A farmer looks out into the barnyard and sees the pigs and the chickens. "I count 70 heads and 180 feet, How many pigs and chickens are there?

Answer:

Because every “x” value has two “y” values.

Step-by-step explanation:

In the graph of a function every value of x has one and only one value of y. So, if we draw a straight line which is parallel to y-axis and it cuts the graph in only one point, this graph will correspond to a function.

Answer:  [tex]11.7<\mu<15.9[/tex]

Step-by-step explanation:

Given : Significance level : [tex]\alpha:1-0.95=0.05[/tex]

Sample size : n=45

Critical value : [tex]z_{\alpha/2}=1.96[/tex]

Sample mean : [tex]\overline{x}=13.8\text{ years}[/tex]

Standard deviation : [tex]\sigma=7.3\text{ years}[/tex]

The confidence interval for population mean is given by :-

[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\\\\=13.8\pm(1.96)\dfrac{7.3}{\sqrt{45}}\\\\\approx13.8\pm2.1\\\\=(13.8-2.1,\ 13.8-2.1)=(11.7,\ 15.9)[/tex]

Hence, the 95% confidence interval estimate for the average number of years served by all Supreme Court justices is [tex]11.7<\mu<15.9[/tex]

This sequence has generating function

[tex]F(x)=\displaystyle\sum_{k\ge0}k^3x^k[/tex]

(if we include [tex]k=0[/tex] for a moment)

Recall that for [tex]|x|<1[/tex], we have

[tex]\displaystyle\frac1{1-x}=\sum_{k\ge0}x^k[/tex]

Take the derivative to get

[tex]\displaystyle\frac1{(1-x)^2}=\sum_{k\ge0}kx^{k-1}=\frac1x\sum_{k\ge0}kx^k[/tex]

[tex]\implies\dfrac x{(1-x)^2}=\displaystyle\sum_{k\ge0}kx^k[/tex]

Take the derivative again:

[tex]\displaystyle\frac{(1-x)^2+2x(1-x)}{(1-x)^4}=\sum_{k\ge0}k^2x^{k-1}=\frac1x\sum_{k\ge0}k^2x^k[/tex]

[tex]\implies\displaystyle\frac{x+x^2}{(1-x)^3}=\sum_{k\ge0}k^2x^k[/tex]

Take the derivative one more time:

[tex]\displaystyle\frac{(1+2x)(1-x)^3+3(x+x^2)(1-x)^2}{(1-x)^6}=\sum_{k\ge0}k^3x^{k-1}=\frac1x\sum_{k\ge0}k^3x^k[/tex]

[tex]\implies\displaystyle\frac{x+4x^3+x^3}{(1-x)^4}=\sum_{k\ge0}k^3x^k[/tex]

so we have

[tex]\boxed{F(x)=\dfrac{x+4x^3+x^3}{(1-x)^4}}[/tex]

Answer:  40000

Step-by-step explanation:

The formula to find the sample size is given by :-

[tex]n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2[/tex], where p is the prior estimate of the population proportion.

Here we can see that the sample size is inversely proportion withe square of margin of error.

i.e. [tex]n\ \alpha\ \dfrac{1}{E^2}[/tex]

By the equation inverse variation, we have

[tex]n_1E_1^2=n_2E_2^2[/tex]

Given : [tex]E_1=0.05[/tex]   [tex]n_1=1000[/tex]

[tex]E_2=0.025[/tex]

Then, we have

[tex](1000)(0.05)^2=n_2(0.025)^2\\\\\Rightarrow\ 2.5=0.000625n_2\\\\\Rightarrow\ n_2=\dfrac{2.5}{0.000625}=4000[/tex]

Hence, the sample size will now have to be 4000.

The new sample size will have to be approximately 4000.

The formula to calculate the sample size (n) needed to estimate a population proportion with a given maximum allowable error (E) and confidence level (usually 95% or 1.96 standard deviations for a two-tailed test) is given by:

[tex]\[ n = \left(\frac{z \times \sigma}{E}\right)^2 \][/tex]

Given that the original maximum allowable error was 0.05 and the sample size calculated was 1000, we can set up the equation:

[tex]\[ 1000 = \left(\frac{1.96 \times 0.5}{0.05}\right)^2 \][/tex]

Now, we want to find the new sample size when the maximum allowable error is reduced to 0.025. The new sample size can be calculated by:

[tex]\[ n_{new} = \left(\frac{z \times \sigma}{E_{new}}\right)^2 \][/tex]

Since \( z \) and[tex]\( \sigma \)[/tex] remain constant, and only \( E \) changes, the relationship between the original sample size and the new sample size is inversely proportional to the square of the ratio of the original error to the new error:

[tex]\[ n_{new} = n_{old} \times \left(\frac{E_{old}}{E_{new}}\right)^2 \] \[ n_{new} = 1000 \times \left(\frac{0.05}{0.025}\right)^2 \] \[ n_{new} = 1000 \times \left(\frac{0.05}{0.025}\right)^2 \] \[ n_{new} = 1000 \times \left(2\right)^2 \] \[ n_{new} = 1000 \times 4 \] \[ n_{new} = 4000 \][/tex]

Therefore, the new sample size will have to be approximately 4000 to reduce the maximum allowable error to 0.025."

As per the question,

Let a be any positive integer and b = 4.

According to Euclid division lemma , a = 4q + r

where 0 ≤ r < b.

Thus,

r = 0, 1, 2, 3

Since, a is an odd integer, and

The only valid value of r = 1 and 3

So a = 4q + 1 or 4q + 3

Case 1 :- When a = 4q + 1

On squaring both sides, we get

a² = (4q + 1)²

   = 16q² + 8q + 1

   = 8(2q² + q) + 1

   = 8m + 1 , where m = 2q² + q

Case 2 :- when a = 4q + 3

On squaring both sides, we get

a² = (4q + 3)²

   = 16q² + 24q + 9

   = 8 (2q² + 3q + 1) + 1

   = 8m +1, where m = 2q² + 3q +1

Now,

We can see that at every odd values of r, square of a is in the form of 8m +1.

Also we know, a = 4q +1 and 4q +3 are not divisible by 2 means these all numbers are odd numbers.

Hence , it is clear that square of an odd positive is in form of 8m +1

Answer:

The answer is:  {x | x is a day of the week}

Step-by-step explanation:

The mathematical expression {x | x is a day of the week}, uses set theory notation and can be translated as: The set of all x such that x is a day of the week. Since the original set contains all days of the week, therefore it is equivalent to the expression  {x | x is a day of the week}.

Final answer:

The correct equivalent set to { Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} is {x | x is a day of the week}, as both sets contain the days of the week.

Explanation:

The student has asked which option is equivalent to the set {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}. This set represents the seven days of the week. Therefore, the correct option that is equivalent to this set is {x | x is a day of the week}. All other options listed represent different sets with no connection to the days of the week. When comparing sets for equivalence, we look for a one-to-one correspondence between the members of each set, which is only present in the option directly referring to days of the week.

Answer:

a) The elements are [tex]\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6}[/tex]

b) The elements are (-∞ ...-1,0,1,2,..∞).

c) The elements are 2 and 3.

Step-by-step explanation:

To find : List all element of the following sets ?

Solution :

a) [tex]\{\frac{1}{n}| n\in \{ 3 , 4 , 5 , 6 \} \}[/tex]

Here, The function is [tex]f(n)=\frac{1}{n}[/tex]

Where, [tex]n\in \{ 3 , 4 , 5 , 6 \}[/tex]

Substituting the values to get elements,

[tex]f(3)=\frac{1}{3}[/tex]

[tex]f(4)=\frac{1}{4}[/tex]

[tex]f(5)=\frac{1}{5}[/tex]

[tex]f(6)=\frac{1}{6}[/tex]

The elements are [tex]\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6}[/tex]

b) [tex]\{x\in \mathbb{Z} | x=x+1\}[/tex]

Here, The function is [tex]f(x)=x+1[/tex]

Where, [tex]x\in \mathbb{Z}[/tex] i.e. integers (..,-2,-1,0,1,2,..)

For x=-2

[tex]f(-2)=-2+1=-1[/tex]

For x=-1

[tex]f(-1)=-1+1=0[/tex]

For x=0

[tex]f(0)=0+1=1[/tex]

For x=1

[tex]f(1)=1+1=2[/tex]

For x=2

[tex]f(2)=2+1=3[/tex]

The elements are (-∞ ...-1,0,1,2,..∞).

c) [tex]\{n\in \mathbb{P}| \text{n is a factor of 24}\}[/tex]

Here, The function is n is a factor of 24.

Where, n is a prime number

Factors of 24 are 1,2,3,4,6,8,12,24.

The prime factor are 2,3.

The elements are 2 and 3.

Step-by-step explanation:

Proposition If a, b [tex]\in[/tex] [tex]\mathbb{Z}[/tex], then [tex]a^{2}-4b \neq2[/tex]

You can prove this proposition by contradiction, you assume that the statement is not true, and then show that the consequences of this are not possible.

Suppose the proposition If a, b [tex]\in[/tex] [tex]\mathbb{Z}[/tex], then [tex]a^{2}-4b \neq2[/tex] is false. Thus there exist integers If a, b [tex]\in[/tex] [tex]\mathbb{Z}[/tex] for which [tex]a^{2}-4b=2[/tex]

From this equation you get [tex]a^{2}=4b+2=2(2b+1)[/tex] so [tex]a^{2}[/tex] is even. Since [tex]a^{2}[/tex] is even, a is even, this means [tex]a=2d[/tex] for some integer d. Next put [tex]a=2d[/tex] into [tex]a^{2}-4b=2[/tex]. You get [tex] (2d)^{2}-4b=2[/tex] so [tex]4(d)^{2}-4b=2[/tex]. Dividing by 2, you get [tex]2(d)^{2}-2b=1[/tex]. Therefore [tex]2((d)^{2}-b)=1 [/tex], and since [tex](d)^{2}-b[/tex] [tex]\in[/tex] [tex]\mathbb{Z}[/tex], it follows that 1 is even.

And that is the contradiction because 1 is not even.  In other words, we were wrong to assume the proposition was false. Thus the proposition is true.

Answer:

(A)  1.52 m²

Step-by-step explanation:

As per the given data of the question,

Height of a person = 5 feet

As we know that 1 feet = 30.48 cm

∴ Height = 152.4 cm

Weight of a person = 120 pounds

And we know that 1 pound = 0.453592 kg

∴ Weight = 54.4311 kg

The Mosteller formula to calculate body surface area (BSA):

[tex]BSA(m^{2})=\sqrt{\frac{Height (cm)\times Weight(kg)}{3600}}[/tex]

Therefore,

[tex]BSA=\sqrt{\frac{Height (cm)\times Weight(kg)}{3600}}[/tex]

[tex]BSA=\sqrt{\frac{152.4\times  54.4311}{3600}}[/tex]

[tex]BSA= 1.517 m^{2} = 1.52 m^{2}[/tex]

Hence, the body surface area of a person =  1.52 m²

Therefore, option (A) is the correct option.

Answer:

Step-by-step explanation:

A steam and leaf plot is the arrangement of numerical data into different groups with place value.  For eg, 17,20,21 is shown as

stem  leaf

1         7

2       0,1

A histogram is a bar chart that described frequency distribution.

A stem and leaf plot displays more information than a histogram.

Hence we have the correct answers are:

A stem-and-leaf display describes the individual observations.

A stem-and-leaf display has slightly more information than a histogram.

A stem-and-leaf display provides detailed individual data points and their distribution, while a histogram offers aggregated data into bins, showing the overall data distribution without individual details.

The differences between a histogram and a stem-and-leaf display are significant in how they present data. A stem-and-leaf display retains the individual data values and is beneficial for small datasets, showing the exact values and the frequency of data for each "stem" which provides a clear view of the distribution shape. On the contrary, a histogram groups data into contiguous bins, providing a visual representation of data distribution, showing the spread and most frequent values but without detailing individual data points. Therefore, a stem-and-leaf display has slightly more information than a histogram because it describes individual observations, unlike a histogram that aggregates data into bins.

The confidence interval for population mean is given by :-

[tex]\hat{p}\pm E[/tex], where [tex]\hat{p}[/tex] is sample proportion and E is the margin of error .

[tex]E=z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

Given : Significance level : [tex]\alpha:1-0.95=0.05[/tex]

Sample size : n= 64

Critical value : [tex]z_{\alpha/2}=1.96[/tex]

Sample proportion: [tex]\hat{p}=\dfrac{22}{64}\approx0.344[/tex]

[tex]E=(1.96)\sqrt{\dfrac{0.344(1-0.344)}{64}}\approx0.1164[/tex]

Hence, the margin of error = 0.1164

Now, the 95% theory-based confidence interval for the population proportion will be :

[tex]0.344\pm0.1164\\\\=(0.344-0.1164,\ 0.344+0.1164)=(0.2276,\ 0.4604)[/tex]

Hence, the  99% confidence interval is [tex](0.2276,\ 0.4604)[/tex]

When constructing a 95% theory-based confidence interval for the proportion of people that can correctly identify the different cola, the interval ranges from about 0.225 to 0.463. The margin of error is approximately 0.118.

This question pertains to a theory-based confidence interval for the population proportion. In this case, the proportion (p) is the number of people who correctly identified the different cola, which is 22 out of 64, or 0.34375. First, we need to calculate the standard error (SE), which is the square root of [ p(1-p) / n ], where n is the sample size. So, SE = sqrt[ 0.34375(1-0.34375) / 64 ] ≈ 0.0602.

The 95% confidence interval can be calculated as p ± Z * SE, where Z is the Z-score from the standard normal distribution corresponding to the desired level of confidence. For a 95% confidence interval, Z = 1.96. Plug the values into the equation gives us the interval [0.34375 - 1.96(0.0602), 0.34375 + 1.96(0.0602)] which is approximately [0.225, 0.463].

The margin of error is the difference between the endpoint of the interval and the sample proportion, which can be calculated as Z*SE. So the margin of error = 1.96(0.0602) ≈ 0.118.

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Answer:

  (a)  0.270 . . . . as you know

  (b)  0.356

Step-by-step explanation:

(a) p(35-44 | neither) = (35-44 & neither)/(neither total) = 210/777 ≈ 0.270

__

(b) p(neither | 35-44) = (neither & 35-44)/(35-44 total) = 210/590 ≈ 0.356

Step-by-step explanation:

When two dies are tossed, possible outcomes are (1,1) (1,2) ,(1,3) (1,4), (1,5) (1,6),(2,1) (2,2) (2,3) (2,4) (2,5) (2,6), (3,1)(3,2)(3,3)(3,4),(3,5)(3,6)(4,1) (4,2),(4,3)(4,4)(4,5)(4,6)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

so the total sample space =36

From de possible outcome, probability of getting a die that five does not appear on either side is given by total outcome of not getting five from either side/sample space

=30/36 = 5/6

And also probability of getting the difference of numbers to be 2 are (3,1)(4,2) (5,3) (6,4) =4outcomes

the probability of getting a difference of 2 is given by outcome/sample space 4/36 = 1/9

probability of getting the summation of two numbers to be equal to or greater than ten are; Outcome/sample space.

the outcome are (4,6) (5,5) (5,6)(6,4)(6,5)and (6,6)= 6outcomes

=6/36 =1/12.

The probabilities for the three events are approximately 0.6944, 0.2222, and 0.1667 respectively. These figures were achieved by comparing the number of favorable outcomes to the total number of outcomes when rolling two dice.

The subject of this question is probability which involves using numbers for calculation. When you roll two dice, there are 36 possible outcomes.

Event C: The sum of the numbers is 10 or more happens in 6 cases (5 and 5, 6 and 4, 4 and 6, 6 and 5, 5 and 6, 6 and 6). So the probability is 6/36, which simplifies to 1/6 or around 0.1667 in decimal approximation.

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Answer:

angle of elevation will be [tex]2.14^o.[/tex]

Step-by-step explanation:

Given,

height of tower = 150 ft

height of tower = 1200 ft

So, total height of peak of tower = 1200 + 150

                                                       = 1350 ft

distance of user from cell tower = 5 miles

                                                     = 5 x 5280 feet

                                                     = 26,400 feet

Since the height of user from sea level = 400 ft

so, height of peak of tower with respect to user = 1350 - 400 ft

                                                                                = 950 ft

If the angle of depression is assumed as [tex]\theta[/tex], then we can write

[tex]tan\theta\ =\ \dfrac{\textrm{height of peak of tower w.r.t user}}{\textrm{distance of user from tower}}[/tex]

[tex]=>\ tan\theta\ =\ \dfrac{950}{26400}[/tex]

[tex]=>\ tan\theta\ =\ 0.374[/tex]

 [tex]=>\ \theta\ =\ 2.14^o[/tex]

So, the angle of elevation will be [tex]2.14^o.[/tex]

Answer with Step-by-step explanation:

We are  given that  a, b, c and x are elements in the group G.

We have to find the value of x in terms of a, b and c.

a.[tex]x^2a=bxc^{-1}[/tex]

[tex]x^2ac=bxc^{-1}c=bx[/tex]

[tex]x^{-1}x^2ac=x^{-1}bx=b[/tex] ([tex]x^{-1}bx=b[/tex])

[tex]xac=b[/tex]

[tex]xacc^{-1}=bc^{-1}[/tex]

[tex]xa=bc^{-1}[/tex]    ([tex]cc^{-1}=[/tex])

[tex]xaa^{-1}=bc^{-1}a^{-1}[/tex]

[tex]x=bc^{-1}a^{-1}[/tex]

b.[tex]acx=xac[/tex]

[tex]acxc^{-1}=xacc^{-1}=xa[/tex]  ([tex]cc^{-1}=1,cxc^{-1}=x[/tex])

[tex]axa^{-1}=xaa^{-1}[/tex]  ([tex]aa^{-1}=1,axa^{-1}=x[/tex])

[tex]x=x[/tex]

Identity equation

Hence, given equation has infinite solution and satisfied for all values of a and c.

Answer:

27.19°

Step-by-step explanation:

According to the picture attached, we can find the distance between the two vectors using cosine law

[tex]a^{2} =b^{2} +c^{2} -2ab*cosA\\a=\sqrt{b^{2} +c^{2} -2ab*cosA} \\\\a=\sqrt{2.1^{2} +8.9^{2} -2(2.1)(8.9)*cos21}\\a=6.98\\\\[/tex]

Then we can get C angle by applying one more time cosine law between a and b

[tex]c^{2} =a^{2} +b^{2} -2ab*cosC\\\\c^{2} -a^{2} -b^{2}= -2ab*cosC\\\\\frac{c^{2} -a^{2} -b^{2}}{-2ab}=cosC\\ \\CosC=\frac{8.9^{2} -6.98^{2} -2.1^{2}}{-2*6.98*2.1}\\ \\CosC=-0.89\\\\ArcCos(-0.89)=C\\\\C=152.81[/tex]

We can see that the C angle is complement of the angle we are looking for, so we take away 180 degrees to get the answer

[tex]180=C+?\\\\180-C=?\\\\180-152.81=C\\\\27.19=C[/tex]

27.19 degrees is our answer!

Answer:

80 %

Step-by-step explanation:

Hi there!

To find the percent of hot dogs with mustard we must divide the number of hotdogs with mustard by the number of total hotdogs, and multiply this number by 100:

[tex]P = \frac{N_{withMustard}}{N_{total}}*100= 100*(4/5) = 80[/tex]

Greetings!

Answer:

5 rows.

Step-by-step explanation:

Imagine having 10 in each row. If you had 5 rows that means you have 5 groups of 10.

Answer:

Step-by-step explanation:

10 goes into 50, 5 times. I know this because if you multiply 10x5 it gives you 50!

Answer:

[tex]\sqrt2[/tex] is irrational

Step-by-step explanation:

Let us assume that [tex]\sqrt2[/tex] is rational. Thus, it can be expressed in the form of fraction [tex]\frac{x}{y}[/tex], where x and y are co-prime to each other.

[tex]\sqrt2[/tex] = [tex]\frac{x}{y}[/tex]

Squaring both sides,

[tex]2 = \frac{x^2}{y^2}[/tex]

Now, it is clear that x is an even number. So, let us substitute x = 2u

Thus,

[tex]2 = \frac{(2u)^2}{y^2}\\y^2 = 2u^2[/tex]

Thus, [tex]y^2[/tex]is even, which follows the fact that y is also an even number. But this is a contradiction as x and y have a common factor that is 2 but we assumed that the fraction [tex]\frac{x}{y}[/tex]  was in lowest form.

Hence, [tex]\sqrt2[/tex] is not a rational number. But [tex]\sqrt2[/tex] is a an irrational number.

Answer:

The number of books in the box is 60.

Step-by-step explanation:

Since it is given that the number of books in the box is divisible by 2,3,4,5 and 6.

So, the number of books in the box is multiple of these numbers.

Thus we have to find Least Common Multiple (L.C.M.) of these number

L.C.M. of (2,3,4,5,6) = 60

Thus the number of books in the box is multiples of 60 i.e. 60, 120, 180, 240,... etc.

The other factor that can we add in statement so there is only one possible solution is: "The number of books in the box is smallest number divisible by 2,3,4,5 and 6".

Answer:

The number is 60

Step-by-step explanation:

So the first way to solve this would be to multiply the greatest numbers in the sequence, you have 5 and 6, the result is 30, since 30 is not divisible by 4 you need to find the next number that is divisible by 5 and 6, that would be 60, since 60 is divisible by 4, then that is the answer, 60 is the first number that is divisible by 2, 3, 4, 5, and 6.

Answer:

The distance is:  

[tex]\displaystyle\frac{3\sqrt{142}}{10}[/tex]

Step-by-step explanation:

We re-write the equation of the line in the format:

[tex]\displaystyle\frac{x+2}{3}=\frac{y+\frac{3}{2}}{2}=\frac{z+\frac{4}{3}}{\frac{5}{3}} [/tex]

Notice we divided the fraction of y by 2/2, and the fraction of z by 3/3.

In that equation, the director vector of the line is built with the denominators of the equation of the line, thus:

[tex]\displaystyle\vec{v}=\left< 3, 2, \frac{5}{3}\right> [/tex]

Then the parametric equations of the line along that vector and passing through the point (-2, 3, -4) are:

[tex]x=-2+3t\\y=3+2t\\\displaystyle z=-4+\frac{5}{3}t[/tex]

We plug them into the equation of the plane to get the intersection of that line and the plane, since that intersection is the image on the plane of the point (-2, 3, -4)  parallel to the given line:

[tex]\displaystyle x+y+z=3\to -2+3t+3+2t-4+\frac{5}{3}t=3[/tex]

Then we solve that equation for t, to get:

[tex]\displaystyle \frac{20}{3}t-3=3\to t=\frac{9}{10}[/tex]

Then plugging that value of t into the parametric equations of the line we get the coordinates of the intersection:

[tex]\displaystyle x=-2+3\left(\frac{9}{10}\right)=\frac{7}{10}\\\displaystyle y=3+2\left(\frac{9}{10}\right)=\frac{24}{5} \\\displaystyle z=-4+\frac{5}{3}\left(\frac{9}{10}\right)=-\frac{5}{2}[/tex]

Then to find the distance we just use the distance formula:

[tex]\displaystyle d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}[/tex]

So we get:

[tex]\displaystyle d=\sqrt{\left(-2-\frac{7}{10}\right)^2+\left(3-\frac{24}{5}\right)^2+\left(-4 +\frac{5}{2}\right)^2}=\frac{3\sqrt{142}}{10}[/tex]

Answer:

The average of the provided grades are 86.75

Step-by-step explanation:

Consider the provided information.

On three examinations, you have grades of 85, 78, and 84. In order to get an A your average must be at least 90.

In the last exam you get 100 marks now calculate the average by using the formula:

[tex]\frac{\text{Sum of observations}}{\text{Number of observations}}[/tex]

[tex]\frac{85+78+84+100}{5}[/tex]

[tex]\frac{347}{5}[/tex]

[tex]86.75[/tex]

86.75 is less than 90 so you will not get A.

The average of the provided grades are 86.75

Answer:

Amplitude=2

Period=[tex]\frac{\pi}{3}[/tex]

Step-by-step explanation:

We are given that [tex]y=2sin6x[/tex]

We have to find the value of period and amplitude of the given function

We know that [tex]y=a sin(bx+c)+d [/tex]

Where a= Amplitude of  function

Period of sin function  =[tex]\frac{2\pi}{\mid b \mid}[/tex]

Comparing with the given function

Amplitude=2

Period=[tex]\frac{2\pi}{6}=\frac{\pi}{3}[/tex]

Hence, period of given  function=[tex]\frac{\pi}{3}[/tex]

Amplitude=2

Answer:

$12000 should be invested in a fund with a 4% return.

Step-by-step explanation:

Consider the provided information.

An amount of $15,000 is invested in a fund that has a return of  6%.

We need to calculate how much money is invested in a fund with a 4% return if the  total return on both investments is $1380.

Let $x should be invested in a fund with a 4% return.

The above information can be written as:

[tex]1380=15000\times 6\%+x\times 4\%[/tex]

[tex]1380=15000\times \frac{6}{100}+x\times \frac{4}{100}[/tex]

[tex]1380=150\times6+x\times 0.04[/tex]

[tex]1380-900=0.04x\\480=0.04x\\x=12000[/tex]

Hence, $12000 should be invested in a fund with a 4% return.

Answer:

a) [tex]c'(t) = (2 Cos(t), -2 Sin(t), 9e^t) [/tex]

b) [tex]c'(t) = (2 Cos(t), -2 Sin(t), 9e^t) [/tex]

Step-by-step explanation:

We are given in the question:

[tex]c(t) = (2 Sin(t), 2 Cos(t), 9e^t)[/tex]

F(x,y,z) = (y, -x, z)  

a) [tex]c'(t) [/tex]

We differentiate with respect to t.

[tex]c'(t) = (2 Cos(t), -2 Sin(t), 9e^t) [/tex]

b) F(c(t))

This is a composite function.

[tex]F(c(t)) = F(2 Sin(t), 2 Cos(t), 9e^t)[/tex]

[tex]= (2 Cos(t), -2 Sin(t), 9e^t)[/tex]

Answer:

Step-by-step explanation:

It is true that for any given odd integer, square of that integer will also be odd.

i.e if [tex]m[/tex] is and odd integer then [tex]m^{2}[/tex] is also odd.

In the given proof the expansion for [tex](2k + 1)^{2}[/tex] is incorrect.

By definition we know,

[tex](a+b)^{2} = a^{2} + b^{2} + 2ab[/tex]

∴ [tex](2k + 1)^{2} = (2k)^{2} + 1^{2} + 2(2k)(1)\\(2k + 1)^{2} = 4k^{2} + 1 + 4k[/tex]

Now, we know [tex]4k^{2}[/tex] and [tex]4k[/tex] will be even values

∴[tex]4k^{2} + 1 + 4k[/tex] will be odd

hence [tex](2k + 1)^{2}[/tex] will be odd, which means [tex]m^{2}[/tex] will be odd.

The error in the proof is in the expansion of the square of an odd integer. The correct expansion is 4k² + 4k + 1, and adding 1 to an even number 4k² + 4k results in an odd number, proving that m² is odd when m is an odd integer.

The proposed proof has a mistake in expanding the square of an odd integer. The correct expansion of (2k + 1)² is 4k² + 4k + 1, not 4k² + 1 as stated in the proof. To correct the proof:

The distribution of swim times from the data given would most likely be right-skewed, as the mean is larger than the median and the Interquartile Range (IQR) suggests the data is concentrated towards the middle.

From the given data about the 50-meter freestyle event, one can deduce probable distribution shape of the swim times. Notably, the mean of 43.70 seconds significantly exceeds the median of 40.15 seconds. This fact suggests a possible right skewed distribution, with longer swim times occurring less frequently but affecting the mean more strongly due to their higher values. It's called right-skewed because the 'tail' of the distribution curve extends more towards the right.

We can also examine the Interquartile Range (IQR), which measures spread in the middle 50% of the data. This is found by subtracting the lower quartile (first 25% of data) from the upper quartile (last 25% of data). An IQR of 4.98 seconds signifies much of the data is bunched in the middle of the distribution rather than at the ends, reinforcing the notion of a skewed distribution.

Thus, without a graphical representation, the swim times would be expected to exhibit a right-skewed distribution, presenting a positive skewness in the data.

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The expected shape of the distribution of swim times would be right-skewed.

To determine the expected shape of the distribution, we compare the mean and median of the swim times:

 - The mean time is 43.70 seconds, which is greater than the median time of 40.15 seconds.

- The standard deviation is 8.07 seconds, which is relatively large compared to the mean, indicating a wide spread of times.

- The interquartile range (IQR) is 4.98 seconds, which is relatively small compared to the standard deviation, suggesting that the middle 50% of the data is more tightly clustered.

In a perfectly symmetric distribution, the mean and median would be equal. However, when the mean is greater than the median, it suggests that there are some outliers or a longer tail on the right side of the distribution, pulling the mean up. The relatively large standard deviation in comparison to the IQR reinforces this idea, as it indicates there are some times that are significantly higher than the majority of the times, which are more closely packed around the median.

Therefore, the solution to the system of equations is:

x=0

y=-3t+4

z=t

We can solve the system of equations using Gaussian elimination with back-substitution. Here's how:

Steps to solve:

1. Eliminate x from the second and fourth equations:

x+y+3z=4

0=0 (2x+5y+15z=20)-(x+2y+6z=8)

3y+9z=12

-x+2y+6z=8

2. Eliminate y from the fourth equation:

x+y+3z=4

0=0

3y+9z=12

3y+9z=12 (3y+9z=12)-(3y+9z=12)

0=0

3. Since the last equation is always true, we can ignore it.

4. Solve the remaining equations:

x+y+3z=4

0=0

3y+9z=12

From the second equation, we know that y=-3z+4. Substituting this into the first equation, we get:

x+(-3z+4)+3z=4

x+4=4

x=0

Now that we know x=0, we can substitute it back into the third equation to solve for z:

3(-3z+4)+9z=12

-9z+12+9z=12

12=12

This equation is always true, so there are infinitely many solutions. We can express x, y, and z in terms of the parameter t as follows:

x=0

y=-3z+4

z=t

Complete Question:

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x, y, and z in terms of the parameter t.)

3x + 3y + 9z = 12

x + y + 3z = 4

2x + 5y + 15z = 20

-x + 2y + 6z = 8

(x, y, z) =

The smallest of five consecutive numbers, whose sum equals 320, is 62. This is determined by setting up an equation where the sum of these numbers equals 320, then solving for the smallest number 'n'.

The subject at hand involves understanding patterns within consecutive numbers (numbers that follow each other in order, without gaps). In the given example of four consecutive even numbers 2, 4, 6 and 8, we see that the difference between them is constantly 2.

When it comes to the sum of five consecutive numbers equating to 320, let's presume the first (and smallest) number is 'n'. Therefore, the consecutive numbers would be n, (n+1), (n+2), (n+3) and (n+4). Their sum ought to total 320, so we write the expression n + (n+1) + (n+2) + (n+3) + (n+4) = 320. By simplifying, we receive 5n+10 = 320.

Further simplifying, we subtract 10 from both sides to afford: 5n = 310. Divide 310 by 5 to isolate 'n', which results in 'n' equals 62. Consequently, the smallest of the five consecutive numbers is 62.

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