Write a Variable equation for each sentence
Danika's new running route is 4 miles longer than
her old route.

The total ticket sales revenue for the stadium was $1,001,258.

To find the total ticket sales revenue, we need to multiply the number of tickets sold at each price point by the corresponding ticket price, and then add up the results.

For the 4,000 tickets sold at $75 each, the total revenue would be:

4,000 x $75 = $300,000

For the 5,350 tickets sold at $62 each, the total revenue would be: 5,350 x $62 = $331,700

And for the 7,542 tickets sold at $49 each, the total revenue would be: 7,542 x $49 = $369,558

To find the total revenue, we simply add up these three amounts:

$300,000 + $331,700 + $369,558 = $1,001,258

Assume that all of the tickets were sold and that there were no discounts or promotions applied to any of the ticket prices.

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The two cases weigh 102,100 decigrams together.

First, we need to convert dekagrams to decigrams, since the question asks for the weight in decigrams.

1 dekagram = 10 grams

1 gram = 10 decigrams

Therefore, 1 dekagram = 100 decigrams

Now, let's calculate the weight of the two cases in decigrams:

Weight of tomato cans case = 563 dekagrams x 100 decigrams/dekagram = 56,300 decigrams

Weight of soup cans case = 458 dekagrams x 100 decigrams/dekagram = 45,800 decigrams

The weight of the two cases together is the sum of the two weights:

Total weight = 56,300 decigrams + 45,800 decigrams = 102,100 decigrams

Therefore, the two cases together weigh 102,100 decigrams.

To solve this problem, we need to first convert the weight of the two cases from dekagrams to decigrams so we can add them together.

1 dekagram = 10 grams

1 gram = 10 decigrams

So,

563 dekagrams = 5630 grams

5630 grams = 56300 decigrams

and

458 dekagrams = 4580 grams

4580 grams = 45800 decigrams

Now we can add the weights of the two cases in decigrams:

56300 decigrams + 45800 decigrams = 102,100 decigrams

Therefore, the two cases weigh 102,100 decigrams together.

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To find the volume of Tammy's storage shed, which is in the shape of a rectangular prism, we need to use the formula: Volume = Length × Width × Height.

Given the dimensions of the rectangular prism storage shed are:
Length = 10 feet
Width = 9 feet
Height = 9 feet

We can calculate the volume as follows:

Step 1: Multiply the length, width, and height.
Volume = 10 × 9 × 9

Step 2: Perform the multiplication.
Volume = 810 cubic feet

So, Tammy's storage shed has a volume of 810 cubic feet.

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Answer:

The cost before rebates is: $69.95 + $49.95 = $119.90

The total rebate amount is: $20 + $30 = $50

The cost of two envelopes is: 2 x $0.20 = $0.40

The cost of two stamps is: 2 x $0.394 = $0.788

The total cost after rebates and including expenses is: $119.90 - $50 + $0.40 + $0.788 = $70.078

Rounding to two decimal places, the total cost after rebates and including expenses is $70.08

The actual rebate after expenses is: $50 - $0.40 - $0.788 = $48.812

Rounding to two decimal places, the actual rebate after expenses is $48.81

Here, we can say with 95% confidence that the annual profit for the new restaurant in Kamloops with the given characteristics will fall between $1,065,200 and $1,264,800.

Based on the given characteristics of the new restaurant in Kamloops, we can estimate the annual profit by subtracting the total costs from the total revenue generated by the restaurant.
Total revenue = 15,000 covers x $100 per cover = $1,500,000
Total costs = $150k food cost + $85k overhead costs + $100k lab = $335,000
Annual profit = Total revenue - Total costs = $1,500,000 - $335,000 = $1,165,000
To compute the 95% prediction interval for annual profit, we need to consider the variability in the data and assume that the annual profit follows a normal distribution. Assuming a standard deviation of $50,000, the 95% prediction interval can be calculated as follows:  95% prediction interval = $1,165,000 +/- (1.96 x $50,000) = $1,065,200 to $1,264,800
Therefore, we can say with 95% confidence that the annual profit for the new restaurant in Kamloops with the given characteristics will fall between $1,065,200 and $1,264,800.

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Considering all three factors, consumers can make more informed decisions about the true value of a car beyond just the initial cost of purchase.

When buying a car, the value of the car is not just determined by the initial cost of purchase, but also by its reliability, low cost of ownership, and performance. To measure the value of a car, Consumer Reports developed a statistic called the "value score". The value score is based on three factors:

Five-year owner costs: These costs include expenses such as depreciation, fuel, maintenance, and repairs, and so on. To measure this, Consumer Reports uses an average cost per mile (Cost/mile) driven over a period of 5 years, assuming a national average of 12,000 miles per year.

Overall road-test scores: These scores are based on more than 50 tests and evaluations and are rated on a scale of 0-100, with higher scores indicating better performance, comfort, convenience, and fuel economy.

Predicted-reliability ratings: These ratings are based on data collected from Consumer Reports' Annual Auto Survey and are rated on a scale of 1-5, with higher scores indicating better predicted reliability.

Using these three factors, Consumer Reports calculates a value score for each car. A value score of 1.0 is considered average, while a value score of 2.0 is considered twice as good as average, and a value score of 0.5 is considered half as good as average.

By considering all three factors, consumers can make more informed decisions about the true value of a car beyond just the initial cost of purchase.

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In the above isosceles trapezoid,

Since R is the midpoint of HN, HR = RN. Similarly, since T is the midpoint of KJ, KT = TJ.

Let's use these properties to write expressions for HJ and NK in terms of x:

HJ = 5x + 9

NK = 3x - 2

Since NKJH is an isosceles trapezoid, we know that HJ = NK + 2RT. Substituting the expressions we found earlier, we get:

5x + 9 = (3x - 2) + 2(3x + 6)

Simplifying this equation gives:

5x + 9 = 9x + 10

Subtracting 5x from both sides gives:

4 = 4x

Dividing both sides by 4 gives:

x = 1

Now that we know x, we can find the values of HJ, NK, and RT:

HJ = 5x + 9 = 14 cm

NK = 3x - 2 = 1 cm

RT = 3x + 6 = 9 cm

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The answer is option d.

For the first question, we want to find a 90% confidence interval for the proportion of shoppers in the population who use a browser other than Firefox or Chrome. We can use the formula:

CI = p ± z*√(p(1-p)/n)

where p is the sample proportion, z is the z-score for the desired confidence level (90% corresponds to a z-score of 1.645), and n is the sample size.

From the data in the Heavenly Sales worksheet, we see that 12 out of 50 shoppers used a browser other than Firefox or Chrome. Therefore, the sample proportion is p = 12/50 = 0.24.

Substituting the values into the formula, we get:

CI = 0.24 ± 1.645*√(0.24(1-0.24)/50)

CI = 0.24 ± 0.126

CI = (0.114, 0.366)

Therefore, the 90% confidence interval for the proportion of shoppers in the population who use a browser other than Firefox or Chrome is between 0.114 and 0.366.

For the second question, we want to find a 90% confidence interval for the population mean amount of time a customer spends on the website. We can use the formula:

CI = x ± t*(s/√n)

where x is the sample mean, t is the t-score for the desired confidence level and degrees of freedom (we use a t-score instead of a z-score because we don't know the population standard deviation), s is the sample standard deviation, and n is the sample size.

From the data in the Heavenly Sales worksheet, we see that the sample mean amount of time spent on the website is x = 25.375 minutes, the sample standard deviation is s = 2.773 minutes, and the sample size is n = 50. To find the t-score, we need to determine the degrees of freedom, which is n-1 = 49. Using a t-table or calculator function, we find that the t-score for a 90% confidence level and 49 degrees of freedom is 1.676.

Substituting the values into the formula, we get:

CI = 25.375 ± 1.676*(2.773/√50)

CI = 25.375 ± 1.482

CI = (23.893, 26.857)

Therefore, the 90% confidence interval for the population mean amount of time a customer spends on the website is between 23.893 and 26.857 minutes. The answer is option d.

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This means that there exists a nonzero vector x such that Ax=0x, which implies that λ=0 is an eigenvalue of A with a corresponding eigenvector x.

a. If Ax-Ax for some vector x, then à is an eigenvalue of A. - True.

This statement is true because if Ax = λ.x for some vector x, then we can rewrite Ax-Ax = λ.x - λ.x as (A-I)x = 0. This means that the matrix A-I is singular, and therefore its determinant is 0. So, we have det(A-I) = 0, which implies that λ = 1 is an eigenvalue of A.

b. A matrix A is not invertible if and only if 0 is an eigenvalue of A. - False.

This statement is false because a matrix A is not invertible if and only if its determinant is 0, which means that the equation Ax = 0 has a nontrivial solution. This implies that 0 is an eigenvalue of A, but the converse is not necessarily true.

c. If 0 is an eigenvalue of A, then the equation Ax-ox has only the trivial solution. The equation Ax-0x is equivalent to the equation Ax-o and Ax-o - True.

This statement is true because if λ=0 is an eigenvalue of A, then we have (A-0I)x = Ax = 0x, which means that the matrix A-0I is singular, and therefore its determinant is 0. So, we have det(A-0I) = 0, which implies that the equation Ax = 0 has a nontrivial solution. However, if A is invertible, then the only solution to the equation Ax=0 is the trivial solution, which means that Ax-0x = Ax = 0x has only the trivial solution.

d. If 0 is an eigenvalue of A, then there are nontrivial solutions to the equation Ax=0x. The equation Ax-0x is equivalent to the equation Ax=0 - True.

This statement is true because if λ=0 is an eigenvalue of A, then we have (A-0I)x = Ax = 0x, which means that the matrix A-0I is singular, and therefore its determinant is 0. So, we have det(A-0I) = 0, which implies that the equation Ax = 0 has a nontrivial solution. This means that there exists a nonzero vector x such that Ax=0x, which implies that λ=0 is an eigenvalue of A with a corresponding eigenvector x.

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The mean of this distribution is (a+b)/2 = $500, and the standard deviation is (b-a)/sqrt(12) = $288.68. The probability of winning more than $600 on one spin of the wheel is the area under the uniform distribution curve from $600 to $1000, which is (1000-600)/(1000-0) = 0.4 or 40%.

In a uniform distribution, all possible outcomes have an equal probability of occurring, and the range of values is defined by the minimum value (a) and the maximum value (b). In this case, the range is from $0 to $1000, so a=0 and b=1000.

The mean of a uniform distribution is the average of the minimum and maximum values, which is (a+ b)/2 = $500. The standard deviation of a uniform distribution is calculated using the formula (b-a)/sqrt(12), which gives a value of $288.68 for this distribution.

To find the probability of winning more than $600, we need to calculate the area under the uniform distribution curve from $600 to $1000. Since the total area under the curve is 1, we can calculate the probability by dividing the width of the interval by the total width of the distribution, which is (1000-600)/(1000-0) = 0.4 or 40%.

Therefore, the probability of winning more than $600 on one spin of the wheel is 0.4.

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a.  The probability that exactly 3 U.S. adults out of 12 favor the use of unmanned drones by police agencies is 0.218.

b. The probability that at least 4 U.S. adults out of 12 favor the use of unmanned drones by police agencies is 0.684.

c. The probability that less than 8 U.S. adults out of 12 favor the use of unmanned drones by police agencies is 0.968.

This problem requires the use of the binomial distribution, where we have:

n = 12 (number of trials, or U.S. adults randomly selected)

p = 0.21 (probability of success, or favoring the use of unmanned drones by police agencies)

(a) To find P(x=3), the probability that exactly 3 U.S. adults out of 12 favor the use of unmanned drones by police agencies, we can use the binomial probability formula:

P(x=3) = (12 choose 3) * 0.21^3 * (1-0.21)^(12-3)

P(x=3) = 0.218

(b) To find P(x≥4), the probability that at least 4 U.S. adults out of 12 favor the use of unmanned drones by police agencies, we can use the complement rule and the binomial cumulative distribution function:

P(x≥4) = 1 - P(x<4)

P(x≥4) = 1 - P(x=0) - P(x=1) - P(x=2) - P(x=3)

P(x≥4) = 1 - (12 choose 0) * 0.21^0 * (1-0.21)^(12-0) - (12 choose 1) * 0.21^1 * (1-0.21)^(12-1) - (12 choose 2) * 0.21^2 * (1-0.21)^(12-2) - P(x=3)

P(x≥4) = 0.684

(c) To find P(x<8), the probability that less than 8 U.S. adults out of 12 favor the use of unmanned drones by police agencies, we can use the binomial cumulative distribution function:

P(x<8) = P(x≤7)

P(x<8) = P(x=0) + P(x=1) + P(x=2) + ... + P(x=7)

P(x<8) = (12 choose 0) * 0.21^0 * (1-0.21)^(12-0) + (12 choose 1) * 0.21^1 * (1-0.21)^(12-1) + (12 choose 2) * 0.21^2 * (1-0.21)^(12-2) + ... + (12 choose 7) * 0.21^7 * (1-0.21)^(12-7)

P(x<8) = 0.968

So the probability that less than 8 U.S. adults out of 12 favor the use of unmanned drones by police agencies is 0.968.

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Answer:

14. C6H12O6 + 6O2 -> 6CO2 + 6H2O

Yes, a special form of a combustion reaction called respiration.

15. Cu(s) + AgNO3(aq) => Ag(s) + CuNO3(aq)

Step-by-step explanation:

Are you asking for equations?

Answer:

BOTH OF THEM ARE TRUE

14:(C6H12O6) burns in oxygen to produce carbon dioxide and water vapor as described in the following equation: C6H12O6 + 6O2 → 6H2O + 6CO2.

15:silver crystal form on the surface of the copper. Additionally, highly soluble copper (I) nitrate is generated.

The max-flow min-cut theorem helps to demonstrate that the maximum number of independent 1's in the matrix H equals the minimum number of lines in a cover.

To show that the maximum number of independent 1's in a 0-1 matrix H with $n_{1}$ rows and $n_{2}$ columns equals the minimum number of lines in a cover, we can use the max-flow min-cut theorem on an appropriately defined network.

First, create a bipartite graph G, where one set of vertices represents the rows and the other set represents the columns of the matrix H. Connect an edge between a row vertex and a column vertex if there is a 1 in the corresponding entry of the matrix H.

Next, add a source vertex s connected to all row vertices and a sink vertex t connected to all column vertices. Assign capacities of 1 to all edges in the network.

Now, apply the max-flow min-cut theorem on this network. The maximum flow in the network represents the maximum number of independent 1's in the matrix H. The minimum cut corresponds to the minimum number of lines in a cover of H, as it separates the source and sink while minimizing the number of crossing edges.

Hence, the max-flow min-cut theorem helps to demonstrate that the maximum number of independent 1's in the matrix H equals the minimum number of lines in a cover.

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-75 lies between -74 and -76.

In mathematics, integers are a set of whole numbers that include positive numbers, negative numbers, and zero. Integers do not include any fractions or decimal points. The set of integers is denoted by the symbol "Z". Integers can be represented on a number line where positive integers are located to the right of zero and negative integers are located to the left of zero.

To determine between which two integers -75 lies, consider the following:

Step 1: Identify the closest integer greater than -75. In this case, that would be -74.

Step 2: Identify the closest integer less than -75. In this case, that would be -76.

So, without using a calculator, -75 lies between the two integers -76 and -74.

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The rule for the translation by the vector (0,5), plus a reflection around the line y = 5 is given as follows:

(x, y) -> (x, |y - 5| + 5).

The general coordinates of the point in a coordinate plane is given as follows:

(x,y).

The rule for the translation by the vector (0,5) is obtained as follows:

(x, y) -> (x + 0, y + 5) -> (x, y + 5).

For the reflection about the line y = 5, the x-coordinate remains constant, while the y-coordinate is moved on the opposite direction to y = 5, as follows:

(x, y) -> (x, |y - 5| + 5).

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The z-score of the norminal distribution is 10.14.

Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values.

To calculate the z-score of the norminal distribution, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

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Answer:

[tex]x = \sqrt{ {19}^{2} - {15.2}^{2} } = 11.4[/tex]

Surface area =

2(1/2)(11.4)(15.2) + 16(11.4) + 16(19) + 16(15.2)

= 173.28 + 182.4 + 304 + 243.2

= 902.88 square yards

The standard deviation of the population is 8.

We have,

The mean of the sampling distribution is the mean of the population, so we have:

[tex]$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i = \mu$[/tex]

where [tex]$\bar{x}$[/tex] is the sample mean, n is the sample size, [tex]x_1[/tex] are the individual samples, and [tex]$\mu$[/tex] is the population mean.

In this case,

We are given the sample size n = 100 and the sample mean [tex]\bar{x}[/tex]

We also know that the sampling distribution comes from a normally distributed population.

The standard error of the mean.

[tex]$SE = \frac{s}{\sqrt{n}}$[/tex]

where s is the sample standard deviation.

The standard error of the mean represents the standard deviation of the sampling distribution.

In this case,

We don't have the sample standard deviation s, but we can estimate it using the sample variance:

[tex]$s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2$[/tex]

Substituting the values we have:

[tex]$s^2 = \frac{1}{99}\sum_{i=1}^{100}(x_i - 145)^2 = 64$[/tex]

Therefore:

[tex]$s = \sqrt{64} = 8$[/tex]

Substituting this value into the standard error equation:

[tex]$SE = \frac{s}{\sqrt{n}} = \frac{8}{\sqrt{100}} = 0.8$[/tex]

The standard deviation of the population is related to the standard error by the following equation:

[tex]$SE = \frac{\sigma}{\sqrt{n}}$[/tex]

Rearranging this equation, we get:

[tex]$\sigma = SE \times \sqrt{n} = 0.8 \times \sqrt{100} = 8$[/tex]

Therefore,

The standard deviation of the population is 8.

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Using opposite sides or angles which must be congruent we can prove that a quadrilateral is a parallelogram.

There are several theorems that can be used to show that a quadrilateral is a parallelogram, depending on the given information. Here are a few:

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The Pythagorean Theorem is a fundamental concept that relates to the sides of a right-angled triangle, and it can also be used to understand the relationship between the areas of the squares constructed on the sides of the triangle.

The area of a square is given by the formula A = s², where s is the length of one of its sides. Therefore, the areas of the three squares are:

Area of the square with side a = a²

Area of the square with side b = b²

Area of the square with side c = c²

Now, let's compare the areas of the squares. We can start by subtracting the area of the square with side a from the area of the square with side c:

c² - a²

Using the Pythagorean Theorem, we know that c² = a² + b². Substituting this into the above expression, we get:

c² - a² = (a² + b²) - a² = b²

This tells us that the difference between the area of the square with side c and the area of the square with side a is equal to the area of the square with side b. In other words:

c² - a² = b²

This is known as the Pythagorean identity. It states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. We can also rearrange this identity to obtain the following:

c² = a² + b²

This is the Pythagorean Theorem that we are familiar with. Therefore, we can conclude that the relationship between the areas of the squares constructed on the sides of a right-angled triangle is given by the Pythagorean identity: the difference between the area of the square on the hypotenuse and the area of the square on the shorter side is equal to the area of the square on the other shorter side.

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The equation 2x + 3 = 5 + 2x has infinitely many solutions. So, correct option is A.

To see why, we can simplify the equation by subtracting 2x from both sides, which gives us:

3 = 5

This is a contradiction since 3 is not equal to 5. However, notice that when we subtracted 2x from both sides, we eliminated the variable x from the equation. This means that the original equation 2x + 3 = 5 + 2x is actually equivalent to the identity 3 = 5, which is always false.

Since this equation is always false, it has no solutions that make it true. However, we can also say that it has infinitely many solutions since any value of x will make the equation false. Therefore, option A is the correct answer.

Options B, C, and D all have unique solutions, since we can simplify them to the form x = some number. For example, option B simplifies to x = -2, option C simplifies to 0 = 0 (which is always true), and option D simplifies to x = 5.

So, correct option is A.

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A graph that represents the inequality y < x² + 4x include the following: A. graph A.

In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the first graph of a quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).

Since the leading coefficient (value of a) in the given quadratic function y < x² + 4x is positive 1, we can logically deduce that the parabola would open upward and the solution would be below the line because of the less than inequality symbol. Also, the value of the quadratic function f(x) would be minimum at -4.

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The simplified form of the expression is [tex]7x^5 - 3x^4 - x^3 - x^2 + 4x.[/tex]

The given expression is a polynomial expression, which can be simplified by combining the like terms. The like terms have the same variable and the same exponent. The given expression can be rearranged and combined as follows:

To simplify the given expression, we need to combine the like terms.

Starting with the x^5 term, we see that there is only one term with [tex]x^5[/tex]which is [tex]7x^5.[/tex]

Moving on to the[tex]x^4[/tex]terms, we have two terms with[tex]x^4,[/tex] namely [tex]3x^4[/tex]and [tex]-3x^4[/tex], which add up to 0. Therefore, we can eliminate the[tex]x^4 t[/tex]erms from the expression.

[tex]7x^5 + 2x^3 - 5x^2 + 4x^2 + 6x - 2x - 3x^4 - 3x^3[/tex]

[tex]= 7x^5 - 3x^4 + 2x^3 - 3x^3 - 5x^2 + 4x^2 + 6x - 2x[/tex] (rearranging the terms)

[tex]= 7x^5 - 3x^4 - x^3 - x^2 + 4x[/tex] (combining the like terms)

Therefore, the simplified form of the expression is [tex]7x^5 - 3x^4 - x^3 - x^2 + 4x.[/tex]

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The number of wheels in the 8 tricycles are 24.

The given equation is w=3×t.

Where, w is total number of wheels and t is number of tricycles.

Here, t=8

Substitute t=8 in w=3×t, we get

w=3×8

w=24 wheels

Therefore, the number of wheels in the 8 tricycles are 24.

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The probability of winning first prize is 1 in 324,632 and winning the second prize is 1 in 52,360.

(a) The probability of winning the first prize in Cash 5 is the probability of correctly choosing all 5 numbers out of the 35 possible numbers. This can be calculated using the formula for combinations, which is

[tex]^{n} C_{r}[/tex]  = n! / r!(n-r)! where n is the total number of items and r is the number of items to be chosen.

For Cash 5, the formula is:

[tex]^{35} C_{5}[/tex] = 35!÷5!(35-5)! = 324,632
So, the probability of winning first prize is 1 in 324,632.

(b) The probability of winning the second prize in Cash 5 is the probability of correctly choosing 4 out of the 5 numbers and not choosing the fifth number (which would be the Cash Ball).

This can be calculated using the formula for combinations again, but this time with n = 35 (the total number of numbers) and r = 4 (the number of numbers to be chosen). Then, the probability of not choosing the Cash Ball is 30/34 (since there are 30 numbers left after the Cash Ball is chosen out of the remaining 34 numbers).

So the formula is:

[tex]^{35} C_{5} (\frac{30}{34}) = 52,360[/tex]

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Answer:

To find the equation of the tangent line to the curve y = x^2 Arccos(3x) at x = 0.2, we need to find the slope of the tangent line at that point and then use the point-slope form of the equation of a line.

First, we find the derivative of y with respect to x:

dy/dx = 2x Arccos(3x) - x^2 / sqrt(1 - (3x)^2)

Next, we evaluate the derivative at x = 0.2 to find the slope of the tangent line at that point:

dy/dx | x=0.2 = 2(0.2) Arccos(3(0.2)) - (0.2)^2 / sqrt(1 - (3(0.2))^2)

dy/dx | x=0.2 = 0.6865

So the slope of the tangent line at x = 0.2 is approximately 0.6865.

Now we can use the point-slope form of the equation of a line to find the equation of the tangent line. We know that the line passes through the point (0.2, 0.04 Arccos(0.6)), so we have:

y - y1 = m(x - x1)

y - 0.04 Arccos(0.6) = 0.6865(x - 0.2)

Simplifying and rearranging, we get:

y = 0.6865x - 0.1261

Therefore, the equation of the tangent line to the curve y = x^2 Arccos(3x) at x = 0.2 is y = 0.6865x - 0.1261.

The equation of the tangent line to the curve y = x^2 Arccos(3x) at x = 0.2 is approximately y = 0.653x - 0.113.

To find the equation of the tangent line to the curve at x = 0.2, we need to find the slope of the tangent line at that point, and then use the point-slope form of a line to write the equation.

First, we need to find the derivative of the curve:

y = x^2 Arccos(3x)

Taking the derivative with respect to x:

y' = 2x Arccos(3x) - x^2 (1/sqrt(1 - 9x^2))

Now, we can evaluate y' at x = 0.2 to find the slope of the tangent line at that point:

y'(0.2) = 2(0.2) Arccos(3(0.2)) - (0.2)^2 (1/sqrt(1 - 9(0.2)^2))

≈ 0.653

So the slope of the tangent line at x = 0.2 is approximately 0.653.

Next, we need to find the y-coordinate of the point on the curve where x = 0.2:

y = (0.2)^2 Arccos(3(0.2))

≈ 0.012

So the point on the curve where x = 0.2 is (0.2, 0.012).

Now we can use the point-slope form of a line to write the equation of the tangent line:

y - y1 = m(x - x1)

where m is the slope we just found, and (x1, y1) is the point on the curve where x = 0.2.

Substituting in the values we found:

y - 0.012 = 0.653(x - 0.2)

Simplifying:

y = 0.653x - 0.113

So the equation of the tangent line to the curve y = x^2 Arccos(3x) at x = 0.2 is approximately y = 0.653x - 0.113.

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Based on the given premises, we know that either the butler, maid, or cook committed the crime. If it was the cook, poison was used, and if poison was used, it wasn't done swiftly, but we know that the crime was swift. Therefore, we can conclude that the cook is not the culprit. This leaves us with the butler and the maid. However, there is no information given about the maid that would implicate her in the crime.

On the other hand, the butler is the only suspect left who has not been ruled out by the given premises. Therefore, we can conclude that the butler did it. However, to prove his guilt, we need more information or evidence. The given premises only allow us to eliminate the other suspects and narrow down the list of possible culprits to one.

In order to prove the butler's guilt, we would need additional information or evidence that directly implicates him in the crime. This could come in the form of eyewitness testimony, forensic evidence, or a confession. Without further information, we cannot definitively prove that the butler is guilty.
To prove that the butler is guilty, we can use the given statements and logical reasoning. Here is a step-by-step explanation for the sequent:

1. Either it was the butler or the maid, or it was the cook. [(BvM)vC]
2. If the cook did it, poison was used. [(CvM)-->P]
3. If it was done with poison then it wasn't done swiftly, but it was swift. [(P--> ~S)&S]

From statement 3, we know that it was done swiftly (S), so it can't be poison (~P):
4. ~P (from 3)

Now, since it wasn't poison, it means the cook couldn't have done it (~C):
5. ~C (from 2 and 4, using Modus Tollens)

We're left with either the butler or the maid:
6. BvM (from 1 and 5, using Disjunction Elimination)

Since we know it was done swiftly, and the only available options are the butler and the maid, we can conclude:
7. B (Butler)

So, we must conclude that the butler did it. The butler is guilty based on the given sequent and logical reasoning.

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This is because as the sample size increases, the confidence interval becomes more precise and thus narrower.

When the level of confidence and sample standard deviation remains the same, a confidence interval for a population mean based on a sample of n = 100 will be narrower than a confidence interval for a population mean based on a sample of n = 50. This is because larger sample sizes typically result in more precise estimates of the population mean, leading to a smaller margin of error and therefore a narrower confidence interval.
This is because as the sample size increases, the confidence interval becomes more precise and thus narrower.

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To achieve your goal of $1,000,000 in today's dollars in 40 years with a 9% annual return on your investments, you need to deposit approximately $37.10 into your saving account each week. To determine how much you need to deposit into your saving account each week to achieve your goal of $1,000,000 in today's dollars in 40 years, follow these steps:

Step:1. Adjust for inflation: First, estimate the future value of $1,000,000 in 40 years by considering an average annual inflation rate of 3% (a common assumption). Use the formula:
Future Value = Present Value * (1 + Inflation Rate) ^ Number of Years
Future Value = $1,000,000 * (1 + 0.03) ^ 40
Future Value ≈ $3,262,037
Step:2. Calculate the weekly deposit amount: Since you expect to earn a 9% annual return on your investments, convert this to a weekly rate by dividing by 52 (assuming compounded weekly):
Weekly Interest Rate = (1 + 0.09) ^ (1/52) - 1
Weekly Interest Rate ≈ 0.00165
Step:3. Determine the number of weekly deposits over the 40-year period:
Number of Weeks = 40 Years * 52 Weeks/Year
Number of Weeks = 2,080
Step:4. Use the future value of an annuity formula to calculate the weekly deposit amount:
Weekly Deposit = Future Value / (((1 + Weekly Interest Rate) ^ Number of Weeks) - 1) / Weekly Interest Rate
Weekly Deposit = $3,262,037 / (((1 + 0.00165) ^ 2,080) - 1) / 0.00165
Weekly Deposit ≈ $37.10
In conclusion, to achieve your goal of $1,000,000 in today's dollars in 40 years with a 9% annual return on your investments, you need to deposit approximately $37.10 into your saving account each week.

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