For what values do maximum r-values occur on the graph the polar equation
r=-3 + 4 cos theta? Note that the maximum r-value occurs at a point that is the maximum distance from the pole.

Answer:

[tex]n=\frac{0.3(1-0.3)}{(\frac{0.01}{1.96})^2}=8067.36[/tex]  

And rounded up we have that n=8068

Step-by-step explanation:

The margin of error for the proportion interval is given by this:  

[tex] ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]    (a)  

We want for this case a 95% of confidence desired, our significance level would be given by [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex]. And the critical value would be given by:

[tex]z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96[/tex]

The margin of error for this case is [tex]ME =\pm 0.01[/tex] and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex]   (b)  

The estimated proportion for this case is [tex]\hat p=0.3[/tex]. And replacing into equation (b) the values from part a we got:

[tex]n=\frac{0.3(1-0.3)}{(\frac{0.01}{1.96})^2}=8067.36[/tex]  

And rounded up we have that n=8068

Answer:

The results are

Face     Count         Face          Count

1               6                  4                9

2              12                 5                5

3              8                 6               20

These results are not fair, because they clearly shows that the dice was tricked. We deduct this by observing that he obtained the face 6, 20 times out of 48, which represents 42% of chances to get that result.

Now, a fair dice must have equal probabilities per side, or an approximation to that at least. In this case, we don't have such equality.

So, next time your little brother roll the dice, we probably will get 6.

To check the fairness of a die, use statistical testing. Expect each face to appear equally in a fair die. Discrepancies suggest possible bias, but randomness can make results vary. Conduct a hypothesis test, like Chi-square, for a more statistically sound conclusion.

To verify the fairness of the die, we need to conduct a hypothesis test. With a fair six-sided die, each face (numbered 1 to 6) should have an equal probability of appearing on any given roll. That means, if you roll the die 48 times, each face should theoretically appear about 8 times (48 rolls divided by 6 faces).

If the observed frequency of each face being rolled significantly varied from 8, it would suggest that the die might be biased. However, keep in mind that random chance can also result in each face not appearing exactly 8 times. Therefore, a hypothesis test would have more statistical power if more repetitions were performed.

Conduct the hypothesis testing by comparing your observed frequencies to the expected frequencies using a Chi-square test. This would provide a more quantitative and concrete conclusion about whether the die is biased or fair.

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Answer:

17.5%

Step-by-step explanation:

Convert fraction (ratio) 7 / 40 Answer: 17.5%

Answer:

17.5

Step-by-step explanation:

7/40 = 17.5

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Answer:

We conclude that political science classes require less writing time than history classes at 0.01 level of significance.

Step-by-step explanation:

We are given that the 55 randomly selected political science classes assigned an average of 16.3 pages of essay writing for the course. The standard deviation for these 55 classes was 4.8 pages.

The 54 randomly selected history classes assigned an average of 18.9 pages of essay writing for the course. The standard deviation for these 54 classes was 5 pages.

Let [tex]\mu_1[/tex] = average writing time required by political science classes.

[tex]\mu_2[/tex] = average writing time required by history classes.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu_1\geq\mu_2[/tex]      {means that political science classes require more or equal writing time than history classes}

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu_1<\mu_2[/tex]      {means that political science classes require less writing time than history classes}

The test statistics that would be used here Two-sample t test statistics as we don't know about the population standard deviation;

                     T.S. = [tex]\frac{(\bar X_1-\bar X_2)-(\mu_1-\mu_2)}{s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }[/tex]   ~ [tex]t__n_1_-_n_2_-_2[/tex]

where, [tex]\bar X_1[/tex] = sample average pages of essay writing for political science classes = 16.3 pages

[tex]\bar X_2[/tex] = sample average pages of essay writing for history classes = 18.9 pages

[tex]s_1[/tex] = sample standard deviation for political science classes = 4.8 pages

[tex]s_2[/tex] = sample standard deviation for history classes = 5 pages

[tex]n_1[/tex] = sample of political science classes = 55

[tex]n_2[/tex] = sample of history classes = 54

Also,  [tex]s_p=\sqrt{\frac{(n_1-1)s_1^{2}+(n_2-1)s_2^{2} }{n_1+n_2-2} }[/tex]  =  [tex]\sqrt{\frac{(55-1)\times 4.8^{2}+(54-1)\times 5^{2} }{55+54-2} }[/tex] = 4.90

So, test statistics  =  [tex]\frac{(16.3-18.9)-(0)}{4.9 \sqrt{\frac{1}{55}+\frac{1}{54} } }[/tex]  ~ [tex]t_1_0_7[/tex]

                               =  -2.77

The value of t test statistics is -2.77.

Now, at 0.01 significance level the t table gives critical value of -2.365 at 107 degree of freedom for left-tailed test.

Since our test statistics is less than the critical value of t, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that political science classes require less writing time than history classes.

Answer:

The sample proportion of highway bridges that need repaired is 0.6.

Step-by-step explanation:

The sample proportion of highway bridges that need repaired is the number of bridges in need of repair divided by the number of bridges around the country.

In this question:

75 bridges in need of repair, out of 125. So

75/125 = 0.6

The sample proportion of highway bridges that need repaired is 0.6.

Answer:

gallons of punch in all  = 4.5 gallons

Step-by-step explanation:

Morgan poured 3/4 gallons of punch into each of 6 bottles . This means Morgan poured 3/4 gallons of punch in each of the available 6 bottle .

He pours 3/4 gallons of punch in 1 bottle . For 6 bottles the gallons of punch poured is calculated below

1 bottle = 3/4 gallons of punch

6 bottles = ? gallons of punch

cross multiply

gallons of punch in all = 3/4 × 6

gallons of punch in all  = 18/4

gallons of punch in all  = 9/2 gallons

gallons of punch in all  = 4.5 gallons

to get the variable alone you need to add three to both sides

x-3=7

+3  +3

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

x= 10

Answer:

Step 1: Bring like terms together Step 2: Add variable to both sides Step 3: Simplify Step 4: Add the constant to both sides Step 5: Simplify  Step 6: Divide by the coefficient Step 7: Solve Step 8: Test your solution Answer: 10

Step-by-step explanation:

Answer:

b++nfr

Step-by-step explanation:

jjkiki

Answer:

(3^6)/2

Step-by-step explanation:

make sure everything as the numerator is in parenthesis

Answer:

  (f◦g)(x) = 6x +9

Step-by-step explanation:

Use g(x) as the argument to f(x) and simplify.

  (f◦g)(x) = f(g(x)) = f(3x +5) = 2(3x +5) -1

  (f◦g)(x) = 6x +9

Answer:

See explaination

Step-by-step explanation:

Please kindly check attachment for the step by step solution of the given problem.

The attached file have the solved problem.

Answer: B

Step-by-step explanation:
just did it

Answer:

81.25

Step-by-step explanation:

45.55 + 35.7

Answer:I hope that helps

Step-by-step explanation:

Answer:22

Step-by-step explanation:

diameter=d

Area=121π

area of circle=π x (d/2)^2

121π=π x d^2/4

121π=πd^2/4

Cross multiplying we get

121π x4=πd^2

484π=πd^2

Divide both sides by π

484π/π=πd^2/π

d^2=484

Taking the square roots of both sides we get

d=√(484)

d=22 diameter=22

Answer:

1/4

Step-by-step explanation:

[tex](\dfrac{3(3)^{-2}(-1)^6}{2(3)^{-1}(-1)^5})^2=[/tex]

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

[tex](-\dfrac{1}{2})^2=[/tex]

[tex]\frac{1}{4}[/tex]

Hope this helps!

Answer:

Sec(x)

Step-by-step explanation:

Tan(x) = Sin(x)/Cos(x)

Csc(x) = 1/Sin(x)

Sin(x)        1

--------- . ---------

Cos(x)    Sin(x)

Sin(x) cancel

   1

--------- =

Cos(x)

Sec(x)

The simplified expression is,

⇒ sec x

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ tan x × csc x

Now,

We can simplify as;

⇒ tan x × csc x

⇒ ( sin x/ cos x ) × (1 / sin x)

⇒ 1 / cos x

⇒ sec x

Thus, The simplified expression is,

⇒ sec x

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Maria and her friends rides for 56 minutes

Start time = 01:09

End time = 02:05

∴ Ride Time = 60-4

∴ Ride Time = 56 minutes

Therefore the total ride time that Maria and her friend rode is 56 minutes

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Final answer:

87 1/3% is approximately 17433/10000 as a fraction after converting 87.33% to a fraction with a denominator of 100 and simplifying.

Explanation:

To convert 87 1/3% into a fraction, we first recognize that 1/3% is equal to 0.33% (approximately), so 87 1/3% is approximately 87.33%. Next, we write this percent as a fraction with a denominator of 100 to represent the percent: 87.33/100.

To simplify further, we can convert the decimal to a fraction, where 0.33 is approximately 33/100, and add this to 87 to get:

Thus, 87 1/3% is approximately 17433/10000 when expressed as a fraction.

Answer:

A. a rectangle and a triangle

Answer:

a) rectangle and triangle, then add both to fin the total area.

Step-by-step explanation:

Answer:

[tex](a)\dfrac{92}{117}[/tex]

[tex](b)\dfrac{8}{39}[/tex]

[tex](c)\dfrac{25}{117}[/tex]

Step-by-step explanation:

Number of Men, n(M)=24

Number of Women, n(W)=3

Total Sample, n(S)=24+3=27

Since you cannot appoint the same person twice, the probabilities are without replacement.

(a)Probability that both appointees are men.

[tex]P(MM)=\dfrac{24}{27}X \dfrac{23}{26}=\dfrac{552}{702}\\=\dfrac{92}{117}[/tex]

(b)Probability that one man and one woman are appointed.

To find the probability that one man and one woman are appointed, this could happen in two ways.

P(One man and one woman are appointed)[tex]=P(MW)+P(WM)[/tex]

[tex]=(\dfrac{24}{27}X \dfrac{3}{26})+(\dfrac{3}{27}X \dfrac{24}{26})\\=\dfrac{72}{702}+\dfrac{72}{702}\\=\dfrac{144}{702}\\=\dfrac{8}{39}[/tex]

(c)Probability that at least one woman is appointed.

The probability that at least one woman is appointed can occur in three ways.

P(at least one woman is appointed)[tex]=P(MW)+P(WM)+P(WW)[/tex]

[tex]P(WW)=\dfrac{3}{27}X \dfrac{2}{26}=\dfrac{6}{702}[/tex]

In Part B, [tex]P(MW)+P(WM)=\frac{8}{39}[/tex]

Therefore:

[tex]P(MW)+P(WM)+P(WW)=\dfrac{8}{39}+\dfrac{6}{702}\\$P(at least one woman is appointed)=\dfrac{25}{117}[/tex]

Answer:

[tex]d = 14,600\,km[/tex]

Step-by-step explanation:

The distanced travelled is equal to twice the distance between points A and B. The distance that was walked in meters is:

[tex]d = 2\cdot (7.3\,km)\cdot \left(\frac{1000\,m}{1\,km} \right)[/tex]

[tex]d = 14,600\,km[/tex]

Final answer:

The total distance walked on the Red Trail, back and forth, when converted from kilometers to meters, is 14,600 meters.

Explanation:

The question asks for the total distance walked in meters if a person walks from point A to point B and back on the Red Trail, where the distance between point A and point B is 7.3 kilometers.

Firstly, we know that 1 kilometer equals 1,000 meters. Therefore, to find the distance in meters for a single trip from point A to point B, we multiply 7.3 kilometers by 1,000:

Since the trip was made back and forth, the total distance covered is twice the distance of a single trip:

Therefore, the total distance walked back and forth on the Red Trail, in meters, is 14,600 meters.

To find the polar coordinates (r, θ) of a point given in Cartesian coordinates (x, y), we can use the equations (r, θ) = (√(x² + y²), atan2(y, x)).

To find the polar coordinates (r, θ) of a point given in Cartesian coordinates (x, y), we can use the following equations:

(r, θ) = (√(x² + y²), atan2(y, x))

(a)(i) For the Cartesian coordinates (6, -6),

(r, θ) = (√(6² + (-6)²), atan2(-6, 6))

(r, θ) = (√(36 + 36), atan2(-6, 6))

(r, θ) = (6√2, -45°)

(a)(ii) For the Cartesian coordinates (6, -6),

(r, θ) = (-√(6² + (-6)²), atan2(-6, 6) + π)

(r, θ) = (-6√2, -45° + π)

(b)(i) For the Cartesian coordinates (-1, 3),

(r, θ) = (√((-1)² + 3²), atan2(3, -1))

(r, θ) = (√(1 + 9), atan2(3, -1))

(r, θ) = (√10, 116.57°)

(b)(ii) For the Cartesian coordinates (-1, 3),

(r, θ) = (-√((-1)² + 3²), atan2(3, -1) + π)

(r, θ) = (-√10, 116.57° + π)

Answer:

The point estimate of this proportion is [tex]\pi = 0.5857[/tex]

The 95% confidence interval to estimate the proportion of Americans who feel that the environment is a major issue with them is (0.5584, 0.6130).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

[tex]n = 1255, \pi = \frac{735}{1255} = 0.5857[/tex]

The point estimate of this proportion is [tex]\pi = 0.5857[/tex]

95% confidence level

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5857 - 1.96\sqrt{\frac{0.5857*0.4143}{1255}} = 0.5584[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5857 + 1.96\sqrt{\frac{0.5857*0.4143}{1255}} = 0.6130[/tex]

The 95% confidence interval to estimate the proportion of Americans who feel that the environment is a major issue with them is (0.5584, 0.6130).

Answer:

The best point of estimate is given by:

[tex]\hat p =\frac{X}{n}= \frac{735}{1255}=0.586[/tex]

[tex]0.586 - 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.559[/tex]

[tex]0.586 + 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.613[/tex]

And we have 95% of confidence that the true proportion of Americans who feel that the environment is a major issue with them is between 0.559 and 0.613

Step-by-step explanation:

The interest is the real proportion of Americans who feel that the environment is a major issue with them. The best point of estimate is given by:

[tex]\hat p =\frac{X}{n}= \frac{735}{1255}=0.586[/tex]

Confidence interval

The confidence interval for the true proportion of interest is given by this:  

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

We want an interval at 95% of confidence, so then the significance level is [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex]. And the critical value for this case are:

[tex]z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96[/tex]

Replacing into the confidence interval formula we got:

[tex]0.586 - 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.559[/tex]

[tex]0.586 + 1.96\sqrt{\frac{0.586(1-0.586)}{1255}}=0.613[/tex]

And we have 95% of confidence that the true proportion of Americans who feel that the environment is a major issue with them is between 0.559 and 0.613

Answer:

Question1:volume=6600cm^3

Question2:cost=$441.18

Step-by-step explanation:

Question 1:

volume=length x width x height

Volume=15 x 20 x 22

Volume=6600cm^3

Question 2:

area of kitchen=length x width

Area of kitchen=18 x 19

Area of kitchen=342ft^2

$1.29 for 1ft^2

$b for 342ft^2

$b=342 x1.29

$b=$441.18

Answer:

N = 16 measurements

Step-by-step explanation:

A method produces a random error for each measurement of 4%

A percent error of equal to or less than 1% is required.

We want to find out the minimum number of measurements that must be collected.

The standard error is given by

[tex]SE = \frac{S}{\sqrt{N} } \\[/tex]

Where s is the standard deviation and N is the number of measurements.

We are given standard deviation equal to 4% and SE equal to 1%

So re-arranging the above equation for N

[tex]\sqrt{N} = \frac{S}{SE} \\N = (\frac{S}{SE})^{2}\\N = (\frac{0.04}{0.01})^{2}\\N = 16[/tex]

Therefore, a minimum 16 number of measurements are needed.

Final answer:

The expected value of flipping a coin twice (X) is 1 head, with a standard deviation of 0.7071. The expected value of flipping a coin 2,000 times (Y) is 1,000 heads, but we cannot calculate P(940 ≤ Y ≤ 1,060) from the given information.

Explanation:

The question involves the concept of probability, expected value (E), and standard deviation (SD) in the context of flipping coins. We'll address each part of the query using basic probability and statistics principles.

P(940 ≤ Y ≤ 1,060) would require the use of the Central Limit Theorem for large sample sizes; however, with the information given, we cannot calculate this probability without additional statistical tables or tools, so we must refuse to answer this part.

The expected value of X, E(X), when flipping a coin twice would be the sum of products of each outcome's value and its probability. E(X) = (0 * 0.25) + (1 * 0.5) + (2 * 0.25) = 1.

The standard deviation of X requires calculating the variance first, then the square root of the variance. σ(X) = √[((0-1)^2 * 0.25) + ((1-1)^2 * 0.5) + ((2-1)^2 * 0.25)] = √(0.5) = 0.7071.

The expected value of Y, E(Y), when flipping a coin 2,000 times is simply 0.5 * 2,000 = 1,000 because the probability of a head for each flip is 0.5.

The probability P(940 ≤ Y ≤ 1,060) is approximately 99.63%, E(X) equals to 1, and the standard deviation of X is approximately 0.707. E(Y) is 1000.

To understand the probability and expectations for tossing fair coins, we will address your questions step by step.

P(940 ≤ Y ≤ 1,060) involves using the normal approximation to the binomial distribution.

For large numbers of coin tosses, the binomial distribution can be approximated by a normal distribution.

Since we are tossing the coin 2,000 times, our mean (μ) and variance (σ²) need to be calculated.

Hence σ ≈ 22.36.

Now, we standardize the range [940, 1060].

From the Z-table, P(-2.68 ≤ Z ≤ 2.68) is approximately 0.9963, or 99.63%.

E(X) represents the expected number of heads when tossing 2 coins. Since each coin is fair and has a probability of 0.5 for heads,

Standard Deviation = √(n * p * (1 - p)). Here, n = 2, p = 0.5.

E(Y) is the expected number of heads when tossing 2,000 coins. Similarly,

Therefore, probability P(940 ≤ Y ≤ 1,060) is 99.63%, E(X) is 1, and the standard deviation is 0.707. E(Y) is 1000.

The manager needs to select dimensions for a right rectangular prism that when multiplied together equal 98 to achieve a volume of 98 cubic feet. Possible dimensions could be 7 feet by 7 feet by 2 feet, or 14 feet by 7 feet by 1 foot.

The question pertains to calculating the volume of right rectangular prisms (boxes) and understanding the relationship between their dimensions and volume. To find a box with a volume of 98 cubic feet, one can use the formula for the volume of a right rectangular prism, which is Volume = length * width  * height. The manager needs to identify the dimensions that, when multiplied together, equal 98. For instance, if a box has dimensions of 7 feet, 7 feet, and 2 feet, the volume would be 7  * 7 * 2 = 98 cubic feet, as required. Other possible dimensions could be 14 feet by 7 feet by 1 foot, as 14 * 7 * 1 also gives 98 cubic feet.

Answer:

[tex]\theta = tan^{-1}(\frac{y}{x})[/tex]

Step-by-step explanation:

If we are given components of a vector then we can find the angle between them.

Suppose we are given a vector v

[tex]v = (x, y)[/tex]

Where x is the horizontal component and y is the vertical component.

The angle can be found by using

[tex]tan(\theta)=\frac{y}{x}\\\\\theta = tan^{-1}(\frac{y}{x})[/tex]

The magnitude of the vector v can be found using

[tex]v = \sqrt{x^{2}+y^{2}}[/tex]

Example:

Lets do a quick example:

[tex]v = (2, 4)[/tex]

The angle of the vector is

[tex]tan(\theta)=\frac{4}{2}\\\\\theta = tan^{-1}(\frac{4}{2})\\\\\theta = 63.43^{\circ}[/tex]

The magnitude of the vector is

[tex]v = \sqrt{x^{2}+y^{2}}\\\\v = \sqrt{2^{2}+4^{2}}\\\\v = \sqrt{4+16}\\\\v = \sqrt{20}\\\\v = 4.47[/tex]

Answer:

the answer is 5

Step-by-step explanation:

the

Answer:

14/10=1.4

7/1.4=5

1.4x5=7

Step-by-step explanation:

In the given equation 14/10 divided by a certain number equals seven, the missing number is 0.2

The subject of this question is Mathematics and it appears to be suitable for a Middle School grade level. You're asking for a number that, when 14/10 or 1.4 is divided by it, the result is seven.

So, 14/10 divided by 0.2 equals seven.

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