3x^2-12x-15 what’s the minimum value ?

25. The area of the region that lies inside the first curve and outside the second curve is 4[√3/2 - 2π/3] square units.

29. The area of the region that lies inside both curves is approximately 1.648 square units.

A 3D solid shape called a cylinder is formed by connecting two parallel and identical bases with a curving surface. The shape of the bases is similar to a disc, and the axis of the cylinder runs through the middle or connects the two circular bases.

25. To find the area of the region that lies inside the first curve and outside the second curve, we need to find the points where the two curves intersect, and then integrate the difference in the areas between the two curves from one intersection point to the other.

The two curves are given by:

r² = 8 cos θ   (first curve)

r = 2           (second curve)

To find the intersection points, we substitute r = 2 into the first equation and solve for θ:

2² = 8 cos θ

cos θ = 1/2

θ = ±π/3

So the two curves intersect at θ = π/3 and θ = -π/3. To find the area between the curves, we integrate the difference in the areas between the two curves from θ = -π/3 to θ = π/3:

A = ∫[-π/3,π/3] [(1/2)r² - 2²] dθ

Using the equation r² = 8 cos θ, we can simplify this to:

A = ∫[-π/3,π/3] [(1/2)(8 cos θ) - 4] dθ

A = ∫[-π/3,π/3] (4 cos θ - 4) dθ

A = 4 ∫[-π/3,π/3] (cos θ - 1) dθ

[tex]A = 4 [sin \theta - \theta]_{(-\pi/3)^{(\pi/3)[/tex]

A = 4 [sin(π/3) - π/3 - (sin(-π/3) + π/3)]

A = 4 [√3/2 - 2π/3]

Therefore, the area of the region that lies inside the first curve and outside the second curve is 4[√3/2 - 2π/3] square units.

29. To find the area of the region that lies inside both curves, we need to determine the points where the two curves intersect and then integrate the area enclosed between the curves over the appropriate range of polar angles.

The two curves are given by:

r = 3 cos(θ)  (first curve)

r = sin(θ)    (second curve)

To find the intersection points, we substitute r = 3 cos(θ) into the equation r = sin(θ) and solve for θ:

3 cos(θ) = sin(θ)

tan(θ) = 3

θ = tan⁻¹(3)

The intersection point lies on the first curve when θ = tan⁻¹(3), so we need to integrate the area enclosed between the curves from θ = 0 to θ = tan⁻¹(3).

The area enclosed between the curves at any angle θ is given by the difference in the areas of the circles with radii r = sin(θ) and r = 3 cos(θ). Thus, the area enclosed between the curves is:

A = ∫[0,tan⁻¹(3)] [(1/2)(3 cos(θ))² - (1/2)(sin(θ))²] dθ

Simplifying, we get:

A = ∫[0,tan⁻¹(3)] [9/2 cos²(θ) - 1/2 sin²(θ)] dθ

Using the identity cos(2θ) = cos²(θ) - sin²(θ), we can simplify this to:

A = ∫[0,tan⁻¹(3)] [(9/2)(cos²(θ) - (1/2)) + (1/2)cos²(2θ)] dθ

We can evaluate the first term of the integrand using the identity cos²(θ) = (1 + cos(2θ))/2, and the second term using the identity cos²(2θ) = (1 + cos(4θ))/2:

A = ∫[0,tan⁻¹(3)] [(9/4)(1 + cos(2θ)) - (1/4)(1 + cos(4θ))] dθ

Integrating each term separately, we get:

[tex]A = [(9/4)\theta + (9/8)sin(2\theta) - (1/16)sin(4\theta)]_{0^{(tan^-1(3))[/tex]

Simplifying and evaluating, we get:

A = (9/4)tan⁻¹(3) + (9/8)sin(2tan⁻¹(3)) - (1/16)sin(4tan⁻¹(3))

Using the identity sin(2tan⁻¹(3)) = 6/10 and simplifying, we get:

A = (9/4)tan⁻¹(3) + (27/40) - (3/40)tan⁻¹(3)

Therefore, the area of the region that lies inside both curves is approximately 1.648 square units.

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Answer:

y ≥ 2x - 3

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (0, -3) (2,1)

We see the y increase by 4 and the x increase by 2, so the slope is

m = 4/2 = 2

Y-intercept is located at (0, -3)

Because the graph is on top left, so the equation will be y ≥ 2x - 3

The missing value is 14, and option d is correct.

Consider the data table:

x      y

1      6

2     10

3      __

Data represent a linear  function.

To find:

The missing value.

Solution:

Let the missing value be p.

Slope Formula:

m= y2-y1/x2-x1

Data represent a linear function. So, the slope always remains the same.

10-6/2-1 = p -10/3-2

4= p- 10

Adding 10 on both sides, we get

p = 14

Therefore, the missing value is 14, and option d is correct.

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From the table shown below, there are 40% of the people from the survey that were teachers.

Through the use of given information, the numbers can be used to fill in the frequency table as follows:

Vote                YES for later start       Vote NO for later start       Total

Students                 30                                 15                                    45

Teachers                 20                                10                                    30

Total                         50                                25                                      75

b) To be able to know the percentage of people that were surveyed and who were teachers, we  have to divide the number of teachers by the total number of people that were said to have been surveyed:

Percentage of teachers

= 30 /75  x 100%

= 40%

Therefore, based on the above, 40% of the people were surveyed and they were teachers.

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Cohen's d is 0.44, which suggests a medium effect size

Null hypothesis: H0: µ = 4.5 and Alternative hypothesis: Ha: µ > 4.5 (one-tailed test)

The sample mean is M = 5.2 and the sample size is n = 31. The population standard deviation is unknown, so we use the t-distribution.

The standard error of the mean is:

[tex]SE=\frac{\sqrt{\frac{SS}{n-1} } }{\sqrt{n} } = \frac{\sqrt{\frac{-170}{30} } }{\sqrt{31} } = 0.328[/tex]

The t-statistic is:

[tex]t= (\frac{M-µ}{SE}) = (\frac{5.2-4.5}{0.328}) = 2.13[/tex]

Using a one-tailed t-test with a .01 level of significance and 30 degrees of freedom, the critical value is 2.756. Since the obtained t-value (2.13) is less than the critical t-value (2.756), we fail to reject the null hypothesis.

Since we failed to reject the null hypothesis, we cannot conclude that the medication significantly increased flexibility.

Cohen's d can be calculated as:

[tex]d= \frac{(M-µ}{SD} = \frac{5.2-4.5}{\sqrt{\frac{SS}{n-1} } } = \frac{0.84}{1.9} = 0.44[/tex]

Therefore, Cohen's d is 0.44, which suggests a medium effect size.

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There are a total of possible values of for which is divisible by .

We want to find the number of values of for which is divisible by . Let's consider the possible remainders when is divided by .

If , then is divisible by , since the last digit of any even number is 0, 2, 4, 6, or 8, all of which are divisible by .

If , then is not divisible by , since the last digit of any odd number is 1, 3, 5, 7, or 9, none of which are divisible by .

Therefore, we can assume that is even, and write as , where is an -digit number and is a digit from 0 to 9. We can then write:

We know that is divisible by , so we need to find the values of for which is divisible by . This is equivalent to finding the values of for which is divisible by , since is relatively prime to .

We can rewrite the equation above as:

This shows that is divisible by if and only if is divisible by . Since and are relatively prime, this occurs if and only if both and are divisible by . In other words, we need to find the values of such that both and are divisible by .

There are 5 even digits (0, 2, 4, 6, and 8) that can be chosen for , and 10 digits (0 to 9) that can be chosen for . Thus, there are a total of possible choices for the pair (). We need to determine how many of these pairs result in both and being divisible by .

For to be divisible by , we need the sum of its digits to be divisible by . Since is even, this is equivalent to requiring the sum of the digits of to be divisible by . This means that we can choose any combination of the even digits (0, 2, 4, 6, and 8) to fill the digits of , with no restrictions. There are 5 choices for each digit of , for a total of possible -digit numbers that are divisible by .

For to be divisible by , we need to be divisible by . Since is relatively prime to , this is equivalent to requiring to be divisible by . Since is an -digit number, it follows that is an -digit number. Thus, we need to choose the first digits of to be divisible by .

There are 10 choices for each of the first digits of , and 5 choices for the last digit (since it must be even). Thus, there are a total of possible -digit numbers that have the first digits divisible by .

To count the number of pairs () that result in both and being divisible by , we can use the multiplication principle: we multiply the number of choices for by the number of choices for , since these choices are independent of each other. Thus, the total number of pairs () is:

Therefore, there are a total of possible values of for which is divisible by .

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The equivalent expression of the fraction 1/{5(n+45)} as,

1/{5(n+45)} = 1/ (5n + 225)

Equivalent expression of 1/{5(n+45)} can be expressed as,

By applying distributive property of addition in the denominator of the given fraction 1/{5(n+45)} , we get

5(n + 45) = 5·n + 5·45

⇒ 5(n + 45) = 5n + 225

The numerator of fraction 1/{5(n+45)} can be written in equivalent expression as,

1 = 1·1 = 1

Thus, we can write the equivalent expression of the fraction 1/{5(n+45)} as,

1/{5(n+45)} = 1/ (5n + 225)

Equivalent expressions are defined as expressions which work in same way even though they look different from each other.

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Answer:

the midpoint would be (-2,-2)

Step-by-step explanation:

Answer:

7/4, 9/4, 11/4, 13/4

Step-by-step explanation:

+ In an arithmetic sequence, to find the pattern you must subtract a term from the term after it to find the common difference.

3/4 subtracted from 5/4 is 2/4

1/4 subtracted from 3/4 is 2/4

+ So the common difference is 2/4 (Aka 1/2, but you want to keep the same denominator)

+ Therefore, between each new term, you add 2/4

1/4, 3/4, 5/4, 7/4, 9/4, 11/4, 13/4... and so on

The number of liters it would take to cover 120 km is 8 liters.

Looking at the graph that shows the liters consumed per kilometer, or rather the number of kilometers per liter, we see that each liter enables to car to go for 15 km.

This means that if we should want to go 120 km, the number of liters needed would be:

= Distance to cover / Kilometers per liter

Distance to cover = 120 km

Kilometers per liter = 15 km

The liters needed are:

= 120 / 15

= 8 liters

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a) The accumulated sales for the first 9 days is approximately $9,359.49.

b) The sales from the 2nd day through the 5th day is approximately $6,022.25.

To find the accumulated sales for the first 9 days, we need to integrate the given rate of change of sales with respect to time:

S'(t) = [tex]23e^t[/tex]

Integrating both sides with respect to t, we get:

S(t) = ∫S'(t) dt = ∫[tex]23e^t[/tex]dt = [tex]23e^t[/tex] + C

where C is the constant of integration.

To find the value of C, we use the initial condition that the sales at day 0 (i.e., the starting point) is $0:

S(0) = 0 = 23e^0 + C

Therefore, C = -23.

Substituting this value of C, we get:

S(t) = [tex]23e^t[/tex] - 23

a) To find the accumulated sales for the first 9 days, we need to evaluate S(9) - S(0):

[tex]S(9) - S(0) = (23e^9 - 23) - (23e^0 - 23) = 23(e^9 - 1) ≈ $9,359.49[/tex]

Therefore, the accumulated sales for the first 9 days is approximately $9,359.49.

b) To find the sales from the 2nd day through the 5th day, we need to evaluate S(5) - S(2):

[tex]S(5) - S(2) = (23e^5 - 23) - (23e^2 - 23) = 23(e^5 - e^2) ≈ $6,022.25[/tex]

Therefore, the sales from the 2nd day through the 5th day is approximately $6,022.25.

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The probability, provided that the windshield is a rectangle with the dimensions 28 inches by 54 inches and his line of site through the windshield is a rectangle with the dimensions 30 inches by 24 inches is 0.476

Continuous Probability is used for this information. Probability = Area of line of sight / total area of windshield

Probability = (30*24)/(28*54)

Probability = 0.476

The probability will be 0.476.

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The probability that a defective component will not be identified as defective before it is shipped  is 0.42 or 42%.

Let's consider the events:

A: the component is defective

B1: the component is identified as defective in the first test

B2: the component is identified as defective in the second test

We want to find the probability that a defective component will not be identified as defective before it is shipped, which is equivalent to the probability that neither B1 nor B2 occur.

Using the complement rule, we can find the probability of the complement event (at least one test identifies the component as defective) and subtract from 1:

P(not identified) = 1 - P(B1 or B2)

Since the tests are independent, we can use the multiplication rule:

P(B1 and B2) = P(B1) * P(B2 | B1)

Since the component can only be identified as defective in the second test if it was not identified as defective in the first test, we have:

P(B2 | B1) = P(B2)

Therefore,

P(B1 and B2) = P(B1) * P(B2)

= P(A) * P(B1 | A) * P(B2 | A')

= 0.3 * 0.7 * 0.9

= 0.189

Using the addition rule for the probability of the union of two events:

P(B1 or B2) = P(B1) + P(B2) - P(B1 and B2)

= P(A) * (P(B1 | A) + P(B2 | A') - P(B1 | A) * P(B2 | A'))

= 0.3 * (0.7 + 0.1 - 0.7 * 0.1)

= 0.58

Therefore,

P(not identified) = 1 - P(B1 or B2)

= 1 - 0.58

= 0.42

So the probability that a defective component will not be identified as defective before it is shipped is 0.42 or 42%.

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To determine the probability that a person owns a Dodge or has four-wheel drive, we need to know the total number of people being considered and how many of them meet either of these criteria. Without this information, we cannot provide an accurate answer.

To calculate the probability that a person owns a Dodge or has four-wheel drive, you need to consider the individual probabilities of each event and the overlapping probability of both events occurring. Let's denote the events as follows:

- P(D): Probability of owning a Dodge
- P(F): Probability of having a four-wheel drive
- P(D ∩ F): Probability of both owning a Dodge and having a four-wheel drive

Using the formula for the probability of either event occurring:

P(D ∪ F) = P(D) + P(F) - P(D ∩ F)

Without specific values for these probabilities, it is impossible to give a numerical answer. However, you can use the above formula once you have the relevant data.

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We can conclude that the marketing manager's claim is true.

To determine whether the marketing manager's claim is true, we need to conduct a hypothesis test.

Let p be the proportion of all children aged 8-12 who have cell phones. The marketing manager claims that p > 0.5, while the national consumers group survey found that 39/54 or p' = 0.722 have cell phones.

The null hypothesis is that the true proportion of children with cell phones is less than or equal to 0.5:

H0: p ≤ 0.5

The alternative hypothesis is that the true proportion of children with cell phones is greater than 0.5:

Ha: p > 0.5

We will conduct a one-tailed hypothesis test with a level of significance of 0.05.

Under the null hypothesis, the sample proportion follows a binomial distribution with parameters n = 54 and p = 0.5. The standard error of the sample proportion is given by:

SE = √[p(1-p)/n] = √[0.5(1-0.5)/54] = 0.070

The test statistic is calculated as:

z = (p' - p) / SE = (0.722 - 0.5) / 0.070 = 3.14

The critical value for a one-tailed test with a level of significance of 0.05 is 1.645, using the standard normal distribution table.

Since the test statistic (z = 3.14) is greater than the critical value (1.645), we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that more than half of the children aged 8-12 have cell phones.

Therefore, we can conclude that the marketing manager's claim is supported by the data from the survey.

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To find the number of zeros at the end of 2022!, we need to count the number of factors of 10 in its prime factorization. Since 10 = 2 x 5, we need to find the number of pairs of 2's and 5's that multiply to make 10.

First, we can count the number of factors of 5 in 2022!. There are 404 multiples of 5, 80 multiples of 25, and 16 multiples of 125. Therefore, there are 404 + 80 + 16 = 500 factors of 5.

Next, we need to count the number of factors of 2 in 2022!. This is equivalent to counting the number of multiples of 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024 that are less than or equal to 2022. We can simplify this by noticing that each power of 2 is a multiple of the previous power of 2. So we only need to count the multiples of 2, 4, and 8, and then multiply by the number of each power of 2 that divides into 2022.

There are 1011 multiples of 2, 505 multiples of 4, and 252 multiples of 8. The highest power of 2 that divides into 2022 is 2^1, so we only need to consider the factors of 2. Therefore, there are 1011 + 505 + 252 = 1768 factors of 2.

Since we need to find the number of pairs of 2's and 5's, the number of zeros at the end of 2022! is equal to the minimum of the number of factors of 2 and the number of factors of 5, which is 500. Therefore, there are 500 zeros at the end of 2022!.

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Step-by-step explanation:

Assuming the order matters....i.e.  they stand up one at a time

  (question does not state how the 4 are chosen)

9 choices for first

8 choices for second

7 choices for third

6 choices for fourth

9 x 8 x 7 x 6 = 3024 ways

  this is   9 P 4  =  9!/5! = 3024

The correct answer is option D, the two-tailed single-sample t-test.

To determine which test should be used in this scenario, we need to consider the following factors:

Type of data: The data collected from the CT scans are continuous data.

Sample size: The sample size is small (11 in total).

Relationship between samples: The data from Jimmy's hippocampus is independent from that of his family members.

Based on these factors, we can eliminate options C and B, which both involve dependent samples.

Next, we need to determine whether we are comparing Jimmy's hippocampus volume to a known value or to the average volume of his family members. If we were comparing Jimmy's hippocampus to a known value (e.g. the population average), we would use a one-sample t-test (option A). However, since we are comparing Jimmy's hippocampus volume to the average volume of his family members, we need to use a two-sample t-test.

Therefore, the correct answer is option D, the two-tailed single-sample t-test.

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The amount of the ordinary simple annuity of $1500 deposited each month for 4 years at 6.1% per year compounded monthly will be approximately $74,552.34.

To determine the amount of an ordinary simple annuity, we can use the formula for the future value of an ordinary annuity:

FV = P * [(1 + r)^n - 1] / r

Where:

FV is the future value of the annuity

P is the monthly payment amount

r is the monthly interest rate

n is the total number of compounding periods

In this case, the monthly payment amount (P) is $1500, the interest rate (r) is 6.1% per year compounded monthly, and the total number of compounding periods (n) is 4 years multiplied by 12 months in a year, which equals 48 months.

First, we need to calculate the monthly interest rate (r) by dividing the annual interest rate by 12 and converting it to a decimal:

r = 6.1% / 12 / 100 = 0.00508333

Now we can substitute the values into the formula to calculate the future value (FV):

FV = 1500 * [(1 + 0.00508333)^48 - 1] / 0.00508333

Calculating this expression gives us:

FV ≈ $74,552.34

Therefore, the amount of the ordinary simple annuity of $1500 deposited each month for 4 years at 6.1% per year compounded monthly will be approximately $74,552.34.

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Answer:42

Step-by-step explanation:

The number: x

--> x * 25% = 30

--> x * 60% = 70

So to find x --> 30 : 25% or 30 * 4 = 120

--> 120 * 35% = 42

The particular solution of the differential equation is given by:

y/x = sqrt(0.7652), or y = x sqrt(0.7652), which is equivalent to option (c).

We can rewrite the given differential equation as:

xyy' = x^2 + y^2

Dividing both sides by y^2, we get:

x/y + y'/y = x/y * (x/y + 1/y')

Letting v = y/x, we have:

y' = v + xv'

Substituting this in the above equation, we get:

x/y + v + xv'/y = x/y * (x/y + 1/(v + xv'/y))

Simplifying this equation, we get:

v' = -(v/x) * [(1+v^2)^(1/2) - v]

This is a separable differential equation. We can separate the variables and integrate:

(1+v^2)^(-1/2) dv = -dx/x

Integrating both sides, we get:

ln|x| - ln|x0| = -arcsin(v)

where x0 is the initial value of x, which is given to be 1. Simplifying this equation, we get:

ln|x| = -arcsin(v) = -arcsin(y/x)

Taking exponentials of both sides, we get:

|x| = e^(-arcsin(y/x))

Squaring both sides, we get:

x^2 = e^(-2arcsin(y/x))

Simplifying this equation, we get:

y/x = sqrt((e^(2arcsin(y/x))-1)/(e^(2arcsin(y/x))+1))

Using the initial condition y = 1 when x = 1, we get:

y/x = sqrt((e^(2arcsin1)-1)/(e^(2arcsin1)+1)) = sqrt((e^(pi/2)-1)/(e^(pi/2)+1))

Simplifying this expression, we get:

y/x = sqrt((e^(pi/2)-1)/(e^(pi/2)+1)) = sqrt((2.71828^(pi/2)-1)/(2.71828^(pi/2)+1)) = sqrt(0.7652)

Therefore, the particular solution of the differential equation is given by:

y/x = sqrt(0.7652), or y = x sqrt(0.7652), which is equivalent to option (c).

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Answer:

[tex] v = \frac{1}{3} h\pi \: r { }^{2} \\ = \frac{1}{3} \times 3 \times \pi \times2 ^{2} \\ \frac{1}{3 } \times 3 \times \pi \times 4 \\ \frac{1}{3} \times 12\pi \\ 4\pi \: cm {}^{3} is \: the \: answer[/tex]

the answer is 4 pie cm cube

may I get branliest

The variables and gives a measure of the strength of the linear relationship that is independent of the units in which the variables are measured.

The formula is the population correlation coefficient, denoted by the Greek letter rho (ρ).

The population correlation coefficient is a measure of the strength and direction of the linear relationship between two variables, denoted by X and Y. It ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship.

The formula for the population correlation coefficient is:

ρ = Cov(X,Y) / (σX * σY

where Cov(X,Y) is the covariance between X and Y, and σX and σY are the standard deviations of X and Y, respectively.

The numerator, Cov(X,Y), measures the degree to which X and Y vary together. It is positive when above-average values of X tend to be associated with above-average values of Y, and negative when above-average values of X tend to be associated with below-average values of Y.

The denominator, σX * σY, is a normalizing factor that puts the covariance on a standardized scale. It represents the spread of X and Y around their respective means.

By dividing the covariance by the product of the standard deviations, the population correlation coefficient removes the scale effects of the variables and gives a measure of the strength of the linear relationship that is independent of the units in which the variables are measured.

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Number of ways to choose three of these people to stand up is,

⇒ 84

We have to given that;

Nine people sit in chairs in a room.

And, We have to choose three of these people to stand up.

Hence, Number of ways to choose three of these people to stand up is,

⇒ ⁹C₃

⇒ 9! / 3! (9 - 3)!

⇒ 9! / 3! 6!

⇒ 9×8×7 / 3×2

⇒ 84

Thus,  Number of ways to choose three of these people to stand up is,

⇒ 84

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Answer: 504

Step-by-step explanation: i took the quiz

The better deal is given by the equation A = 18 ounces for 6.60

Given data ,

Let the equation be represented as A

Now , For 18 oz for $6.60

Price per ounce = Total cost / Total ounces = $6.60 / 18 oz ≈ $0.3667 per oz

And , for 12 oz for $4.75

Price per ounce = Total cost / Total ounces = $4.75 / 12 oz ≈ $0.3958 per oz

Comparing the two price per ounce values, we can see that the price per ounce for 18 oz for $6.60 is lower than the price per ounce for 12 oz for $4.75

Hence , the better deal is 18 oz for $6.60

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The two numbers are x = 22.5 and y = 9.5. To find the two numbers, x and y, we will solve the given equations (A and B) simultaneously.

Equation A: x + y = 32
Equation B: x - y = 13

Step 1: Add Equation A and Equation B together to eliminate the 'y' variable.
(x + y) + (x - y) = 32 + 13
2x = 45

Step 2: Divide both sides by 2 to isolate 'x'.
2x / 2 = 45 / 2
x = 22.5

Step 3: Substitute the value of 'x' in Equation A to find the value of 'y'.
22.5 + y = 32

Step 4: Subtract 22.5 from both sides to isolate 'y'.
y = 32 - 22.5
y = 9.5

The two numbers are x = 22.5 and y = 9.5.

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it is 95% confident that the true proportion of heads in a large number of tosses of this coin lies between 0.4969 and 0.5165.

a) The proportion of heads obtained in the experiment is

[tex]\frac{5067}{10000} = 0.5067[/tex]. The null hypothesis is that the coin is unbiased, which means the probability of getting a head is 0.5. The standard error (SE) of the proportion is given by

[tex]SE = \sqrt{\frac{p(1-p)}{n} } = \sqrt{\frac{(0.5)(0.5)}{10000} } = 0.005[/tex]

The test statistic is [tex]Z=\frac{(P-0.5)}{SE} = \frac{0.5067-0.5}{0.005} = 1.34[/tex]. The one-tailed p-value for this test is P(Z > 1.34) = P(Z < -1.34) = 0.0901 (assuming a significance level of 0.1). Since this p-value is greater than the significance level, we fail to reject the null hypothesis and conclude that the data do not provide evidence that the coin is biased.

b) The margin of error (m) for a 95% confidence interval is given by                                 m = 2SE = 0.01. The critical value for a 95% confidence interval is z= 1.96. Therefore, the 95% confidence interval for the true proportion of heads is given by P +/- mz' = 0.5067 +/- 0.01*1.96 = (0.4969, 0.5165).

In summary, we can say that based on the given information, the data do not provide evidence that the coin is biased and we are 95% confident that the true proportion of heads in a large number of tosses of this coin lies between 0.4969 and 0.5165.

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The argument is valid as it follows the definition of the Fourier transform for both ranges of the function f(t).

(a) To convert the argument into symbolic notation, let's denote the Fourier transform of f(t) as F(w):

f(t) = sin(3t), for k ≤ |t| ≤ 2k

0, for |t| > 2k

F(w) = (1/2) * [(sin(2kw - 3) - sin(kw - 3)) / (kw - 3) + (sin(kw + 3) - sin(2kw + 3)) / (kw + 3)]

(b) To show that the argument is valid, we need to demonstrate that the expression for F(w) derived above satisfies the definition of the Fourier transform:

F(w) = (1/√(2π)) * ∫[from -∞ to +∞] f(t) * e^(-iwt) dt

Let's examine the validity of the argument:

For k ≤ |t| ≤ 2k:

In this range, the function f(t) is sin(3t). We substitute f(t) = sin(3t) into the integral expression and evaluate it to obtain the expression for F(w).

For |t| > 2k:

In this range, the function f(t) is 0. Since the Fourier transform of a zero function is also zero, F(w) = 0 in this case.

Therefore, the argument is valid as it follows the definition of the Fourier transform for both ranges of the function f(t).

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