What is an explicit formula for the sequence 5,8,11,14, ........ ?

By using the given geometric figure and splitting it into three rectangular blocks, we can prove the factoring formula for the sum of cubes, a³+b³=(a+b)(a² - ab+b²).

To prove the factoring formula for the sum of cubes, a³+b³=(a+b)(a² - ab+b²), we can use the geometric figure provided.

First, let's split the figure into three rectangular blocks. One block has dimensions a, b, and a+b, while the other two blocks have dimensions a, b, and a.

Now, let's calculate the volume of the entire figure. We know that the volume is equal to the sum of the volumes of each rectangular block. The volume of the first block is (a)(b)(a+b) = a²b + ab². The volume of the second and third blocks is (a)(b)(a) = a²b.

Adding these volumes together, we have a²b + ab² + a²b = 2a²b + ab².

Next, let's factor out the common terms from this expression. We can factor out ab to get ab(2a + b).

Now, let's compare this expression with the formula we want to prove, a³+b³=(a+b)(a² - ab+b²). Notice that a³+b³ can be written as ab(a²+b²), which is equivalent to ab(a² - ab+b²) + ab(ab).

Comparing the terms, we see that ab(a² - ab+b²) matches the expression we obtained from the volume calculation, while ab(ab) matches the remaining term.

Therefore, we can conclude that a³+b³=(a+b)(a² - ab+b²) based on the volume calculation and the fact that the two expressions match.

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The probability that the duration is at least 2 hours is 0.435 and for 3 hours is 0.611, probability that the duration is between 2 and 3 hours is 0.176.

The probability that the duration of a particular rainfall event at Toronto Pearson International Airport is at least 2 hours can be calculated using the exponential distribution with a mean of 2.725 hours. To find this probability, we need to calculate the cumulative distribution function (CDF) of the exponential distribution.

The CDF of an exponential distribution is given by: CDF(x) = 1 - exp(-λx), where λ is the rate parameter. In this case, since the mean is 2.725 hours, we can calculate the rate parameter λ as [tex]1/2.725.[/tex]
a) To find the probability that the duration is at least 2 hours, we need to calculate CDF(2) = 1 - exp[tex](-1/2.725 * 2).[/tex]
b) To find the probability that the duration is at most 3 hours, we can calculate CDF(3) = 1 - exp[tex](-1/2.725 * 3).[/tex]
c) To find the probability that the duration is between 2 and 3 hours, we can subtract the probability calculated in part (a) from the probability calculated in part (b).

For example, if we calculate the CDF(2) to be 0.435 and the CDF(3) to be 0.611, then the probability of the duration being between 2 and 3 hours is [tex]0.611 - 0.435 = 0.176.[/tex].

In summary: a) The probability that the duration is at least 2 hours is 0.435.
b) The probability that the duration is at most 3 hours is 0.611.
c) The probability that the duration is between 2 and 3 hours is 0.176.

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Given question is incomplete. Hence, the complete question is :

Data collected at Toronto Pearson International Airport suggests that an exponential distribution with mean value 2.725 hours is a good model for rainfall duration (Urban Stormwater Management Planning with Analytical Probabilistic Models, 2000, p. 69).

a. What is the probability that the duration of a particular rainfall event at this location is at least 2 hours? At most 3 hours? Between 2 and 3 hours?

f(x) = (x - 3)/(x + 2)

As the equation is basically a fraction the only thing that can be out of domain is if the denominator is equal to 0, so let's see when the denominator can be 0

x + 2 = 0

x = -2

So -2 is out of domain and all the other numbers are inside the domain.

Answer:

[tex]-2 \implies \sf no[/tex]

 [tex]0 \implies \sf yes[/tex]

 [tex]3 \implies \sf yes[/tex]

Step-by-step explanation:

Given rational function:

[tex]f(x)=\dfrac{x-3}{x+2}[/tex]

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

A rational function is not defined when its denominator is zero.

Therefore, to find when the given function f(x) is not defined, set the denominator to zero and solve for x:

[tex]x+2=0 \implies x=-2[/tex]

Therefore, the domain is restricted to all values of x except x = -2.

This means that the domain of f(x) is (-∞, 2) ∪ (2, ∞).

In conclusion:

To determine a cubic polynomial with integer coefficients that has [tex]$\sqrt[3]{2} \sqrt[3]{4}$[/tex]as a root, we can use the fact that if $r$ is a root of a polynomial, then $(x-r)$ is a factor of that polynomial.

In this case, let's assume that $a$ is the unknown cubic polynomial. Since[tex]$\sqrt[3]{2} \sqrt[3]{4}$[/tex] is a root, we have the factor[tex]$(x - \sqrt[3]{2} \sqrt[3]{4})$[/tex].
Now, we need to rationalize the denominator. Simplifying [tex]$\sqrt[3]{2} \sqrt[3]{4}$, we get $\sqrt[3]{2^2 \cdot 2} = \sqrt[3]{8} = 2^{\frac{2}{3}}$.[/tex]
Substituting this back into our factor, we have $(x - 2^{\frac{2}{3}})$. To find the other two roots, we need to factor the cubic polynomial further. Dividing the cubic polynomial by the factor we found, we get a quadratic polynomial. Using long division or synthetic division, we find that the quadratic polynomial is [tex]$x^2 + 2^{\frac{2}{3}}x + 2^{\frac{4}{3}}$.[/tex]Now, we can find the remaining two roots by solving this quadratic equation using the quadratic formula or factoring. The resulting roots are Simplifying these roots further will give us the complete cubic polynomial with integer coefficients that has[tex]$\sqrt[3]{2} \sqrt[3]{4}$[/tex] as a root.

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A cubic polynomial with integer coefficients that has [tex]\sqrt[3]{2} \sqrt[3]{4}[/tex] as a root is [tex]x^{3} - 6x^{2} + 12x - 8$[/tex].

To determine a cubic polynomial with integer coefficients that has  [tex]\sqrt[3]{2} \sqrt[3]{4}[/tex] as a root, we can start by recognizing that the expression  [tex]\sqrt[3]{2} \sqrt[3]{4}[/tex] can be simplified.

First, let's simplify [tex]\sqrt[3]{4}[/tex]. We know that [tex]\sqrt[3]{4}[/tex] is the cube root of 4. Therefore, [tex]\sqrt[3]{4} = 4^{\frac{1}{3}}[/tex].

Next, let's simplify [tex]\sqrt[3]{2}[/tex]. This can be written as [tex]2^{\frac{1}{3}}[/tex] since [tex]\sqrt[3]{2}[/tex] is also the cube root of 2.

Now, let's multiply [tex]\sqrt[3]{2} \sqrt[3]{4}[/tex]:
[tex](2^{\frac{1}{3}}) (4^{\frac{1}{3}})[/tex].

Using the property of exponents [tex](a^m)^n = a^{mn}[/tex], we can rewrite the expression as [tex](2 \cdot 4)^{\frac{1}{3}}[/tex]. This simplifies to [tex]8^{\frac{1}{3}}[/tex].

Now, we know that [tex]8^{\frac{1}{3}}[/tex] is the cube root of 8, which is 2.

Therefore, [tex]\sqrt[3]{2} \sqrt[3]{4} = 2[/tex].

Since we need a cubic polynomial with [tex]\sqrt[3]{2} \sqrt[3]{4}[/tex] as a root, we can use the root and the fact that it equals 2 to construct the polynomial.

One possible cubic polynomial with [tex]\sqrt[3]{2} \sqrt[3]{4}[/tex] as a root is [tex](x-2)^{3}[/tex]. Expanding this polynomial, we get [tex]x^{3} - 6x^{2} + 12x - 8[/tex].

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To find the left-rectangle approximation of the shaded region.

To find the left-rectangle approximation of the shaded region using 5 rectangles, we can follow these steps:

1. Determine the width of each rectangle. Since we are using 5 rectangles, we divide the total width of the shaded region by 5.
2. Calculate the left endpoint of each rectangle. We start from the leftmost point of the shaded region and add the width of each rectangle to find the left endpoint of the next rectangle.
3. Calculate the area of each rectangle. Multiply the width of each rectangle by the height of the shaded region.
4. Sum up the areas of all the rectangles to find the total approximate area of the shaded region using the left-rectangle approximation.

Please note that without the specific values of the width and height of the shaded region, I cannot provide the numerical answer. However, by following the steps above, you will be able to find the left-rectangle approximation of the shaded region.

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When designing a simulation using a random number generator to predict scores, the simulated data is likely to be different from the actual statistics from last season.

This is because the simulation relies on random numbers, which introduce an element of randomness into the predictions.

Additionally, the simulation might not capture all the variables and factors that affect scores during a game. Therefore, the simulated data will likely have variations and may not perfectly match the actual statistics from last season.

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There is no speed for the skater that would allow the cyclist to travel 75 kilometers while the skater travels 45 kilometers in the same amount of time.

To find the speed of the skater, let's denote the speed of the skater as "x" kilometers per hour. Since the cyclist traveled 12 kilometers per hour faster than the skater, the speed of the cyclist would be "x + 12" kilometers per hour.

We can use the formula: speed = distance/time to solve this problem.

For the cyclist:
Speed of cyclist = 75 kilometers / t hours

For the skater:
Speed of skater = 45 kilometers / t hours

Since both the cyclist and the skater traveled for the same amount of time, we can set up an equation:

75 / t = 45 / t

Cross multiplying, we get:
75t = 45t

Simplifying, we have:
30t = 0

Since the time cannot be zero, we have no solution for this equation. This means that the given information in the question is not possible and there is no speed for the skater that satisfies the conditions.

There is no speed for the skater that would allow the cyclist to travel 75 kilometers while the skater travels 45 kilometers in the same amount of time.

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The least amount we can charge for each CD to make a $100 profit depends on the number of CDs sold. The revenue per CD will decrease as the number of CDs sold increases.

According to Problem 2, we want to find the minimum amount we can charge for each CD to make a $100 profit. To determine this, we need to consider the cost and revenue associated with selling CDs.

Let's say the cost of producing each CD is $5. We can start by calculating the total revenue needed to make a $100 profit. Since the profit is the difference between revenue and cost, the revenue needed is $100 + $5 (cost) = $105.

To find the minimum amount we can charge for each CD, we need to divide the total revenue by the number of CDs sold. Let's assume we sell x CDs. Therefore, the equation becomes:

Revenue per CD * Number of CDs = Total Revenue
x * (Revenue per CD) = $105

To make it simpler, let's solve for the revenue per CD:
Revenue per CD = Total Revenue / Number of CDs
Revenue per CD = $105 / x

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Answer:

20 Dimes and 30 nickels

Step-by-step explanation:

Let n =  the number of nickels

Let d = the number of dimes.

.05n + .1d = 3.50  Multiply through by 100 to remove the decimal

5n + 10d = 350

n = d + 10

Substitute d + 10 for n in the first equation.

5n + 10d = 350

5(d  10) + 10d = 350  Distribute the 5

5d + 50 + 10d = 350  Combine the d's

15d + 50 = 350  Subtract 50 from both sides

15d = 300 Divide both sides by 15

d = 20

The number of dimes is 20.

Substitute 20 for d

n = d + 10

n = 20 + 10

n = 30

The number of nickels is 30.

Helping in the name of Jesus.

To find the sum of all possible values of the greatest common divisor (GCD) of 3n+4 and n, we need to consider the possible values of n.

Let's start by writing down the given expression: 3n+4.

The GCD of 3n+4 and n will be the largest positive integer that divides both 3n+4 and n.

To find the GCD, we can use the Euclidean algorithm.

Step 1: Divide 3n+4 by n:
3n+4 = 3n + (n + 4)

Step 2: Divide n by (n+4):
n = 1*(n+4) - 4

Step 3: Repeat the process until we reach a remainder of 0.
(n+4) = 1*(4) + 0

Since we have reached a remainder of 0, the GCD of 3n+4 and n is the divisor in the last step, which is 4.

Now, we need to consider the range of positive integers for n. Let's assume n takes on the values 1, 2, 3, ..., 250.

For each value of n, the GCD will be 4. So, the sum of all possible values of the GCD is:

4 + 4 + 4 + ... + 4 (250 times)

We can simplify this as 4 * 250, which equals 1000.

Therefore, the sum of all possible values of the GCD of 3n+4 and n, as n ranges over the positive integers, is 1000.

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To prove that the set L = {x ∈ X | f(x) < c} is open in the topological space X, we can show that for any point x in L, there exists an open neighbourhood N of x such that N is entirely contained in L.

Let x be an arbitrary point in L. This means that f(x) < c. Since f is continuous, for any ε > 0, there exists a δ > 0 such that if y is any point in X and d(x, y) < δ, then |f(x) - f(y)| < ε.

Let's choose ε = c - f(x). Since f(x) < c, we have ε > 0. By the continuity of f, there exists δ > 0 such that if d(x, y) < δ, then |f(x) - f(y)| < ε.

Now, consider the open ball B(x, δ) centred at x with radius δ. Let y be any point in B(x, δ). Then, d(x, y) < δ, which implies |f(x) - f(y)| < ε = c - f(x). Adding f(x) to both sides of the inequality gives f(y) < f(x) + c - f(x), which simplifies to f(y) < c. Thus, y is also in L.

Therefore, we have shown that for any point x in L, there exists an open neighbourhood N (in this case, the open ball B(x, δ)) such that N is entirely contained in L. Hence, the set L is open in the topological space X.

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Yes, it is true that when the population distribution is normal, the sampling distribution of the mean of x is also normal for any sample size n.

This is known as the Central Limit Theorem, which states that when independent random variables are added, their normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.The Central Limit Theorem is important in statistics because it allows us to make inferences about the population mean using sample statistics. Specifically, we can use the standard error of the mean to construct confidence intervals and conduct hypothesis tests about the population mean, even when the population standard deviation is unknown.

Overall, the Central Limit Theorem is a fundamental concept in statistics that plays an important role in many applications.

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By visually analyzing the scatter plot, you can gain insights into the relationship between population and the number of licensed drivers. Keep in mind that scatter plots are just one way to visualize data, and additional analysis may be needed to draw definitive conclusions.

To make a scatter plot, you would plot the population on the x-axis and the number of licensed drivers on the y-axis. Each point on the graph represents a specific data point from the table.

First, label the x-axis as "Population" and the y-axis as "Licensed Drivers". Then, plot each data point on the graph by finding the corresponding population value on the x-axis and the corresponding number of licensed drivers value on the y-axis.

Make sure to use a consistent scale on both axes to accurately represent the data. It's important to evenly space the intervals on each axis and label them accordingly.

After plotting all the data points, you can observe the overall pattern or trend in the scatter plot. It might show a positive correlation if the points are generally going upwards from left to right, indicating that as the population increases, the number of licensed drivers also tends to increase. Alternatively, it might show a negative correlation if the points are generally going downwards from left to right, indicating an inverse relationship between population and licensed drivers.

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The inverse function f⁻¹(x) is given by f⁻¹(x) = (4 + x)/x.

To determine the inverse function f⁻¹(x) of the function f(x) = 4/(x - 1), we need to find the value of x when given f(x).

The equation of the function: f(x) = 4/(x - 1).

Replace f(x) with y:

y = 4/(x - 1).

Swap x and y in the equation:

x = 4/(y - 1).

Multiply both sides of the equation by (y - 1) to eliminate the fraction:

x(y - 1) = 4.

Expand the equation: xy - x = 4.

Move the terms involving y to one side:

xy = 4 + x.

Divide both sides by x:

y = (4 + x)/x.

Therefore, the inverse function f⁻¹(x) is f⁻¹(x) = (4 + x)/x.

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The 95% confidence interval for μ is approximately $144.32 to $175.68.

To find a 95% confidence interval for μ, we can use the formula:
Confidence interval = sample mean ± (critical value * standard error)

Step 1: Find the critical value for a 95% confidence level. Since the sample size is large (n > 30), we can use the z-distribution. The critical value for a 95% confidence level is approximately 1.96.

Step 2: Calculate the standard error using the formula:
Standard error = standard deviation / √sample size

Given that the standard deviation is $64 and the sample size is 64, the standard error is 64 / √64 = 8.

Step 3: Plug the values into the confidence interval formula:
Confidence interval = $160 ± (1.96 * 8)

Step 4: Calculate the upper and lower limits of the confidence interval:
Lower limit = $160 - (1.96 * 8)
Upper limit = $160 + (1.96 * 8)

Therefore, the 95% confidence interval for μ is approximately $144.32 to $175.68.

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The odds in favor of the event are a/(b - a).

To find the odds in favor of an event, we need to determine the ratio of favorable outcomes to unfavorable outcomes.

In this case, the probability of the event is given as a/b. To find the odds, we need to express this probability as a ratio of favorable outcomes to unfavorable outcomes.

Let's assume that the number of favorable outcomes is x and the number of unfavorable outcomes is y.

According to the given information, the probability of the event is x/(x+y) = a/b.

To find the odds in favor of the event, we need to express this probability as a ratio.

Cross-multiplying, we get bx = a(x+y).

Expanding, we have bx = ax + ay.

Moving the ax to the other side, we get bx - ax = ay.

Factoring out the common factor, we have x(b - a) = ay.

Finally, dividing both sides by (b - a), we find that x/y = a/(b - a).

Therefore, the odds in favor of the event are a/(b - a).

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To find the probability of a company having both a website and a section in the newspaper, we can use the formula for conditional probability.

Let's denote the events as follows:
A: A company has a section in the newspaper
B: A company has a website

We are given the following probabilities:
P(A) = 0.43 (Probability of a company having a section in the newspaper)
P(B|A) = 0.84 (Probability of a company having a website given that it has a section in the newspaper)

The probability of both events A and B occurring can be calculated as:
P(A and B) = P(A) * P(B|A)

Substituting in the values we have:
P(A and B) = 0.43 * 0.84
P(A and B) = 0.3612

Therefore, the probability of a company having both a website and a section in the newspaper is 0.3612 or 36.12%.

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Answer:

15 m =16.404 yards

Step-by-step explanation:

15 m = 16.404 yards

For every inch in the scale drawing, it represents 60 inches in the actual gym.

To determine the scale Jorge used to create the scale drawing of the school gym, we can calculate the ratio of the length in the scale drawing to the length of the actual gym.

In the scale drawing, the length of the gym is 17 inches, while the length of the actual gym is 85 feet.

Since there are 12 inches in a foot, we can convert the length of the actual gym from feet to inches:

85 feet * 12 inches/foot = 1020 inches

Now, we can calculate the scale by dividing the length in the scale drawing by the length of the actual gym:

17 inches / 1020 inches = 1/60

Therefore, the scale that Jorge used to create the scale drawing of the school gym is 1:60.

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The quadratic equation with roots 4-3i and 4+3i is x² + 8x + 25 = 0.

To find the quadratic equation with roots 4-3i and 4+3i, we can use the sum and product of roots formulas.

The sum of the roots is given by -b/a, so in this case, -b/a = -8/a = -8/1 = -8.

The product of the roots is given by c/a, so in this case, c/a = (4-3i)(4+3i)/1 = (16-9i²)/1 = (16-9(-1))/1 = (16+9)/1 = 25/1 = 25.

Now, we can use these values to form the quadratic equation. Since a=1, the quadratic equation is:

x² - (sum of roots)x + product of roots = 0

Substituting the values, we have:

x² - (-8)x + 25 = 0

Simplifying further, we get:

x² + 8x + 25 = 0

Therefore, the quadratic equation with roots 4-3i and 4+3i is:

x² + 8x + 25 = 0.

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The system of equations represented by this matrix is:-1x + 2y = -6 1x + 1y = 7, "x" and "y" represent the variables in the system of equations.

The matrix -1 2 -6 1 1 7 represents a system of equations.

To write the system of equations, we can use the matrix entries as coefficients for the variables.
The first row of the matrix corresponds to the coefficients of the first equation, and the second row corresponds to the coefficients of the second equation.
The system of equations represented by this matrix is:
-1x + 2y = -6
1x + 1y = 7
"x" and "y" represent the variables in the system of equations.

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The given matrix represents a system of three equations with three variables. The equations are:
-1x + 2y = 6
-6x + y = 1
x + 7y = 7

The given matrix can be written as:

[tex]\left[\begin{array}{cc}-1&2\\-6&1\\1&7\end{array}\right][/tex]

To convert this matrix into a system of equations, we need to assign variables to each element in the matrix. Let's use x, y, and z for the variables.

The first row of the matrix corresponds to the equation:
-1x + 2y = 6

The second row of the matrix corresponds to the equation:
-6x + y = 1

The third row of the matrix corresponds to the equation:
x + 7y = 7

Therefore, the system of equations represented by this matrix is:
-1x + 2y = 6
-6x + y = 1
x + 7y = 7

This system of equations can be solved using various methods such as substitution, elimination, or matrix operations.

In conclusion, the given matrix represents a system of three equations with three variables. The equations are:
-1x + 2y = 6
-6x + y = 1
x + 7y = 7

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The probability that exactly 20 customers will arrive in the next 2 hours is 0.070. The average arrival rate of customers at the bakery is 10 customers per hour. So, in 2 hours, there is an expected arrival of 10 * 2 = 20 customers.

We can use the Poisson distribution to calculate the probability that exactly 20 customers will arrive in the next 2 hours. The Poisson distribution is a probability distribution that describes the number of events that occur in a fixed period of time,

given an average rate of occurrence. In this case, the event is a customer arriving at the bakery and the average rate of occurrence is 10 customers per hour.

The formula for the Poisson distribution is: P(X = k) = (λ^k e^(-λ)) / k!

where:

In this case, we want to calculate the probability that there are 20 events (customers arriving at the bakery) in a period of time with an average rate of occurrence of 10 events per hour (2 hours).

So, we can set λ = 10 and k = 20. We can then plug these values into the formula for the Poisson distribution to get the following probability: P(X = 20) = (10^20 e^(-10)) / 20!

This probability is very small, approximately 0.070. In conclusion, the probability that exactly 20 customers will arrive in the next 2 hours at the bakery is 0.070.

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The simplified form of (√y+√2)(√y - 7√2) is y - 5√2y - 14. Simplifying in mathematics refers to the process of reducing or transforming an expression, equation, or mathematical object into a more concise or manageable form without changing its essential meaning or value.

The goal of simplification is to make mathematical expressions easier to understand, manipulate, and work with.

In various mathematical contexts, simplifying involves applying mathematical rules, properties, and operations to eliminate redundancies, combine like terms, reduce fractions, factorize, cancel out common factors, or rewrite expressions using equivalent forms. By simplifying, we can often reveal underlying patterns, highlight important relationships, and facilitate further analysis or computation.

To simplify the given expression (√y+√2)(√y - 7√2), we can use the distributive property of multiplication over addition.

Expanding the expression, we multiply each term in the first parentheses by each term in the second parentheses:

(√y + √2)(√y - 7√2) = √y * √y + √y * (-7√2) + √2 * √y + √2 * (-7√2)

Simplifying each term, we have:

√y * √y = y

√y * (-7√2) = -7√2y

√2 * √y = √2y

√2 * (-7√2) = -14

Combining the terms, we get:

y - 7√2y + √2y - 14

Simplifying further, we can combine like terms:

y - 5√2y - 14

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Experiment was conducted to study tensile strength of Portland cement using four different mixing techniques. Data was collected to compare performance of these techniques in terms of tensile strength.

In a completely randomized experiment, the four different mixing techniques for Portland cement were randomly assigned to different samples. The tensile strength of each sample was then measured, resulting in a dataset that allows for comparisons between the mixing techniques.

The collected data can be analyzed to determine if there are any significant differences in tensile strength among the mixing techniques. Statistical methods such as analysis of variance (ANOVA) can be applied to assess whether there is a statistically significant variation in tensile strength between the techniques.

The analysis of the data will provide insights into which mixing technique yields the highest tensile strength for Portland cement. It will help identify the most effective method for producing cement with desirable tensile properties. By conducting a completely randomized experiment, researchers aim to eliminate potential biases and confounding factors, ensuring a fair comparison between the different mixing techniques.

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Leah paid $45 for supplies and incurred additional costs for wrapping each item. She was able to sell $75 worth of baked goods.

Leah's bake sale for her favorite charity had some costs involved. She initially paid $45 for supplies at the grocery store. Additionally, she spent about $0.50 for wrapping each individual item. As for the revenue, Leah was able to sell $75 worth of baked goods at the bake sale.

To calculate the total expenses, we can add the cost of supplies to the cost of wrapping each item. The cost of wrapping can be determined by multiplying the number of items by the cost per item. However, we don't have the exact number of items Leah sold, so we cannot provide an accurate calculation.

To determine the profit or loss from the bake sale, we need to subtract the total expenses from the revenue. Since we don't have the exact total expenses, we cannot determine the profit or loss.

In conclusion, Leah paid $45 for supplies and incurred additional costs for wrapping each item. She was able to sell $75 worth of baked goods. However, without knowing the exact expenses, we cannot calculate the profit or loss from the bake sale.

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If the product of two polynomials with integer coefficients is a polynomial with even coefficients, not all of which are divisible by 4, then in one of the polynomials all the coefficients are even, and in the other at least one of the coefficients is odd. This statement is proved.

To prove that if the product of two polynomials with integer coefficients is a polynomial with even coefficients, not all of which are divisible by 4, then in one of the polynomials all the coefficients are even, and in the other at least one of the coefficients is odd, we can use proof by contradiction.

Assume that both polynomials have all even coefficients. In this case, every coefficient in each polynomial would be divisible by 2. When we multiply these polynomials, the resulting polynomial will have all even coefficients, as each term in the product will have even coefficients.

However, since not all of the coefficients in the resulting polynomial are divisible by 4, this means that there must be at least one coefficient that is divisible by 2 but not by 4. This contradicts our assumption that all coefficients in both polynomials are even.

Therefore, our assumption is incorrect. At least one of the polynomials must have at least one odd coefficient.

In conclusion, if the product of two polynomials with integer coefficients is a polynomial with even coefficients, not all of which are divisible by 4, then in one of the polynomials all the coefficients are even, and in the other at least one of the coefficients is odd.

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The approximate probability that you will win a prize is 0.39 or 39%.

If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1/100, the approximate probability that you will win a prize is 0.39 or 39%.

Here's how to calculate it:

Probability of not winning a prize in one lottery = 99/100

Probability of not winning a prize in 50 lotteries = (99/100)^50 ≈0.605

Probability of winning at least one prize in 50 lotteries = 1 - Probability of not winning a prize in 50 lotteries

= 1 - 0.605 = 0.395 ≈0.39 (rounded to two decimal places)

Therefore, the approximate probability that you will win a prize is 0.39 or 39%.

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Therefore, Isaac should use the arithmetic mean to describe the temperatures recorded at noon during the week.

To describe the temperatures recorded by Isaac during one week, we need to choose an appropriate measure of center. The measure of center provides a representative value that summarizes the central tendency of the data.

In this case, since the temperatures do not contain an extreme value and we want a measure that represents the typical or central value of the data, the most suitable measure of center to use is the arithmetic mean or average.

The arithmetic mean is calculated by summing all the values and dividing the sum by the number of values. It provides a balanced representation of the data as it considers every observation equally.

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a. The area of parallelogram ABCD is 2√3.

b. The area of triangle ABC is √3.

c. The distance from point B to line AC is (6/5)√3.

a. To find the area of parallelogram ABCD, we first calculate the vectors AB→ and AC→ using the coordinates of points A, B, and C. The cross product of AB→ and AC→ gives us the area of the parallelogram, which is 2√3.

b. The area of triangle ABC is half the area of the parallelogram, so it is √3.

c. To find the distance from point B to line AC, we use the formula for the distance between a point and a line. We calculate the vectors B - A and B - C, and then take their cross product. The absolute value of the cross product divided by the magnitude of vector A - C gives us the distance. The final result is (6/5)(√6 / √2), which simplifies to (6/5)√3.

Therefore, the area of parallelogram ABCD is 2√3, the area of triangle ABC is √3, and the distance from point B to line AC is (6/5)√3.

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The center of the circle with equation (x-5)²+(y+1)²=81 is (5,-1).

The equation of a circle with center (h,k) and radius r is given by (x - h)² + (y - k)² = r². The equation (x - 5)² + (y + 1)² = 81 gives us the center (h, k) = (5, -1) and radius r = 9. Therefore, the center of the circle is option g. (5,-1).

Explanation:The equation of the circle with center at the point (h, k) and radius "r" is given by: \[(x-h)²+(y-k)^{2}=r²\]

Here, the given equation is:\[(x-5)² +(y+1)² =81\]

We need to find the center of the circle. So, we can compare the given equation with the standard equation of a circle: \[(x-h)² +(y-k)² =r² \]

Then, we have:\[\begin{align}(x-h)² & =(x-5)² \\ (y-k)² & =(y+1)² \\ r²& =81 \\\end{align}\]

The first equation gives us the value of h, and the second equation gives us the value of k. So, h = 5 and k = -1, respectively. We also know that r = 9 (since the radius of the circle is given as 9 in the equation). Therefore, the center of the circle is (h, k) = (5, -1).:

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