 = 153, which means C = 133.

Therefore, there are 133 chickens and 20 pigs in the barnyard.

Answer:

46 minutes.

Step-by-step explanation:

You have a gift certificate worth $20 for one long-distance phone call.

The charge for the first minute = $1.10

Let the other additional minutes that you can talk = x

The charges for the x minutes = 0.42 per minute

the equation will be : 1.10 + 0.42x = 20

0.42x = 20 - 1.10

0.42x = 18.90

x = [tex]\frac{18.90}{0.42}[/tex]

x = 45

1 minute for $1.10 + 45 minutes for $0.42/min.

you can talk for 46 minutes.

The longest phone call you can make with a $20 gift certificate, given the cost structure of $1.10 for the first minute and $0.42 for each additional minute, is approximately 46 minutes.

The subject of this question is Mathematics, specifically related to linear equations and budget constraints in the context of phone call charges. To find the longest time you can talk on the phone using your gift certificate, you'll need to understand the cost structure. The charge is $1.10 for the first minute - that leaves you with $18.90 from the gift certificate for the remaining time ($20 - $1.10). Each additional minute costs $0.42. Now, divide the remaining amount in your gift certificate by the cost per additional minute: $18.90 ÷ $0.42 ≈ 45 minutes. Adding back the first minute, the longest call you can make with the gift certificate is approximately 46 minutes

.

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Answer:  [tex]30^{\circ},\ 60^{\circ},\ 90^{\circ}[/tex]

Step-by-step explanation:

We know that the sum of measure of all the angles of a triangle is 180°.

Given : The measure of the angles of a triangle are x, 2x, and 3x.

Then, the sum of all the angle will be given by :-

[tex]x+2x+3x=180^{\circ}\\\\\Rightarrow\ 6x=180^{\circ}\\\\\Rightarrow\ x=\dfrac{180^{\circ}}{6}=30^{\circ}[/tex]

Then, the measures of angles of the triangle will be : [tex]30^{\circ},\ 2(30^{\circ}),\ 3(30^{\circ})[/tex]

i.e. [tex]30^{\circ},\ 60^{\circ},\ 90^{\circ}[/tex]

Answer:

The cost to rent a trailer for 2.9 ​hours is $27.2.

The cost to rent a trailer for 3 ​hours is $28.

The cost to rent a trailer for 8.5 ​hours is $72.

The cost to rent a trailer for 9 ​hours is $76.

Step-by-step explanation:

It is given that the charge to rent a trailer is $20 for up to 2 hours plus $8 per additional hour or portion of an hour.

Let x be the number of hours.

The cost to rent a trailer for x ​hours is defined as

[tex]C(x)=\begin{cases}20 & \text{ if } x\leq 2 \\ 20+8(x-2) & \text{ if } x>2 \end{cases}[/tex]

For x>2,

[tex]C(x)=20+8(x-2)[/tex]

Substitute x=2.9 in the cost function.

[tex]C(x)=20+8(2.9-2)=27.2[/tex]

The cost to rent a trailer for 2.9 ​hours is $27.2.

Substitute x=3 in the cost function.

[tex]C(x)=20+8(3-2)=28[/tex]

The cost to rent a trailer for 3 ​hours is $28.

Substitute x=8.5 in the cost function.

[tex]C(x)=20+8(8.5-2)=72[/tex]

The cost to rent a trailer for 8.5 ​hours is $72.

Substitute x=9 in the cost function.

[tex]C(x)=20+8(9-2)=76[/tex]

The cost to rent a trailer for 9 ​hours is $76.

All ordered​ pairs, in the form of (hours,​ cost) are (2.9, 27.2), (3,28), (8.5, 72) and (9,76).

The graph of all ordered​ pairs is shown below.

Final answer:

The costs to rent a trailer for 2.9 and 3 hours are both $28, for 8.5 and 9 hours are both $76. To plot this function, one must mark these costs against the rental hours, showing a flat rate for the first two hours and additional charges thereafter.

Explanation:

The question asks for the cost of renting a trailer for varying numbers of hours and then requires plotting a graph with these costs against the hours. The rental system has a flat rate of $20 for the first 2 hours, and an additional charge of $8 for each hour or part of an hour thereafter.

To graph all ordered pairs, plot points for each of the time periods mentioned with their corresponding costs. Note how the graph demonstrates incremental jumps after the first 2 hours, reflecting the additional $8 charge per hour or part thereof.

Answer:

(x,y) = ([tex]\frac{9}{2}[/tex],[tex]\frac{1}{2}[/tex])

Step-by-step explanation:

We have to find point of intersection of two lines.

the given equations of line are:

10x - 4y = 43 - (1)

-3x - 3y = -15 - (2)

Multiplying the first equation by 3 we have:

(10x - 4y = 43)×3 = 30x - 12 y = 129 - (3)

Multiplying second equation by 10 we have :

(-3x - 3y = -15)×10 = -30x -30y = -150 - (4)

Now, adding equation (3) and (4)  we have:

-42y = -21

⇒ y = [tex]\frac{1}{2}[/tex]

Now, putting this value of y in equation (1), we have

10x - 2 = 43

⇒ 10x = 45

⇒x = [tex]\frac{9}{2}[/tex]

Hence, the intersection of given two lines is (x,y) = ([tex]\frac{9}{2}[/tex],[tex]\frac{1}{2}[/tex])

Answer:

[tex]y(x)\ =\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]

Step-by-step explanation:

Given differential equation is

[tex]x^2y"-xy'+y=0[/tex]                     (1)

Let's assume that

[tex]x=e^t[/tex]

[tex]=>\ t\ =\ logx[/tex]

then,

[tex]\dfrac{dx}{dt}=e^t[/tex]

[tex]and\ \dfrac{d^2x}{dt^2}=e^t[/tex]

We can write,

[tex]\dfrac{dy}{dx}=\dfrac{dy}{dt}.\dfrac{dt}{dx}[/tex]

                      [tex]=e^{-t}\dfrac{dy}{dt}[/tex]

Similarly,

[tex]\dfrac{d^2y}{dt^2}=\dfrac{d^2y}{dt^2}.\dfrac{dt^2}{dx^2}[/tex]

                            [tex]=e^{-2t}.\dfrac{d^2y}{dt^2}[/tex]

Putting these values in equation (1), we will get

[tex]e^{2t}.e^{-2t}.\dfrac{d^y}{dt^2}-e^t.e^{-t}\dfrac{dy}{dt}+y=0[/tex]

[tex]=>\dfrac{d^2y}{dt^2}-\dfrac{dy}{dt}+y=0[/tex]

So, the characteristics equation can be given as

[tex]D^2-D+1=0[/tex]

[tex]=>D\ =\ \dfrac{1+\sqrt{1-4}}{2}\ or\ \dfrac{1-1\sqrt{1-4}}{2}[/tex]

[tex]=>D=\ \dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\ or\ \dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}[/tex]

Hence, the general solution of the equation can be give by

[tex]y(t)\ =\ e^{\dfrac{t}{2}}[C_1cos\dfrac{\sqrt{3}}{2}t+C_2sin\dfrac{\sqrt{3}}{2}t][/tex]

Now, by putting the value of t in above solution, we will have

[tex]y(x)\ =\ e^{\dfrac{1}{2}logx}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]

[tex]y(x)=\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]

Hence, the solution of above given differential equation can be given by

[tex]y(x)=\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]

Answer:

The shop’s profit if it sold 150 units of sunscreen yesterday is $1000

Step-by-step explanation:

The total daily cost associated with the sale of this item yesterday was $200.

Cost of 1 unit = $8

So, Cost of 150 units = [tex]8 \times 150[/tex]

                                  = [tex]1200[/tex]  

So, revenue = $1200

Cost = $200

So Profit = Revenue - Cost

Profit = $1200-$200

Profit = $1000

Hence the shop’s profit if it sold 150 units of sunscreen yesterday is $1000

Answer:

The proof will be as follows

      Statement                                  Reason

1. -2(x+1) = 8                                      Given

2.  -2(x+1)/ -2 = 8/-2                          Division Property of equality

3. x+1 = -4                                          Simplifying

4. x+1-1 = -4-1                                      Subtraction property of Equality

5. x = -5                                             Simplifying

The option of Distributive property of equality will not be used ..

Answer:

The area of the region between the curves y=6x^2 and y=4x is 8/27

Step-by-step explanation:

Use the diagram to visualize the problem, the area colored of blue is the one that needs to be found, let's do it in 3 parts:

Part 1: Find the intersection points of the curves

To do this we put both equations in one and solve it for x:

[tex]6x^2=4x[/tex]

[tex]6x^2-4x=0\\2x(3x-2)=0[/tex]

This equation has 2 possible solutions:

x=0 and x=2/3, so the interval for integration is 0 <= x <= 2/3

Part 2: Find the area below each curve

[tex]A_{blue}=\int\limits^0_{2/3} {6x^2} \, dx \\A_{blue}=2x^3[/tex], evaluate in 0 and 2/3

[tex]A_{blue}=\frac{16}{27}[/tex]

[tex]A_{red}=\int\limits^0_{2/3} {4x} \, dx \\A_{red}=2x^2[/tex], evaluate in 0 and 2/3

[tex]A_{red}=\frac{8}{9}[/tex]

Part 3: Substract the area of the blue curve (y=6x^2) to the area of the red curve (y=4x)

[tex]Area=\frac{8}{9}-\frac{16}{27}\\Area=\frac{8}{27}[/tex]

Answer:

83%

Step-by-step explanation:

100% - 17% = 83%

Cutting the number of pages by 17% leaves 83% of the original number of pages.

After reducing the number of pages by 17%, the final version of the book contains 83% of the original number of pages.

If the author of a book was told to cut the number of pages by 17%, we want to determine what percent of the original pages remained in the final version. To do this, we subtract the percentage of the pages cut from 100% (which represents the original number of pages).

100% - 17% = 83%.

Therefore, after a 17% cut, 83% percent of the pages are left in the final version of the book.

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Answer:

First one: Its place value is 600,000.

Second one: Its place value is 600.

Step-by-step explanation:

Answer:

(a) The confidence interval is: 0.0304 ≤ π ≤ 0.0830.

(b) Upper confidence bound = 0.0787

Step-by-step explanation:

(a) The confidence interval for p (proportion) can be calculated as

[tex]p \pm z*\sigma_{p}[/tex]

[tex]\sigma=\sqrt{\frac{\pi*(1-\pi)}{N} }\approx\sqrt{\frac{p(1-p)}{N} }[/tex]

NOTE: π is the proportion ot the population, but it is unknown. It can be estimated as p.

[tex]p=17/300=0.0567\\\\\sigma=\sqrt{\frac{p(1-p)}{N} }=\sqrt{\frac{0.0567(1-0.0567)}{300} }=0.0134[/tex]

For a 95% two-sided confidence interval, z=±1.96, so

[tex]\\LL = p-z*\sigma=0.0567 - (1.96)(0.0134) = 0.0304\\UL =p+z*\sigma= 0.0567 + (1.96)(0.0134) = 0.0830\\\\[/tex]

The confidence interval is: 0.0304 ≤ π ≤ 0.0830.

(b) The confidence interval now has only an upper limit, so z is now 1.64.

[tex]UL =p+z*\sigma= 0.0567 + (1.64)(0.0134) = 0.0787[/tex]

The confidence interval is: -∞ ≤ π ≤ 0.0787.

Final answer:

To calculate a confidence interval for the fraction of defective circuits, use the formula for the confidence interval of a proportion. The 95% two-sided confidence interval for the fraction of defective circuits is 0.0182 ≤ p ≤ 0.0951. The 95% upper confidence bound on the fraction of defective circuits is 0.0951.

Explanation:

To calculate a confidence interval for the fraction of defective circuits, we can use the formula for the confidence interval of a proportion. Let p-hat be the proportion of defectives in the sample, which is equal to 17/300 = 0.0567. We can calculate the standard error as [tex]\sqrt{((p-hat*(1-p-hat))/n)[/tex], where n is the sample size.

Using a 95% confidence level, we can find the critical value from the standard normal distribution, which is approximately 1.96. The lower bound of the confidence interval is given by p-hat - z*(standard error), and the upper bound is given by p-hat + z*(standard error).

Therefore, the 95% two-sided confidence interval for the fraction of defective circuits is 0.0182 ≤ p ≤ 0.0951. The 95% upper confidence bound on the fraction of defective circuits is 0.0951.

Answer:

Yes, a brain volume of [tex] 1392.7 cm^{3}[/tex] is significantly high.

Step-by-step explanation:

Given

The range rule of thumb indicates that the usual values are within 2 standard deviations from the mean:

minimum usual value = mean - 2 * standard deviation = [tex]1094.9 - 2*128.9=837.1 cm^{3}[/tex]

maximum usual value = mean + 2 * standard deviation = [tex]1094.9 + 2*128.9=1352.7 cm^{3}[/tex]

We can see that [tex] 1392.7 cm^{3}[/tex] is not between [tex] 837.1 cm^{3}[/tex] and [tex] 1352.7 cm^{3}[/tex], which indicates that this value is unusually high.

Using the range rule of thumb, it is found that:

----------------------------

The range rule of thumb states that:

----------------------------

----------------------------

[tex]1094.9 - 2(128.9) = 837.1[/tex]

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[tex]1094.9 + 2(128.9) = 1352.7[/tex]

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A similar problem is given at https://brainly.com/question/24126815

Final answer:

There are 108 students in the class, calculated using the data from the class survey with the principle of inclusion-exclusion.

Explanation:

To find out how many students are in the class, we should analyze the survey data provided. We know that:

29 students watched TV on Monday.

24 students watched TV on Tuesday.

25 students watched TV on Wednesday.

13 students watched TV on only Monday.

9 students watched TV on only Tuesday.

10 students watched TV on only Wednesday.

12 students watched TV on all three days.

14 students watched TV on both Monday and Tuesday.

Since 14 watched on both Monday and Tuesday, and 12 of those also watched on Wednesday, there are 14 - 12 = 2 who watched on Monday and Tuesday but not Wednesday.

We can now calculate the total number of students using the principle of inclusion-exclusion:

Start with the total number of students who watched TV each day: 29 + 24 + 25 = 78.

Subtract the students who were counted twice because they watched on two different days: 78 - 2 (from Monday and Tuesday) = 76.

Subtract the students who watched on all three days, as they have been counted three times: 76 - 12 = 64.

Add back the number of students who watched TV on all three days to account for their previous subtraction: 64 + 12 = 76.

Add the students who watched TV on only one specific day to avoid double-counting those who were included in the two and three days' viewership: 76 + 13 (Monday only) + 9 (Tuesday only) + 10 (Wednesday only) = 108.

Therefore, there are 108 students in the class.

Answer:

Described

Step-by-step explanation:

A solution becomes infeasible when no solution exit and which satisfies all the constraints. We will consider two basic types of infeasibility. The 1st we will call continuous infeasibility and the second one is discrete or integer infeasibility. Continuous infeasibility is the one where a non–MIP problem is infeasible. In this case the feasible region defined by the intersecting constraints is empty. Discrete or integer infeasibility is the one  where a MIP problem has a feasible relaxation (note that a relaxation of a MIP is the problem we get when we drop the discreteness required on the variables) but the feasible region of the relaxation contains no solution that satisfies the discreteness requirement.

Final answer:

An infeasible solution occurs when constraints of a problem conflict with each other, making it impossible to find a solution that satisfies all conditions. This is common in linear programming and can happen due to poor design decisions or unreasonable premises. Brainstorming alternative solutions is essential for finding feasible approaches.

Explanation:

An infeasible solution occurs in problem-solving when the constraints of a problem are such that no possible solution satisfies all the conditions. In mathematics and particularly in linear programming, this can happen when the set of constraints creates a situation where no intersection point exists between all the constraint equations - essentially, the requirements for a solution are mutually exclusive and cannot be met simultaneously.

For example, a situation may arise where a design problem contains a premise that produces an unreasonable result, highlighting that some design choices may not align with the physical reality, and thus are infeasible. An extremely large Coulomb force in a physics problem, due to the assumption of an excessively large separated charge, is an instance where the result is obviously incorrect and infeasible based on known physical principles.

When approaching such problems, a common strategy is to brainstorm alternative solutions and explore a wide variety of ideas that could potentially solve the problem within realistic constraints. Some ideas may be discarded as infeasible early on if they conflict with the basic laws of nature or if they stand little chance of practical application. This process is especially useful in engineering and design fields, where practicality and adherence to real-world principles are crucial.

Answer: -2.12°C

Step-by-step explanation:

Let x denotes the reading of the thermometers .

We assume that the readings on the thermometers are normally distributed.

Let a be the reading that separates the rejected thermometers from the others.

Given: Population mean : [tex]\mu=0[/tex]

Standard deviation: [tex]\sigma= 1[/tex]

Also, [tex]P(x<a)=0.017[/tex]

By using the z-table , the z-value corresponds to the p-value (one -tailed)0.017 is [tex]\pm2.12[/tex].

Now, [tex]z=\dfrac{a-\mu}{\sigma}[/tex]

i.e. [tex]\pm2.12=\dfrac{a-0}{1}[/tex]

i.e. [tex]\pm2.12=a[/tex]

For left tailed , [tex]a=-2.12[/tex]

It means the reading that separates the rejected thermometers from the others = -2.12°C.

By DeMoivre's theorem,

[tex]\cos(4\theta)+i\sin(4\theta)=(\cos\theta+i\sin\theta)^4[/tex]

Expanding the right side gives

[tex]\cos^4\theta+4i\cos^3\theta\sin\theta-6\cos^2\theta\sin^2\theta-4i\cos\theta\sin^3\theta+\sin^4\theta[/tex]

Equating imaginary parts tells us

[tex]\sin(4\theta)=4\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta[/tex]

(Not sure what you mean by sin(e) and cos(e)...)

Answer:

a) 59.98

b) 2.99

c) 2.99

d) Significantly High

Step-by-step explanation:

Part a)

Highest speed measured = x = 75.6 Mbps

Average/Mean speed = [tex]\overline{x}[/tex] = 15.62 Mbps

Standard Deviation = s = 20.03 Mbps

We need to find the difference between carrier's highest data speed and the mean of all 50 data​ speeds i.e. x - [tex]\overline{x}[/tex]

x - [tex]\overline{x}[/tex] = 75.6 - 15.62 = 59.98 Mbps

Thus, the difference between​ carrier's highest data speed and the mean of all 50 data​ speeds is 59.98 Mbps

Part b)

In order to find how many standard deviations away is the difference found in previous part, we divide the difference by the value of standard deviation i.e.

[tex]\frac{59.98}{20.03}=2.99[/tex]

This means, the difference is 2.99 standard deviations or in other words we can say, the Carrier's highest data speed is 2.99 standard deviations above the mean data speed.

Part c)

A z score tells us that how many standard deviations away is a value from the mean. We calculated the same in the previous part. Performing the same calculation in one step:

The formula for the z score is:

[tex]z=\frac{x-\overline{x}}{s}[/tex]

Using the given values, we get:

[tex]z=\frac{75.6-15.62}{20.03}=2.99[/tex]

Thus, the Carriers highest data is equivalent to a z score of 2.99

Part d)

The range of z scores which are neither significantly low nor significantly​ high is -2 to + 2. The z scores outside this range will be significant.

Since, the z score for carrier's highest data speed is 2.99 which is well outside the given range, i.e. greater than 2, we can conclude that the  carrier's highest data speed​ is significantly higher.

Final answer:

The highest data speed recorded is 59.98 Mbps above the mean, and this difference equals approximately 2.996 standard deviations. This results in a z-score of 2.996, indicating that the carrier's highest data speed is significantly high.

Explanation:

a. The difference between the carrier's highest data speed and the mean of all 50 data speeds is found by subtracting the mean speed from the highest speed. This is calculated as 75.6 Mbps - 15.62 Mbps = 59.98 Mbps.

b. To find out how many standard deviations this difference is, we divide the difference by the standard deviation of the data speeds: 59.98 Mbps / 20.03 Mbps = 2.996 standard deviations.

c. The z-score for the carrier's highest data speed is calculated by subtracting the mean from the data speed and then dividing by the standard deviation: (75.6 Mbps - 15.62 Mbps) / 20.03 Mbps = 2.996.

d. Since the z-score of 2.996 is greater than 2, it suggests that the carrier's highest data speed is significantly higher than what is considered neither significantly low nor high. In other words, the carrier's highest data speed is significant.

Final answer:

The expected time between successive mosquito bites is calculated using the probabilities of landing and biting, resulting in an average time of 10 seconds between bites.

Explanation:

The expected time between successive mosquito bites can be calculated by considering the probabilities of two independent events: a mosquito landing on your neck and a mosquito bite, given that it has landed. As per the data provided, a mosquito lands with a probability of 0.5 per second, and out of those, it will bite with a probability of 0.2. The probability of getting bitten by a mosquito that has landed is then 0.5 (probability of landing) × 0.2 (probability of biting) = 0.1 per second.

To find the expected time between successive bites, we need to consider the inverse of this probability, which tells us that on average, you can expect to get bitten every 1/0.1 or 10 seconds.

The expected time between bites is 10 seconds.

To find the expected time between successive mosquito bites, we need to consider the probability of a mosquito biting you at any given second.

Given the probabilities:

The combined probability that a mosquito both lands and bites in any given second is the product of the two probabilities:
[tex]P(bite) = P(land) * P(bite\ |\ land) = 0.5 * 0.2 = 0.1.[/tex]

To find the expected time between successive bites, we take the reciprocal of the probability of being bitten in a given second:
[tex]Expected time = \frac{1}{P(bite)} = \frac{1}{0.1} = 10\ seconds[/tex].

The expected time between successive mosquito bites is approximately 10 seconds.

Answer:

Case 1

None of the sides is yellow.

Probability of rolling a yellow on your next toss: zero %

Case 2

All the sides are yellow.

Probability of rolling a yellow on your next toss: 100 %

Case 3  

At least one of the sides is yellow.

[tex]\frac{q}{N}\times100\%[/tex]

where  

N = total number of tosses  in one hour.

q = number of tosses you rolled a yellow.

Step-by-step explanation:

Case 1

None of the sides is yellow.

Probability of rolling a yellow on your next toss: zero %

Case 2

All the sides are yellow.

Probability of rolling a yellow on your next toss: 100 %

Case 3  

At least one of the sides is yellow.

As you do not know if the die is fair or not, the only way to approximate a probability of rolling a yellow is by making a table of frequencies and record the times you have rolled yellow.

If the number of tosses made in an hour is big enough as to draw a conclusion, then according to the Law of Large Numbers, the probability of rolling a yellow in one toss of the die should be

[tex]\frac{q}{N}\times100\%[/tex]

where  

N = total number of tosses

q = number of tosses you rolled a yellow.

Now, suppose you want to roll the dice once more.  

As the event of rolling a die is independent of the previous tosses, this means that the probable outcome of the event does not depend on the previous results. So, the probability remains the same.  

Probability of tossing a yellow on your next toss of the die

[tex]\frac{q}{N}\times100\%[/tex]

where  

N = total number of previous tosses

q = number of tosses you rolled a yellow.

Step-by-step explanation:

A direct proof is a method that takes an statement p, which we assume to be true, and use it to show directly that another statement q is true. So this method has the following steps:

Fact that we need to use:

Every odd integer can be written in the form 2m + 1 for some unique other integer m

Let p be the statement a and b be odd integers and q be the statement that the product of a and b is odd.

Proposition if a and b are odd, then the product of a and b is odd

Proof: Assume that a and b are odd integers, the by definition a = 2m + 1 and b = 2n + 1 for some integers m and n. we will now use this to show that the product of a and b is odd.

[tex]a\cdot b= (2m+1) \cdot (2n+1)\\a\cdot b = 2m\cdot 2n+2m+2n+1\\a\cdot b =4mn+2m+2n+1\\a\cdot b = 2(2mn+2m+2n) +1\\\:If  \:k=2mn+2m+2n\\a\cdot b = 2k+1[/tex]

Hence we have shown that the product of a and b is odd since 2k + 1 is and odd integer. Therefore we have shown that p ⇒ q and so we have completed our proof.

Answer:

Step-by-step explanation:

The proof by the direct method that the product of two odd numbers is odd integer number, is the following:

Let [tex]z_1[/tex] and [tex]z_2[/tex] be two odd integers, then [tex]z_1 = 2a+1[/tex] and [tex]z_2 = 2b +1[/tex], for some integers a and b.

[tex]z_1z_2 = (2a + 1) (2b + 1)\\\\z_1z_2 = 4ab + 2a + 2b + 1\\\\z_1z_2 = 2 (2ab + a + b) +1\\\\z_1z_2 = 2n + 1[/tex]

where [tex]n = 2ab + a + b[/tex], which guarantees that [tex]n[/tex] is an integer number. In this way, [tex]z_1z_2[/tex] is an odd integer.