A food processor packages orange juice in small jars. The weights of the filled jars are approximately normally distributed with a mean of 10.5 ounces and a standard deviation of 0.3 ounce. Find the proportion of all jars packaged by this process that have weights that fall above10.95 ounces.

Answer:[tex]18\sqrt{3}[/tex]

Step-by-step explanation:

Given data

we haven given a parabola and a straight line

Parabola is [tex]{y^2}={-\left ( x-10\right )[/tex]

line is [tex]x=7[/tex]

Find the point of intersection of parabola and line

[tex]y=\pm \sqrt{3}[/tex] when[tex]x=7[/tex]

Area enclosed is the shaded area which is given by

[tex]Area=\int_{0}^{\sqrt{3}}\left ( 10-y^2 \right )dy[/tex]

[tex]Area=_{0}^{\sqrt{3}}10y-_{0}^{\sqrt{3}}\frac{y^3}{3}[/tex]

[tex]Area=10\sqrt{3}-\sqrt{3}[/tex]

[tex]Area=9\sqrt{3}units[/tex]

Required area will be double of calculated because it is symmetrical about x axis=[tex]18\sqrt{3}units[/tex]

To find the area of the region enclosed by the graphs of[tex]x=10-y^2[/tex]and x=7, we need to find the points of intersection between the two equations and then integrate the curve between those points.

To find the area of the region enclosed by the graphs of  [tex]x=10-y^2[/tex] and x=7, we need to find the points of intersection between the two equations. Setting x equal to each other, we have  [tex]10-y^2=7.[/tex]Solving for y, we get y=±√3.

Now we can integrate the curve between the two values of y, as y goes from -√3 to √3. So the area is given by  [tex]\int (10 - y^2 - 7) \, dy[/tex] from -√3 to √3.

Evaluating the integral, we get A=√3*10-2√3/3 ≈ 30.78.

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Answer:    

1.  6 fives.

2.  1 ten and 4 fives.

3.  2 tens and 2 fives.

4.  3 tens.

5.  1 twenty and 2 fives.

6.  1 twenty and 1 ten.

Step-by-step explanation:

Given : A customer is owed $30.00.

To find : How many different combinations of bills,using only five, ten, and twenty dollars bills are possible to give his or her change?

Solution :

We have to split $30 in terms of only five, ten, and twenty dollars.

1) In terms of only five we required 6 fives as

[tex]6\times 5=30[/tex]

So, 6 fives.

2) In terms of only ten and five,

a) We required 1 ten and 4 fives as

[tex]1\times 10+4\times 5=10+20=30[/tex]

So, 1 ten and 4 fives.

b) We required 2 tens and 2 fives as

[tex]2\times 10+2\times 5=20+10=30[/tex]

So, 2 tens and 2 fives

3) In terms of only tens we require 3 tens as

[tex]3\times 10=30[/tex]

So, 3 tens.

4)  In terms of only twenty and five, we required 1 twenty and 2 fives as

[tex]1\times 20+2\times 5=20+10=30[/tex]

So, 1 twenty and 2 fives.

5)  In terms of only twenty and ten, we required 1 twenty and 1 ten as

[tex]1\times 20+1\times 10=20+10=30[/tex]

So, 1 twenty and 1 ten.

Therefore, There are 6 different combinations.

Answer:

There is enough evidence to believe that the average credit card debt has changed in the past 5 years

Step-by-step explanation:

We are to compare the means of two samples. Since only sample std deviations are used, we have to use t test for this hypothesis

H0: Means are equal

Ha: Means are not equal

(Two tailed test at 5% )

Difference between means [tex]M1-M2 = -2587[/tex]

Std deviation combined = 3856

Std error for difference = 460.88

t statistic[tex]= -2587/460.88=-5.613[/tex]

p value =0

Since p <0.05 reject null hypothesis.

There is enough evidence to believe that the average credit card debt has changed in the past 5 years

Step-by-step explanation:

write the equation in slope intercept form

7x + 4y - 20=0

find the slope of corresponding line.

????

then

find the y intercept of corresponding line

(x,y)=???.

Answer: 0.0336

Step-by-step explanation:

Given : The distribution of cholesterol levels in teenage boys is approximately normal with mean :[tex]\mu= 170[/tex]

Standard deviation : [tex]\sigma= 30[/tex]

The formula for z-score is given by :-

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

For x=225

[tex]z=\dfrac{225-170}{30}=1.83[/tex]

The p-value =[tex]P(z>1.83)=1-P(z<1.83)[/tex]

[tex]=1-0.966375=0.033625\approx0.0336[/tex]

The probability that a teenage boy has a cholesterol level greater than 225 =0.0336

To write the equation 5x + 3y = 15 in slope-intercept form, solve for y to get y = (-5/3)x + 5. The slope is -5/3 and the y-intercept is 5.

To convert the equation 5x + 3y = 15 into slope-intercept form, which is y = mx + b, we need to solve for y. Here are the steps:

Answer:

80,00[tex]ft^{2}[/tex]

Step-by-step explanation:

According to my research, the formula for the Area of a rectangle is the following,

[tex]A = L*W[/tex]

Where

Since the building wall is acting as one side length of the rectangle. We are left with 1 length and 2 width sides. To maximize the Area of the parking lot we will need to equally divide the 800 ft of fencing between the Length and Width.

800 / 2 = 400ft

So We have 400 ft for the length and 400 ft for the width. Since the width has 2 sides we need to divide 60 by 2.

400/2 = 200 ft

Now we can calculate the maximum Area using the values above.

[tex]A = 400ft*200ft[/tex]

[tex]A = 80,000ft^{2}[/tex]

So the Maximum area we are able to create with 800 ft of fencing is 80,00[tex]ft^{2}[/tex]

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The largest possible parking lot, given 800 feet of fencing with one side not requiring a fence, would be a rectangular lot with dimensions of 400 feet by 200 feet.

We're dealing with a rectangular parking lot here where one of its sides is bordered by a building, so it doesn't need a fence. We have 800 feet of fence available for the three remaining sides. Let's denote the length of the rectangular parking lot by 'x' and the width by 'y'.

Because we only need to fence three sides, we can establish the following equation based on the total amount of fence available:

x + 2y = 800

To calculate the maximum possible area of a rectangle, we need to use the formula for the area of a rectangle, which is length times width (Area = x * y). However, we want to express the area in terms of a single variable. To do this, we can rearrange our fence equation to solve for y:

y = (800 - x) / 2

Now replace y in the area equation:

Area = x * (800 - x) / 2

For the area to be maximum, the derivative of the area with respect to x must be equal to zero. Differentiating and solving for x, we get the dimensions as x = 400 and y = 200.

So the largest possible parking lot would have dimensions of 400 feet by 200 feet.

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Answer:

2 ties of 4 feet and 1 tie of 5 feet

2 ties of 6 feet and 1 tie of 9 feet

Step-by-step explanation:

Given data

2 ties = 4 feet

2 ties = 6 feet

1 tie = 9 feet

1 tie = 5 feet

to find out

what are the total possible dimensions for each set of flower beds

solution

there are many combination but

the best possible combination are for 1st triangular

when we use 2 ties of 4 feet and 1 tie of 5 feet

and

for 2nd triangular

when we use 2 ties of 6 feet and 1 tie of 9 feet

only these are best combination

Answer:

Number of adults = 7

Number of children = 16

Step-by-step explanation:

Tickets for a play cost 2 pounds for a child, and 4 pounds for an adult.

Let x number of adults and y number of children.

1 child ticket cost = 2 pound

y children ticket cost = 2y pound

1 adult ticket cost = 4 pound

x adults ticket cost = 4x pound

Total number of ticket sales were 60 pounds

Therefore, 4x + 2y = 60  ------------- (1)

One adult brought 4 children with him and the remaining adults each brought 2 children with them.

Remaining number of adult whose brought 2 children = x-1

Number children = 2(x-1)

Total number of children = 2(x-1)+4

Therefore, y=2x+2 ---------------------(2)

System of equation,

 2x + y = 30

-2x + y = 2

Using augmented matrix to solve system of equation.

[tex]\begin{bmatrix}2&1&\ |30\\-2&1&|2\end{bmatrix}\\\\R_2\rightarrow R_2+R_1\\\\\begin{bmatrix}2&1& |30\\0&2&|32\end{bmatrix}\\\\R_2\rightarrow\dfrac{1}{2}R_2\\[/tex]

[tex]\begin{bmatrix}2&1&\ |30\\0&1&|16\end{bmatrix}\\\\R_1\rightarrow R_1-R_2\\\\\begin{bmatrix}2&0&\ |14\\0&1&|16\end{bmatrix}\\\\\\[/tex]

[tex]R_1\rightarrow \dfrac{1}{2}R_1\\\\\begin{bmatrix}1&0&|7\\0&1&|16\end{bmatrix}\\\\[/tex]

Now, we find the value of variable.

[tex]x=7\text{ and }y=16[/tex]

Hence, Number of adults are 7 and Number of children are 16.

Answer:  0.1587

Step-by-step explanation:

Given : The amount dispensed is approximately normally distributed with Mean : [tex]\mu=\ 16[/tex]

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

The formula to calculate the z-score :-

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

For x= 17

[tex]z=\dfrac{17-16}{1}=1[/tex]

The p-value =[tex] P(17<x)=P(1<z)[/tex]

[tex]=1-P(z<1)=1-0.8413447\\\\=0.1586553\approx0.1587[/tex]

The proportion of bottles will have more than 17 ounces = 0.1587

Answer:

Step-by-step explanation:

Given lines in parametric form

line [tex]L_1[/tex]

[tex]\frac{x}{3}=\frac{y-1}{-2} =\frac{z-2}{-3}[/tex]

direction vector of [tex]L_1 v_1=<3,-2,-3 >[/tex]

Line [tex]L_2[/tex]

direction vector of [tex]L_2 v_2=<-9,6,9 >[/tex]

therefore

[tex]v_2=-3v_1[/tex]

thus lines are parallel.

(ii)distance between two lines is

[tex]L_2[/tex] is given by

[tex]\frac{x-3}{-9}=\frac{y-1}{6} =\frac{z-4}{9}[/tex]=s

[tex]\frac{x-3}{-3}=\frac{y-1}{2} =\frac{z-4}{3}[/tex]=3s

[tex]\left | \frac{\begin{vmatrix}x_2-x_1 &y_2-y_1 &z_2-z_1 \\ a_1&b_1&c_1 \\ a_2&b_2&c_2\end{vmatrix}}{\sqrt{\left ( a_1b_2-a_2b_1 \right )^2+\left ( b_1c_2-b_2c_1 \right )^2+\left ( c_1a_2-c_2a_1 \right )^2}}\right |[/tex]

where [tex]a_1[/tex]=3

[tex]b_1[/tex]=-2

[tex]c_1[/tex]=-3

[tex]a_2[/tex]=-9

[tex]b_2[/tex]=6

[tex]c_2[/tex]=9

distance(d)=0 units since value of the matrix

[tex]\begin{vmatrix}x_2-x_1 &y_2-y_1 &z_2-z_1\\ a_1&b_1&c_1 \\a_2&b_2 &c_2 \end{vmatrix}[/tex]

is zero

Answer:

The value of det (3A) is -108.

Step-by-step explanation:

If M is square matrix of order n x n, then

[tex]|kA|=k^n|A|[/tex]

Let as consider a matrix A or order 3 x 3. Using the above mentioned property of determinant we get

[tex]|kA|=k^3|A|[/tex]

We need to find the value of det(3A).

[tex]|3A|=3^3|A|[/tex]

[tex]|3A|=27|A|[/tex]

It is given that the det(A)= -4. Substitute |A|=-4 in the above equation.

[tex]|3A|=27(-4)[/tex]

[tex]|3A|=-108[/tex]

Therefore the value of det (3A) is -108.

Answer:

0.48

Step-by-step explanation:

Probability that the first person chooses a hotel

⁵C₁

[tex]^5C_1=\frac{5!}{(5-1)!1!}\\=\frac{120}{24}=5[/tex]

Probability that the second person chooses a different hotel

⁴C₁

[tex]^4C_1=\frac{4!}{(4-1)!1!}\\=\frac{24}{6}=4[/tex]

because the choice of hotels has reduced by 1 as one hotel is occupied by the first person

Probability that the second person chooses a different hotel

³C₁

[tex]^3C_1=\frac{3!}{(3-1)!1!}\\=\frac{6}{2}=3[/tex]

because the choice of hotels has reduced by 2 as two different hotels are occupied by the first person and second person

∴ The favorable outcomes are =⁵C₁×⁴C₁׳C₁=5×4×3=60

The total number of outcomes=5³=125

∴Probability that they each check into a different hotel=60/125=0.48

We have [tex]\lambda(21)=6[/tex], where [tex]\lambda[/tex] is the Carmichael function. So we have

[tex]10^{5^{101}}\equiv10^{5^{101}\pmod6}\pmod{21}[/tex]

The powers of 5 modulo 6 follow a periodic pattern

[tex]5^1\equiv5\pmod6[/tex]

[tex]5^2\equiv25\equiv1\pmod6[/tex]

[tex]5^3\equiv1\cdot5\equiv5\pmod6[/tex]

[tex]5^4\equiv5^2\equiv1\pmod6[/tex]

and so on, with odd powers of 5 equivalent to 5 modulo 6. So

[tex]10^{5^{101}}\equiv10^{5^{101}\pmod6}\equiv10^5\pmod{21}[/tex]

The rest is easy to deal with. We have

[tex]10^2\equiv16\pmod{21}[/tex]

[tex]10^3\equiv160\equiv13\pmod{21}[/tex]

[tex]10^4\equiv130\equiv4\pmod{21}[/tex]

[tex]10^5\equiv40\equiv19\pmod{21}[/tex]

and so the answer is 19.

Answer: 38

Step-by-step explanation:

                 i.e. [tex]P-Q=n(P)-n(P\cap Q)[/tex]

Let A be the number of students who correctly answered the first question and B be the number of students who correctly answered the second question .

Given : [tex]n(A)=76[/tex]

[tex]n(B)=60[/tex]

[tex]n(A\cap B)=38[/tex]

Then the number of students who answered the first question correctly, but not the second is given by :-

[tex]A-B=n(A)-n(A\cap B)\\\\=76-38=38[/tex]

Hence, the number of students who answered the first question correctly, but not the second is 38.

Answer:

The probability is [tex]\frac{1}{9}[/tex]

Step-by-step explanation:

There are a total of 9 die in the box after she added the "cheat" die. Since there is only 1 "cheat" die in the box and chooses a die at random then the probability of her having chosen the cheat die is [tex]\frac{1}{9}[/tex] . The fact that she rolled two sixes did not affect when she choose the die therefore the probability remains as [tex]\frac{number.of.cheat.die}{total.dice}[/tex].

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With a 95% confidence level, the population mean is estimated to be between approximately 96.85 and 111.15 based on a sample size of 10, a mean of 104, and a standard deviation of 10.

With a sample size (n) of 10, a mean \bar{x}104, and a standard deviation (s) of 10, we can find the 95% confidence interval for the population mean (μ).

First, we calculate the standard error of the mean (SE). The standard error of the mean can be calculated by dividing the standard deviation by the square root of the sample size.

SE = s/√n.  
By substituting s = 10 and n = 10 into the equation, we get SE = 3.162277660168379.

Next, we need to find the critical value (t) for a 95% confidence interval based on a t-distribution. Since we're using a confidence level of 95% and the sample size is 10, which means degree of freedom is n-1=9, the critical value (t) is 2.2621571627409915 based on the t-distribution table.

To calculate the lower bound and the upper bound of the 95% confidence interval, you should subtract and add to the mean the product of the critical value and the standard error respectively.

So,
Lower Bound = \bar{x} - t * SE
Upper Bound = \bar{x} + t * SE

Substituting from our known values, we get:
Lower Bound = 104 - 2.2621571627409915 * 3.162277660168379 = 96.84643094047428
Upper Bound = 104 + 2.2621571627409915 * 3.162277660168379 = 111.15356905952572

So, with a 95% confidence level, the confidence interval estimate of the population mean is (96.84643094047428, 111.15356905952572). This means we are 95% confident that the true population mean lies somewhere between approximately 96.85 and 111.15.

To learn more about standard deviation

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The 95% confidence interval for the population mean IQ score of professional athletes, based on a sample size of 10 with a mean of 104 and standard deviation of 10, is estimated to be between 96.83 and 111.17.

To find the 95% confidence interval estimate of the population mean [tex](\( \mu \))[/tex] given the sample data, we'll use the formula for the confidence interval for a population mean when the population standard deviation is unknown:

[tex]\[ \text{Confidence interval} = \bar{x} \pm t \left( \frac{s}{\sqrt{n}} \right) \][/tex]

Where:

-[tex]\( \bar{x} \)[/tex] is the sample mean,

-  s  is the sample standard deviation,

-  n  is the sample size, and

-  t  is the critical value from the t-distribution for the desired confidence level and degrees of freedom.

Given:

- Sample size  n  = 10

- Sample mean [tex](\( \bar{x} \))[/tex]= 104

- Sample standard deviation  s  = 10

First, we need to find the critical value t  for a 95% confidence level with 9 degrees of freedom (since n - 1 = 10 - 1 = 9 ).

Using a t-table or statistical software, [tex]\( t \approx 2.262 \)[/tex] for a 95% confidence level and 9 degrees of freedom.

Now, let's plug in the values into the formula:

[tex]\[ \text{Confidence interval} = 104 \pm 2.262 \left( \frac{10}{\sqrt{10}} \right) \][/tex]

Now, let's calculate the margin of error:

[tex]\[ \text{Margin of error} = 2.262 \left( \frac{10}{\sqrt{10}} \right) \]\[ \text{Margin of error} \approx 7.17 \][/tex]

Finally, let's calculate the confidence interval:

[tex]\[ \text{Lower bound} = 104 - 7.17 \]\[ \text{Upper bound} = 104 + 7.17 \]\[ \text{Lower bound} \approx 96.83 \]\[ \text{Upper bound} \approx 111.17 \][/tex]

So, the 95% confidence interval estimate of the population mean IQ score of professional athletes is approximately between 96.83 and 111.17.

Answer:

The seasons would become constant. It would be equinox throughout the year.

Step-by-step explanation:

The earth would be in a state of constant equinox i.e., the length of day and night would be same in a particular place.

The season of a place would be what it is when it is normally titled at equinox.

The animal and plant life which depend on the seasons would be affected.

Snow would only occur at parts where it normally snows at equinoxes.

Answer: 34650

Step-by-step explanation:

The number of permutations of n objects, where one object is repeated [tex]n_1[/tex] times , another is repeated [tex]n_2[/tex] times and so on is :

[tex]\dfrac{n!}{n_1!n_2!....n_k!}[/tex]

Given : The number of letters in string MISSISSIPPI = 11

Here I is repeated 4 times, S is repeated 4 times and P is repeated 2 times.

Then , the number of different strings can be made from the letters in MISSISSIPPI, using all letters is given by :-

[tex]\dfrac{11!}{4!4!2!}=34650[/tex]

Therefore , there are 34650 different strings can be made from the letters in MISSISSIPPI.

To find the number of different strings that can be formed from the letters in MISSISSIPPI, we use the formula for permutations of a multi set. The total is 34,650 unique permutations.

The question asks how many different strings can be made from the letters in MISSISSIPPI, using all letters.

This is a problem of calculating permutations of a multi set.

The word MISSISSIPPI has 11 letters with the following counts of each letter: M occurs 1 time, I occurs 4 times, S occurs 4 times, and P occurs 2 times.

To find the total number of unique permutations, we use the formula for the permutations of a multiset:

The formula is:

Permutations = n! / (n1! * n2! * ... * nk!),

where n is the total number of items to arrange, and n1, n2, ..., nk are the counts of each distinct item.

For MISSISSIPPI:

Total permutations = 11! / (1! * 4! * 4! * 2!) = 34650

Therefore, there are 34,650 different strings that can be made from the letters in MISSISSIPPI using all the letters.

Answer: [tex](0.52,\ 0.68)[/tex]

Step-by-step explanation:

The formula for a [tex]\alpha[/tex]- level confidence interval for the population proportion:-

[tex]p\pm z_{\alpha/2}\times\sqrt{\dfrac{p(1-p)}{n}}[/tex]

Given : n = 225 ; p = 0.60 ; [tex]\alpha= 1-0.99=0.01[/tex]

By using the given information , the confidence interval for the population proportion:-

[tex]0.6\pm z_{0.005}\times\sqrt{\dfrac{0.6(1-0.6)}{225}}\\\\=0.6\pm(2.576)(0.0327)\\\\=0.6\pm0.08\\\\=0.6-0.08,\ 0.6+0.08\\\\=(0.52,\ 0.68)[/tex]

Hence, the resulting confidence interval is [tex](0.52,\ 0.68)[/tex] .

To construct a 99% confidence interval for the population proportion, we can use the sample proportion, critical value, and standard error. The resulting confidence interval is (0.516, 0.684).

To calculate a confidence interval for a population proportion, we can use the formula:

CI = sample proportion ± (critical value)(standard error)

Given that the sample proportion is 60% (0.6) and the confidence level is 99%, we need to find the critical value corresponding to a 99% confidence level. Using a normal distribution, the critical value is approximately 2.576. The standard error can be calculated using the formula:

SE = √((sample proportion)(1 - sample proportion) / sample size)

Let's assume the sample size is 225. Plugging these values into the formula, we can calculate the standard error:

SE = √((0.6)(1 - 0.6) / 225) ≈ √(0.24 / 225) ≈ √0.001067 ≈ 0.0326

Now we can calculate the confidence interval:

CI = 0.6 ± (2.576)(0.0326) ≈ 0.6 ± 0.084

Therefore, the 99% confidence interval for the population proportion is approximately (0.516, 0.684).

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Answer: a) 0.2   b) 0.6

c) The event of selecting red tulip is not likely to occur.

The event of selecting purple tulip is likely to occur.

Step-by-step explanation:

Given : Total number of tulips = 100

The number of red tulips = 20

The number of purple tulips =60

The probability that a randomly selected tulip bulb is​ red :-

[tex]\dfrac{\text{Number of red tulips}}{\text{Total tulips}}\\\\=\dfrac{20}{100}=0.2[/tex]

Since 0.2 is less than 0.5.

It means that the event of selecting red tulip is not likely to occur.

The probability that a randomly selected tulip bulb is​ purple :-

[tex]\dfrac{\text{Number of purple tulips}}{\text{Total tulips}}\\\\=\dfrac{60}{100}=0.6[/tex]

Since 0.6 is more than 0.5.

It means that the event of selecting purple tulip is likely to occur.

Final answer:

The probability of selecting a red tulip bulb is 20%, and the probability of selecting a purple tulip bulb is 60%. These probabilities reflect the likelihood of picking a bulb of a particular color at random from the bag.

Explanation:

The question involves calculating the probability of selecting a red or purple tulip bulb from a bag.

Probability of Selecting a Red Tulip Bulb

The probability, P(Red), is calculated by dividing the number of red bulbs by the total number of bulbs:

P(Red) = Number of Red Bulbs / Total Number of Bulbs = 20 / 100 = 0.2

Probability of Selecting a Purple Tulip Bulb

Similarly, the probability, P(Purple), is:

P(Purple) = Number of Purple Bulbs / Total Number of Bulbs = 60 / 100 = 0.6

Interpretation of Probabilities

These probabilities indicate that there is a 20% chance of selecting a red bulb and a 60% chance of selecting a purple bulb from the bag. The higher the probability, the more likely it is to select a bulb of that color at random.

[tex]u(x,y,z)=x^2+yz[/tex]

[tex]\begin{cases}x(p,r,\theta)=pr\cos\theta\\y(p,r,\theta)=pr\sin\theta\\z(p,r,\theta)=p+r\end{cases}[/tex]

At the point [tex](p,r,\theta)=(2,2,0)[/tex], we have

[tex]\begin{cases}x(2,2,0)=4\\y(2,2,0)=0\\z(2,2,0)=4\end{cases}[/tex]

Denote by [tex]f_x:=\dfrac{\partial f}{\partial x}[/tex] the partial derivative of a function [tex]f[/tex] with respect to the variable [tex]x[/tex]. We have

[tex]\begin{cases}u_x=2x\\u_y=z\\u_z=y\end{cases}[/tex]

The Jacobian is

[tex]\begin{bmatrix}x_p&x_r&x_\theta\\y_p&y_r&y_\theta\\z_p&z_r&z_\theta\end{bmatrix}=\begin{bmatrix}r\cos\theta&p\cos\theta&-pr\sin\theta\\r\sin\theta&p\sin\theta&pr\cos\theta\\1&1&0\end{bmatrix}[/tex]

By the chain rule,

[tex]u_p=u_xx_p+u_yy_p+u_zz_p=2xr\cos\theta+zr\sin\theta+y[/tex]

[tex]u_p(2,2,0)=2\cdot4\cdot2\cos0+4\cdot2\sin0+0\implies\boxed{u_p(2,2,0)=16}[/tex]

[tex]u_r=u_xx_r+u_yy_r+u_zz_r=2xp\cos\theta+zp\sin\theta+y[/tex]

[tex]u_r(2,2,0)=2\cdot4\cdot2\cos0+4\cdot2\sin0+0\implies\boxed{u_r(2,2,0)=16}[/tex]

[tex]u_\theta=u_xx_\theta+u_yy_\theta+u_zz_\theta=-2xpr\sin\theta+zpr\cos\theta[/tex]

[tex]u_\theta(2,2,0)=-2\cdot4\cdot2\cdot2\sin0+4\cdot2\cdot2\cos0\implies\boxed{u_\theta(2,2,0)=16}[/tex]

This problem is about using the Chain Rule to compute the partial derivatives of a function with respect to different variables, followed by substitution of specific values into the obtained derivatives.

The problem involves finding partial derivatives using the Chain Rule on the given equations with given parameters: p = 2, r = 2, θ = 0. By substituting the equations for x, y, z into u which gives us u = (prcosθ)² + prsinθ(p+r). The next step is to compute (partial u)/(partial p), (partial u)/(partial r), (partial u)/(partial theta) by using the Chain Rule to find each partial derivative. After computing, you just substitute the given values of p, r, θ into the obtained derivates to get the final answers.

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Answer: (a) 0.8641

(b) 0.1359

Step-by-step explanation:

Given : The monthly worldwide average number of airplane crashes of commercial airlines [tex]\lambda= 3.5[/tex]

We use the Poisson  distribution for the given situation.

The Poisson distribution formula for probability is given by :-

[tex]P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex]

a) The probability that there will be at least 2 such accidents in the next month is given by :-

[tex]P(X\geq2)=1-(P(X=1)+P(X=0))\\\\=1-(\dfrac{e^{-3.5}(3.5)^0}{0!}+\dfrac{e^{-3.5}(3.5)^1}{1!})\\\\=1-(0.1358882254)=0.8641117746\approx0.8641[/tex]

b) The probability that there will be at most 1 accident in the next month is given by :-

[tex]P(X\leq1)=(P(X=1)+P(X=0))\\\\=\dfrac{e^{-3.5}(3.5)^0}{0!}+\dfrac{e^{-3.5}(3.5)^1}{1!}\\\\=0.1358882254\approx0.1359[/tex]

Answer:

$100 * (1 + 6.8%/12)^216 + $100*(1+6.8%/12)^215 + ... + $100*(1+6.8%/12)^1  

Now note that  

x + x^2 + x^3 + ... + x^N = x ( 1 + x + ... + x^(N-1) )  

= x ( (x^N -1)/(x-1) )  

Here, x = 1+6.8%1 = 1.00566666 and N = 216, so  

$100 * ( 1.00566666 ( 1.00566666^216 -1) / 0.00566666 )  

= $ 42398.33  

The total interest earned is $42,398 - $21,600 = $20,798

Step-by-step explanation:

The generic formula used in this compound interest calculator is V = P(1+r/n)^(nt)

V = the future value of the investment

P = the principal investment amount

r = the annual interest rate

n = the number of times that interest is compounded per year

t = the number of years the money is invested for

Answer:

Alfred has 250 socks in his collection.

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

Based on the information given to us we can see that out of All his Brown socks he has 20% in the wash and the rest are in his drawer. Meaning 80% of the Brown socks are in his drawer. So we first need to find how many Brown socks are in the wash. We can solve this using the Rule of Three property as shown in the picture below.

120 drawer   ⇒   80%

      x wash    ⇒  20%

[tex]\frac{120*20}{80} = 30[/tex]

Now that we have the amount of Brown socks in the washer we can add that to the amount in the drawer to find the total amount of Brown socks.

[tex]Br = 120+30\\Br = 150[/tex]

So we now know that there are a total of 150 Brown socks. Since the question states that the Brown socks are 60% of the total we can use the Rule of Three to find the total.

150 Brown   ⇒   60%

    T  Total   ⇒   100%

[tex]\frac{150*100}{60} = T[/tex]

[tex]250 socks = T[/tex]

Finally, we can see that Alfred has 250 socks in his collection.

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Alfred owns a total of 250 socks. This was derived by first finding out that Alfred has 150 brown socks, which represent 60% of his total socks' collection. Thus, the total number of socks Alfred owns is 250.

The question deals with the calculation of a total number of socks owned by Alfred.

If we know that 120 brown socks represent 80% of all of Alfred's brown socks (because 20% of them are in the wash), we can calculate the total number of brown socks. To do this, divide 120 by 0.8, which equals 150. Hence, Alfred has 150 brown socks.

We know from the problem that the brown socks account for 60% of all his socks. Hence the total number of socks (brown and black) is calculated by dividing the total number of brown socks (150) by 0.6. After performing this percentage calculation, we find that Alfred owns 250 socks.

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We are asked to prove by the method of mathematical induction that:

      3n(n+1) is divisible by 6 for all positive integers.

[tex]3n(n+1)=3\times 1(1+1)\\\\i.e.\\\\3n(n+1)=3\times 2\\\\i.e.\\\\3n(n+1)=6[/tex]

which is divisible by 6.

Hence, the result is true for n=1

i.e. 3k(k+1) is divisible by 6.

Let n=k+1

then

[tex]3n(n+1)=3(k+1)\times (k+1+1)\\\\i.e.\\\\3n(n+1)=3(k+1)(k+2)\\\\i.e.\\\\3n(n+1)=(3k+3)(k+2)\\\\i.e.\\\\3n(n+1)=3k(k+2)+3(k+2)\\\\i.e.\\\\3n(n+1)=3k^2+6k+3k+6\\\\i.e.\\\\3n(n+1)=3k^2+3k+6k+6\\\\i.e.\\\\3n(n+1)=3k(k+1)+6(k+1)[/tex]

Since, the first term:

[tex]3k(k+1)[/tex] is divisible by 6.

( As the result is true for n=k)

and the second term [tex]6(k+1)[/tex] is also divisible by 6.

Hence, the sum:

[tex]3k(k+1)+6(k+1)[/tex]  is divisible by 6.

Hence, the result is true for n=k+1

Hence, we may say that the result is true for all n where n belongs to positive integers.

To prove by induction that the expression 3n(n+1) is divisible by 6, we start with the base case of n=1, which is divisible by 6, and then show that if it holds for an integer k, it also holds for k+1. By factoring and using the induction hypothesis, we demonstrate the expression's divisibility by 6 for all positive integers.

Proof by Induction of Divisibility by 6

To prove by induction that 3n(n+1) is divisible by 6 for all positive integers, we follow two steps: the base case and the inductive step.

Base Case

Let's check for n=1:

Inductive Step

Assume the statement holds for a positive integer k, so 3k(k+1) is divisible by 6.

Now, we must show that 3(k+1)((k+1)+1) = 3(k+1)(k+2) is also divisible by 6.

Factoring out the common term we get:

Notice that (k+1) is an integer, hence 3(k+1) is divisible by 3. If k is even, then k+1 is odd, so 3(k+1) is still divisible by 3 but not necessarily by 6. However, if k is odd, k+1 is even and 3(k+1) is divisible by 6. Since the divisible by 3 part is always true, and the divisible by an additional factor of 2 part is true every other time, the sum 6m + 3(k+1) is divisible by 6 regardless of whether k is odd or even. This is because adding a multiple of 6 to either another multiple of 6 or a multiple of 3 always results in a multiple of 6.

Therefore, 3n(n + 1) is divisible by 6 for all positive integers n.

Answer:

Step-by-step explanation:

Given that in the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College.

Table is prepared as follows

                            Bus coll.     Lib Arts coll      Educ. coll        Total

Observed                  90                120                 90                  300

Expected p.c.             35                 35                 30                  100

Expected number     105                105                 90                 300

Percent*300/100

Hence expected frequency for business college = 105

Answer with  explanation:

Let R be a communtative ring .

a and b elements in R.Let a and b are units

1.To prove that ab is also unit in R.

Proof: a and b  are units.Therefore,there exist elements u[tex]\neq0[/tex] and v [tex]\neq0[/tex] such that

au=1 and bv=1 ( by definition of unit )

Where u and v are inverse element  of a and b.

(ab)(uv)=(ba)(uv)=b(au)(v)=bv=1 ( because ring is commutative)

Because bv=1 and au=1

Hence, uv is an inverse element of ab.Therefore, ab is a unit .

Hence, proved.

2. Let a is a unit and b is a zero divisor .

a is a unit then there exist an element u [tex]\neq0[/tex]

such that au=1

By definition of unit

b is a zero divisor then there exist an element [tex]v\neq0[/tex]

such that bv=0 where [tex]b\neq0[/tex]

By definition of zero divisor

(ab)(uv)=b(au)v    ( because ring is commutative)

(ab)(uv)=b.1.v=bv=0

Hence, ab is a zero divisor.

If a is unit and b is a zero divisor then ab is a zero divisor.

Answer:

The amount of water released during the​ week-long flood was 15,180,400,000 cubic feet per second.

Step-by-step explanation:

How many seconds are there in a week?

Each minute has 60 seconds

Each hour has 60 minutes

Each day has 24 hours

Each week has 7 days. So

60*60*24*7 = 604,800

A week has 604,800 seconds.

Water was released at a rate of 25,100 cubic feet per second.

In a week(604,800 seconds)

604,800*25,100 = 15,180,400,000

The amount of water released during the​ week-long flood was 15,180,400,000 cubic feet per second.

Answer:

2100

Step-by-step explanation:

50*18=900

900+1,200=2100

To find the population in 2018, we calculate the total increase from 2000 to 2018 by multiplying the yearly increase of 50 people by 18 years, resulting in an additional 900 people. Adding this to the initial population of 1,200 people gives us a total population of 2,100 people in 2018.

If a population is recorded at 1,200 in the year 2000 and increases at a steady rate of 50 people each year, we can calculate the population in 2018 using a linear growth model. First, we need to determine the number of years between 2000 and 2018, which is 18 years. Next, we multiply the annual increase (50 people) by the number of years (18) to find the total increase over this period.

The calculation would be as follows:

We then add this total increase to the initial population to get the population in 2018:

The population in 2018 would be 2,100 people.