Are Sin(x) and e^x linearly independent? Justify.

Answer: [tex]H_0:\mu=453[/tex]

[tex]H_1:\mu\neq453[/tex]

Step-by-step explanation:

Given : An operator of chocolate-packaging machine monitors the net weights of the packaged boxes by periodically weighing random samples of boxes.

The operator wishes to see if a random sample of 50 boxes gives evidence that the mean is different than 453 grams.

i.e. Claim : [tex]\mu\neq 453[/tex]

We know that the null hypothesis is a kind of claim which always takes equals sign but alternative hypothesis does not.

Thus , the null and alternative hypotheses for the given situation will be :-

[tex]\text{Null hypothesis}\ H_0:\mu=453\\\text{Alternative hypothesis}\ H_1:\mu\neq453[/tex]

Answer:

  y = 11(x -1) +6

Step-by-step explanation:

The slope at the given point is ...

  y' = 14x -3x^2 = 14(1) -3(1^2) = 11

Then the point-slope equation for the line can be written as ...

  y = m(x -h) +k . . . . . . for slope m at point (h, k)

  y = 11(x -1) +6

The equation of the tangent line to the curve y = 7x^2 - x^3 at the point (1,6) is y = 11x - 5.

In

Mathematics

, specifically in Calculus, the equation of the tangent line to a curve can be found by first taking the derivative of the function. The derivative of the function y = 7x^2 - x^3 is y'= 14x - 3x^2. To find the slope of the tangent line at the point (1,6), we substitute x = 1 into the derivative to get y'(1) = 14*1 - 3*1^2  = 11. So the slope of the tangent line at the point (1,6) is 11. Using the point-slope formula, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the line, we substitute to get the equation of the line: y - 6 = 11(x - 1). Simplifying this, we find the equation of the tangent line to be y = 11x - 5.

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Answer:

There are 225 adult tickets.

Step-by-step explanation:

Let the number of adult tickets but represented by A while the number of student tickets be represented by S.

We are told the total is 391, so A+S=391.

There are 59 fewer student tickets (S) sold than adult tickets (T).

So there are 59 more adult tickets which means, A=59+S.

We are going to solve the system:

A+S=391

A=59+S

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

We are going to input the second equation into the first:

A+S=391 with A=59+S

(59+S)+S=391

59+S+S=391

59+2S=391

Subtract 59 on both sides:

2S=391-59

2S=332

Divide both sides by 2:

S=332/2

S=166

So A=59+S given S=166 means A=59+166=225.

There are 225 adult tickets while there is 166 student tickets.

Let's say the adult tickets would be X and the student tickets would be

X - 59.

Let's add them up. X + X - 59 = 391

2x = 450

x = 225

If there are 225 student tickets, we know that there are 63 fewer adult tickets so we take 225 and subtract 59 and we will get an answer of 166.

225 adult tickets were sold.

Answer:

Step-by-step explanation:

Given that X is the height, in inches, of American men. The mean and standard deviation for American men's height are 69 inches and 3 inches, respectively.

X is N(69,3)

If a sample of 25 american men are selected the mean would remain the same.

But std dev becomes 3/sqrt 25=0.60

Hence sample height will follow

N(69, 0.60)

The probability that a bottle of ketchup contains less than 17.5 oz, given a normal distribution with a mean of 17.6 oz and a standard deviation of 0.05 oz, is approximately 2.28%.

The question is asking for the probability that a bottle of ketchup contains less than 17.5 oz, given that the distribution of ketchup amounts in the bottle is normally distributed with a mean of 17.6 oz and a standard deviation of 0.05 oz. In terms of a normal distribution chart, we are looking for the area under the curve to the left of 17.5 oz.

To find this, we need to first convert our 17.5 oz to a standard score (or z-score). The z-score represents how many standard deviations away from the mean a data point is. It is calculated using the formula:

z = (X - μ) / σ

Where X is the data point (17.5 oz), μ is the mean (17.6 oz), and σ is the standard deviation (0.05 oz). Using this formula, we find z = -2.

Now, to find the probability, we can refer to a standard normal distribution table. The area under the curve to the left of z = -2 (which represents a data point of 17.5 oz) is approximately 0.0228, or 2.28%. Therefore, the probability that a bottle of ketchup contains less than 17.5 oz is about 2.28%.

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Answer:

6.67 ft

Step-by-step explanation:

   Let d = distance

 and w = weight

Then d = k/w

 or dw = k

 Let d1 and w1 represent Roger

and d2 and w2 represent Ellen. Then

d1w1 = d2w2

Data:

d1 = 6 ft; w1  = 120 lb

d2 = ?    ; w2 = 108 lb

Calculation:

6 × 120 = 108d2

       720 = 108d2

         d2 = 720/108 = 6.67 ft

Ellen must sit 6.67 ft from the fulcrum.

Ellen must sit approximately 6.67 feet from the fulcrum to balance the seesaw.

To solve this problem, we'll use the principle of inverse proportionality. The relationship between the distance from the fulcrum and the weight of the person can be expressed as:

[tex]\[ \text{Distance} \times \text{Weight} = \text{Constant} \][/tex]

Step 1:

Determine the constant of proportionality.

Since Roger weighs 120 pounds and sits 6 feet from the fulcrum, we can use his information to find the constant:

[tex]\[ 6 \times 120 = 720 \][/tex]

Step 2:

Use the constant to find Ellen's distance from the fulcrum.

Ellen weighs 108 pounds. We'll use the constant to find her distance:

[tex]\[ \text{Distance} \times 108 = 720 \][/tex]

[tex]\[ \text{Distance} = \frac{720}{108} \][/tex]

Step 3:

Calculate Ellen's distance from the fulcrum.

[tex]\[ \text{Distance} \approx 6.67 \text{ feet} \][/tex]

Therefore, Ellen must sit approximately 6.67 feet from the fulcrum to balance the seesaw.

Answer:

The gender of the students in a class

Step-by-step explanation:

Quantitative data is the data which can be counted or measured and expressed in terms of numbers with certain units of measurements with it.

Cholesterol level can be measured hence its quantitative data.

Exams scores of students can be counted and recorded in terms of numbers, hence this is quantitative data.

Gender of students cannot be measured or counted. This is a characteristic and ix expressed in words. Hence, this is not a quantitative data. Such a data is known as qualitative data

Number of siblings can be counted, hence this is quantitative data

Amount of sleep can be measured, hence this is quantitative data.

Final answer:

The one example of qualitative data among the options provided is the gender of the students in a class, as it is a descriptor, not expressed numerically.

Explanation:

The question concerns quantitative and qualitative data. Among the given options, the one that is not an example of quantitative data is the gender of the students in a class. Quantitative data involve numbers and include both discrete and continuous data. Discrete data refer to counts such as the number of siblings, while continuous data involve measurements and can include decimals like weight or amount of sleep.

Cholesterol levels, exam scores, number of siblings, and amount of sleep are all examples of quantitative data because they can be measured or counted and expressed numerically. On the other hand, gender is qualitative data as it is categoric and describes a characteristic or quality that cannot be counted.

Looks like we have

[tex]\vec F(x,y,z)=z^2x\,\vec\imath+\left(\dfrac{y^3}3+\sin z\right)\,\vec\jmath+(x^2z+y^2)\,\vec k[/tex]

which has divergence

[tex]\nabla\cdot\vec F(x,y,z)=\dfrac{\partial(z^2x)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial z}=z^2+y^2+x^2[/tex]

By the divergence theorem, the integral of [tex]\vec F[/tex] across [tex]S[/tex] is equal to the integral of [tex]\nabla\cdot\vec F[/tex] over [tex]R[/tex], where [tex]R[/tex] is the region enclosed by [tex]S[/tex]. Of course, [tex]S[/tex] is not a closed surface, but we can make it so by closing off the hemisphere [tex]S[/tex] by attaching it to the disk [tex]x^2+y^2\le1[/tex] (call it [tex]D[/tex]) so that [tex]R[/tex] has boundary [tex]S\cup D[/tex].

Then by the divergence theorem,

[tex]\displaystyle\iint_{S\cup D}\vec F\cdot\mathrm d\vec S=\iiint_R(x^2+y^2+z^2)\,\mathrm dV[/tex]

Compute the integral in spherical coordinates, setting

[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

so that the integral is

[tex]\displaystyle\iiint_R(x^2+y^2+z^2)\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^1\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{2\pi}5[/tex]

The integral of [tex]\vec F[/tex] across [tex]S\cup D[/tex] is equal to the integral of [tex]\vec F[/tex] across [tex]S[/tex] plus the integral across [tex]D[/tex] (without outward orientation, so that

[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\frac{2\pi}5-\iint_D\vec F\cdot\mathrm d\vec S[/tex]

Parameterize [tex]D[/tex] by

[tex]\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath[/tex]

with [tex]0\le u\le1[/tex] and [tex]0\le v\le2\pi[/tex]. Take the normal vector to [tex]D[/tex] to be

[tex]\dfrac{\partial\vec s}{\partial v}\times\dfrac{\partial\vec s}{\partial u}=-u\,\vec k[/tex]

Then we have

[tex]\displaystyle\iint_D\vec F\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^1\left(\frac{u^3}3\sin^3v\,\vec\jmath+u^2\sin^2v\,\vec k\right)\times(-u\,\vec k)\,\mathrm du\,\mathrm dv[/tex]

[tex]=\displaystyle-\int_0^{2\pi}\int_0^1u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac\pi4[/tex]

Finally,

[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\frac{2\pi}5-\left(-\frac\pi4\right)=\boxed{\frac{13\pi}{20}}[/tex]

To utilize the Divergence Theorem, compute the divergence of the vector field and create a closed surface by including a flat disk S1. As F and dS on S1 are orthogonal, the surface integral over S1 is zero. The surface integral over S is thus equal to the integral over S2.

To evaluate the surface integral using the Divergence Theorem, we must first calculate the divergence of the vector field F(x, y, z). That means we take the partial derivative of the i-component with respect to x, the j-component with respect to y, and the k-component with respect to z. Do this to get the divergence of F.

Next, we turn S into a closed surface by adding the disk S1. By Divergence Theorem, the surface integral over S is equal to the triple integral of the divergence of F over the volume enclosed by S. But, the surface integral over S is the sum of the integrals over S1 and S2.

The integral over S1 is easy to compute because S1 is a flat disk in the xy-plane and F always points in the z-direction, so F and dS are orthogonal. Hence the dot product F· dS equals zero and so the integral over S1 equals zero. Thus, the surface integral over S equals the integral over S2.

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Answer:

The correct answer here is option c) 0.05 mg(milligram)

Step-by-step explanation:

The first step here to calculate the milligram of active ingredient in a 5% solution per milliliter is to assume that Z milligram of active ingredient is present in the 5% solution per milliliter.

   NOTE* -    milligram ( mg) and milliliter (ml)

so therefore, the equation would become,

5\% = \frac{Z mg}{1 ml}

.05 = \frac{Z mg}{1 ml}

.05 x 1 = Z mg

therefore there is .05 mg of active ingredient in the 5% solution per ml.

The correct option is A. 50 mg.  This is calculated by knowing that 5% means 5 grams or 5000 milligrams in 100 millilitres, which divides down to 50 milligrams per millilitre.

To determine how many milligrams of active ingredients are in a 5% solution per millilitres, we need to understand what a 5% solution means. A 5% solution means that there are 5 grams of the active ingredient in every 100 millilitres of the solution.

Let's break it down step-by-step:

→ 5% solution means 5 grams of active ingredient per 100 millilitres (mL) of solution.

→ Convert grams to milligrams: 5 grams = 5000 milligrams.

→ Therefore, 100 mL of solution contains 5000 milligrams of active ingredient.

To find out how many milligrams are in 1 mL, we divide the total milligrams by the total millilitres:

→ 5000 milligrams / 100 mL = 50 milligrams per mL

Thus, the correct answer is A. 50 mg.

Answer:

13

Step-by-step explanation:

For m||n to be true, we would need to have those angles equal, or 7x-21 = 4x+18.

Solve this equation, and get 3x=39 or x=13.

value of  x = 13.

Equal angles are those that are vertically opposed, such as a = d and b = c. Adjacent angles add up to 180 degrees, as in the cases where a + b and a + c. Angles that correspond to one another are equal, such as a=e, b=f, c=g, and d=h. Interior angles combine to make 180 degrees, such as c plus e or d plus f. Equal alternate angles are c = f and d = e.

Given a figure, one line is intersecting the two parallel lines known as m and n. since both given angles are equal alternate angles hence both angles would be the same.

Hence,

4x + 18 = 7x - 21

x =13

Therefore, in the given drawing the value of x needs to be 13.

Learn more about angle here:

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Answer:

Robin's gross pay=$50,400

Step-by-step explanation:

We are given that Robin has 15% of his gross pay directly into his mutual fund account each month.

If Robin  deposited each month =$630

We have to find the gross pay

Let Robin's gross pay =x

15 % of gross pay= 15% of x=[tex]\frac{15}{100}\times x=\frac{15}{100}x[/tex]

We know that 12 month in a year

[tex] \frac{15}{100}x=630\times 12[/tex]

[tex] 15x=630\times 100\times 12[/tex]

Using multiplication property of equality

[tex] x=\frac{756000}{15}[/tex]

Using division property of equality

[tex]x=50400[/tex]

Hence, Robin's gross pay=$50,400

Answer:

d) stigma

Step-by-step explanation:

Secondary deviance marks the start of what Erving Goffman called a deviant career, which results in the acquisition of a stigma.

The correct answer is d) stigma.

According to Erving Goffman, secondary deviance is the process that occurs when an individual who has been labeled as deviant begins to accept the label and act in accordance with it. This acceptance and incorporation of the deviant label into one's self-concept can lead to a deviant career, which is a pattern of behavior that increasingly conforms to the label. The result of this process is the acquisition of a stigma, which is a powerful and discrediting social label that radically changes a person's self-concept and social identity.

To understand this concept, let's break it down:

- Primary deviance is the initial act of rule-breaking that may or may not be known to others. It does not necessarily lead to a deviant career.

- Secondary deviance occurs when others react to the primary deviance by labeling the individual as deviant. This labeling can lead to a self-fulfilling prophecy where the individual begins to adopt the identity of a deviant.

- The term ""stigma"" refers to a mark of disgrace associated with a particular circumstance, quality, or person. Goffman describes stigma as a discrepancy between virtual social identity and actual social identity.

- ""Anomie"" is a concept developed by Émile Durkheim, referring to a state of normlessness in society that can lead to deviant behavior, but it is not directly related to the concept of a deviant career as described by Goffman.

- ""Verstehen"" is a term used by Max Weber, referring to a deep form of understanding that is essential for social science, but it is not related to the acquisition of a label due to deviant behavior.

- A ""criminal record"" is a formal legal document that records a person's criminal history, but it is not the term Goffman used to describe the social label acquired as a result of secondary deviance.

Therefore, the correct term that fits Goffman's description of the result of a deviant career following secondary deviance is ""stigma.""

Answer:

B. f−1(x) = The quantity of 4x plus 3, divided by 5

Step-by-step explanation:

Given

[tex]f(x) = \frac{5x-3}{4}[/tex]

We have to find the inverse of the function

[tex]Let\\f(x) = y\\y=\frac{5x-3}{4}\\4y=5x-3\\4y+3=5x\\\frac{4y+3}{5} =y[/tex]

So, the inverse of f(x) is:

[tex]\frac{4y+3}{5}[/tex]

Hence,

The correct answer is:

B. f−1(x) = The quantity of 4x plus 3, divided by 5 ..

Answer:

Rule of 9 : an integer is divisible by 9 if and only if the sum of its integers is divisible by 9.

Proof :

Let us consider a number x such that,

[tex]a=a_n......a_3a_2a_2[/tex]

Where, [tex]a_0, a_1, a_2......,a_n[/tex] are the the digits of the number x,

So, we can write,

[tex]x=a_0+a_1\times 10 + a_2\times 10^2 +a_3\times 10^3..........a_n\times 10^n[/tex]

Let,

[tex]S=a_0+a_1+a_2+a_3+.......a_n[/tex]

[tex]x-S=a_1(10-1)+a_2(10^2-1)+a_3(10^3-1)+.......a_n(10^n-1)[/tex]

Since, a number is in the form of [tex]10^k-1[/tex], where k is an positive integer, is always divisible by 9,

⇒ [tex]a_1(10-1), a_2(10^2-1), a_3(10^3-1),.......a_n(10^n-1)[/tex] are divisible by 9.

⇒ x-S is divisible by 9,

⇒ If S is divisible by 9 ⇒ x must divisible by 9,

Or if x is divisible by 9 ⇒ S must divisible by 9.

Hence, proved...

The 'rule of 9' states that an integer is divisible by 9 if and only if the sum of its digits is divisible by 9. For example, the integer 243 is divisible by 9 because the sum of its digits is 9, while the integer 175 is not divisible by 9 because the sum of its digits is 13.

The 'rule of 9' states that an integer is divisible by 9 if and only if the sum of its digits is divisible by 9. Let's take an example to prove this rule:

Consider the integer 243. The sum of its digits is 2 + 4 + 3 = 9, which is divisible by 9. Therefore, 243 is divisible by 9.

On the other hand, if we take an integer like 175, the sum of its digits is 1 + 7 + 5 = 13, which is not divisible by 9. Therefore, 175 is not divisible by 9.

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Answer:1.8701

Step-by-step explanation:

Area (A) of trapezium as marked is =[tex]\frac{1}{2}[/tex][tex]\left ( sum\ of\ parallel\ sides\right )\times height[/tex]

A=[tex]\frac{1}{2}[/tex][tex]\left ( 3+6\right )\times 4[/tex]

A=18[tex]in^2[/tex]

Now for Volume we have to multiply by length because area is same across the length.

volume(v)=[tex]A\times length[/tex]

Volume(v)=[tex]18\times 24[/tex]

Volume(v)=432 [tex]in^3[/tex]

Volume(v)=1.8701 gallon

The value is 5 weeks

To find out how many weeks the oil will last with a specific usage per week, divide the total oil amount by the weekly usage, resulting in 5 weeks.

The oil barrel holds 53 3/4 gallons of oil, and 10 3/4 gallons are used every week.

To determine how many weeks the oil will last, you need to divide the total amount of oil by the weekly usage.

Answer: There would be 9 possibilities.

Step-by-step explanation:

Given : The number of regions in the spinner = 3

if we spin the spinner twice, then by using fundamental principal of counting , the number of possibilities is given by :-

[tex]3\times3=9[/tex]

Therefore, there would be 9 possibilities.

Step-by-step explanation:

Given that

[tex]ax^2+bx+c=0[/tex]  

Where a,b,and c are odd integers.

We know that solution of quadratic equation given by following formula

[tex]z=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

Now we have to prove that above solution is a irrational number when a,b and c all are odd numbers.

Let's take a=3  ,b=9 ,c= -3   these are all odd numbers.

Now put the values

[tex]z=\dfrac{-9\pm \sqrt{9^2-4\times 3\times (-3)}}{2\times 3}[/tex]

So

[tex]z=\dfrac{-9\pm \sqrt{117}}{6}[/tex]

We know that square roots,cube roots etc are irrational number .So we can say that above value z is also a irrational number.

When we will put any values of a,b and c (they must be odd integers) oue solution will be always irrational.

Final answer:

The discriminator of the quadratic equation with odd a, b, and c is odd, making its square root and thus the solution z irrational.

Explanation:

Consider the quadratic equation ax2 + bx + c = 0, where a, b, and c are odd integers. To find the solution to this equation, we can use the quadratic formula.

The quadratic formula states that the solutions to the equation ax2 + bx + c = 0 are given by x = (-b ± sqrt(b2 - 4ac))/(2a). We need to consider the discriminant, b2 - 4ac, to prove whether the solution z is rational or irrational.

Since a, b, and c are odd integers, b2 is also an odd number (odd times odd is odd), and 4ac is an even number (even times odd is even). Therefore, b2 - 4ac is the difference between an odd and an even number, which is odd. An even number has a square root that is an integer or rational, but an odd number does not have an integer square root, making it an irrational number. Because the square root of an odd number is irrational, the solutions z must also be irrational.

Answer: 0.3413

Step-by-step explanation:

Given :Mean : [tex]\mu=60\text{ weeks}[/tex]

Standard deviation : [tex]\sigma = 5\text{ weeks}[/tex]

The formula for z -score :

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x= 60 ,

[tex]z=\dfrac{60-60}{5}=0[/tex]

For x= 65 ,

[tex]z=\dfrac{65-60}{5}=1[/tex]

The p-value = [tex]P(0<z<1)=P(z<1)-P(z<0)[/tex]

[tex]= 0.8413447-0.5= 0.3413447\approx0.3413[/tex]

Hence,  the probability that the project will take anywhere between 60 and 65 weeks to complete = 0.3413.

Final answer:

The probability that the project will take anywhere between 60 and 65 weeks to complete is 0.3413

Explanation:

To find the probability that the project will take anywhere between 60 and 65 weeks to complete, we need to calculate the z-scores for both 60 and 65 weeks using the given mean and standard deviation.

The formula to calculate the z-score is z = (x - mean) / standard deviation.

For 60 weeks, the z-score is (60 - 60) / 5 = 0, and for 65 weeks, the z-score is (65 - 60) / 5 = 1.

Next, we use the z-scores to find the corresponding probabilities using a standard normal distribution table or a calculator. The probability that the project will take anywhere between 60 and 65 weeks can be calculated as the difference between the two probabilities: P(60 ≤ x ≤ 65) = P(x ≤ 65) - P(x ≤ 60).

= 0.8413447 - 0.5

= 0.3413

Answer: 0.0038

Step-by-step explanation:

Given : Mean : [tex]\mu=\text{60 minutes}[/tex]

Standard deviation : [tex]\sigma = \text{9 minutes}[/tex]

Sample size = 36

The formula for z -score :

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x= 56 ,

[tex]z=\dfrac{56-60}{\dfrac{9}{\sqrt{36}}}=-2.67[/tex]

The p-value = [tex]P(z<-2.67)[/tex]

[tex]=0.0037925\approx0.0038[/tex]

Hence, the probability that the sample mean of the sampled students is less than 56 minutes  =0.0038.

To find the probability that the sample mean is less than 56, calculate the standard error of the mean, find the z-score corresponding to the value of 56, and then use a z-table to find the probability associated with this z-score. The resulting probability is around 0.0038.

The question relates to the probability that the sample mean of students' time taken to complete a statistics exam is less than 56 minutes, given that the time taken follows a left-skewed distribution with a mean of 60 minutes and standard deviation of 9 minutes.

First, we need to find the standard error of the mean. The standard error of the mean equals the standard deviation divided by the square root of the sample size. In our case, the standard deviation is 9 minutes, and the sample size is 36, so the standard error of the mean is 9/√36 = 1.5 minutes.

Next, we need to find the z-score. A z-score tells us how many standard deviations an element is from the mean. Here, we find the z-score using the formula (X - μ) / σ, where X is the value for which we're finding the z-score, μ is the mean, and σ is the standard deviation. In our case, X = 56 minutes, μ = 60 minutes, and σ = 1.5 minutes. Therefore, the z-score is (56 - 60) / 1.5 = -2.67.

Lastly, we use a z-table to find the probability that a z-score is less than -2.67. If we look this up in a z-table, we get a probability of around 0.0038.

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Answer: Let us assume that our income is $X.

In the first case:

1) We received a raise and our income increased to 135% of its previous amount.

"increased to" denotes that we will multiply our original income by the percentage given.

Therefore our income would be = $X * 1.35 = $1.35

X

In the second case:

2) We received a raise and our income increased by 135%.

"increased by" denotes that we have to add the given percentage of our income to our original income.

Therefore our income would be $X + $X*1.35= $2.35X

[tex]A=\{a,b,c,d\}\\|A|=4\\\mathcal{P}(A)=2^4=16[/tex]

Answer: The value of cos theta is -0.352.

Step-by-step explanation:

Since we have given that

[tex]\vec{u}=1\hat{i}+2\hat{j}-2\hat{k}\and\\\\\vec{w}=\hat{j}+4\hat{k}[/tex]

We need to find the cos theta between u and w.

As we know the formula for angle between two vectors.

[tex]\cos\ \theta=\dfrac{\vec{u}.\vec{w}}{\mid u\mid \mid w\mid}[/tex]

So, it becomes,

[tex]\cos \theta=\dfrac{2-8}{\sqrt{1^2+2^2+(-2)^2}\sqrt{1^2+4^2}}\\\\\cos \theta=\dfrac{-6}{\sqrt{17}\sqrt{17}}=\dfrac{-6}{17}=-0.352\\\\\theta=\cos^{-1}(\dfrac{-6}{17})=110.66^\circ[/tex]

Hence, the value of cos theta is -0.352.

Answer:

The provided statement is not a binomial experiment.

Step-by-step explanation:

Four conditions of a binomial experiment:

There should be fixed number of trials

Each trial is independent with respect to the others

The maximum possible outcomes are two

The probability of each outcome remains constant.

For example:

Binomial distribution: Asking 200 people if they ever visit to new york.

Not Binomial distribution: Asking 200 people how much they earn in a week.

Now, observe the provided information:

We need to determine that, asking 100 people which brand of car they drive is a binomial experiment or not.

Clearly it is not a binomial experiment because the maximum possible outcomes are not two.

Therefore, the provided statement is not a binomial experiment.

Answer:  The required answers are

(a) 69,  (b) 21,  (c) 21  and  (d) 0.

Step-by-step explanation:  We are given that the set A has 48 elements and the set B has 21 elements.

(a) To determine the maximum possible number of elements in A ∪ B.

If the sets A and B are disjoint, that is they do not have any common element. Then, A ∩ B = { }   ⇒   n(A ∩ B) = 0.

From set theory, we have

[tex]n(A\cup B)=n(A)+n(B)-n(A\cap B)=48+21-0=69.[/tex]

So, the maximum possible number of elements in  A ∪ B is 69.

(b) To determine the minimum possible number of elements in A ∪ B.

If the set B is a subset of set A, that is all the elements of set B are present in set A. Then,  n(A ∩ B) = 21.

From set theory, we have

[tex]n(A\cup B)=n(A)+n(B)-n(A\cap B)=48+21-21=48.[/tex]

So, the minimum possible number of elements in  A ∪ B is 21.

(c) To determine the maximum possible number of elements in A ∩ B.

If the set B is a subset of set A, that is all the elements of set B are present in set A. Then, n(A ∩ B) = 21.

So, the maximum possible number of elements in  A ∩ B is 21.

(d) To determine the minimum possible number of elements in A ∩ B.

If the sets A and B are disjoint, that is there is no common element in the sets A and B . Then,  n(A ∩ B) = 0.

So, the maximum possible number of elements in  A ∩ B is 0.

Thus, the required answers are

(a) 69,  (b) 21,  (c) 21  and  (d) 0.

Answer:

194.07 cubic inches.

Step-by-step explanation:

The cardboard is 12x16 before removing a square from each end.  This square is x inches wide.  Thus the 16 inside is shortened by x inches on both sides, or it is now 16-2x inches.  The 12 inside is also reduced by 2x.  The x value is also the height of the box when you fold the sides up.  Thus the volume V = wlh = (16-2x)*(12-2x)*(x) = 4x^3 - 56x^2 + 192x.  

To find the maximum, take the derivative, and find its roots  

V = 4x^3 - 56x^2 + 192x

dV/dx = 12x^2 - 112x + 192

The roots are (14+2(13)^.5)/3 ~= 7.07 and (14-2(13)^.5)/3 ~= 2.26  

The roots would be the possible values of x, the square we cut.  Since 7.07 x 2 = 14.14 inches, this exceeds the 12 inch side, thus x = 2.26 inches.  Thus you cut 2.26 inches from each corner to obtain the maximum volume.  

Cube is 11.48 x 7.48 x 2.26 with a volume of 194.07 cubic inches.

Answer:

Given inequality is,

-12 + 5(9p+3) - 36p < 6p - 12

By distributive property,

-12 + 45p + 15 - 36p < 6p - 12

Combine like terms,

3 + 9p < 6p - 12

Additive property of equality,

15 + 9p < 6p

Subtraction property of equality,

9p < 6p - 15

3p < -15

⇒ p < -5

That is, solution of the given inequality is (-∞, 5)

Answer:

P=0.05 or 5%

Step-by-step explanation:

The formula to calculate the probability in a continuous uniform distribution is:

[tex]P=\frac{d-c}{b-a}[/tex]

Where a is the lower value of the limits of the distribution (a=47.0), b is the upper value of the limit of the distribution (b=57.0), c It is the lowest value in the range from which probability is calculated (c=50.75) and d It is the highest value in the range from which probability is calculated (d=51.25).

[tex]P=\frac{51.25-50.75}{57.0-47.0} =0.05[/tex]

Final answer:

The probability that a class period runs between 50.75 and 51.25 minutes, based on uniform distribution, is 5%.

Explanation:

The probability that a given class period runs between 50.75 and 51.25 minutes can be found using the principles of uniform distribution. For a uniform distribution, the probability of an event occurring between two points (a and b) is given by the formula \((b - a) / (maximum - minimum)\). Here, the class duration minimum is 47.0 minutes and the maximum is 57.0 minutes. So the probability is \((51.25 - 50.75) / (57.0 - 47.0)\), which equals to 0.5 / 10 or 0.05. Thus, the probability is 5%.

The work done by [tex]\vec F[/tex] is

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r[/tex]

where [tex]C[/tex] is the given curve and [tex]\vec r(t)[/tex] is the given parameterization of [tex]C[/tex]. We have

[tex]\mathrm d\vec r=\dfrac{\mathrm d\vec r}{\mathrm dt}\mathrm dt=k(1-2t)\,\vec\imath+\vec\jmath[/tex]

Then the work done by [tex]\vec F[/tex] is

[tex]\displaystyle\int_0^1((t-kt(1-t))\,\vec\imath+kt^2(1-t)\,\vec\jmath)\cdot(k(1-2t)\,\vec\imath+\vec\jmath)\,\mathrm dt[/tex]

[tex]=\displaystyle\int_0^1((k-k^2)t-(k-3k^2)t^2-(k+2k^2)t^3)\,\mathrm dt=-\frac k{12}[/tex]

In order for the work to be 1, we need to have [tex]\boxed{k=-12}[/tex].

In a length-13 string of all uppercase letters, there are 26^13 possible strings. The number of strings that contain the word CHARITY is 26^6, and the number of strings that contain neither CHARITY nor HORSES is 26^13 - 26^6 - 26^7.

(a) To find the number of length-13 strings of all uppercase letters, we can use the formula: number of choices for each position (26) raised to the power of the number of positions (13). So, there are 26^13 possible strings.

(b) To find the number of strings that contain the word CHARITY, we can treat it as one block, which leaves us with 6 other available spaces. So, there are 26^6 strings that contain the word CHARITY.

(c) To find the number of strings that contain neither the word CHARITY nor the word HORSES, we can subtract the number of strings that contain CHARITY or HORSES from the total number of strings. So, the answer is 26^13 - 26^6 - 26^7.

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You correctly calculated the total number of strings in part (a), but part (b) requires multiplying by the number of possible positions for 'CHARITY'. For part (c), the initial calculation is correct after fixing part (b), as the words 'CHARITY' and 'HORSES' cannot both fit in a 13-letter string without overlapping.

The question involves calculating the number of possible strings using combinatorics and the rules of permutation.

For part (a), you correctly calculated the number of length-13 strings of uppercase letters as 26^13, since each position in the string can be occupied by any of the 26 letters of the alphabet.

In part (b), treating CHARITY as a single block and having 6 additional characters is almost correct, but you need to account for the positions where the block can start. There are 7 possible starting positions for 'CHARITY' in a 13-character string, so you need to multiply your answer by 7, giving 7 * 26^6.

For part (c), the calculation is more complex because you have to account for overlapping cases and ensure they are not subtracted twice. Subtracting 26^6 for 'CHARITY' strings and 26^7 for 'HORSES' strings from 26^13 doesn't account for the possibility of having both words in the string. However, since 'CHARITY' and 'HORSES' cannot both fit in a 13-letter string without overlap, your initial method would work here, but you still need to correct part (b) as explained. So the corrected calculation for (c) would remain at 26^13 - 7 * 26^6 - 26^7.

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