10. Solve the differential equation x?dy - y(x + y)dx = 0, by using Bernoulli's equations.

The slant height of the triangular pyramid is approximately 19.88 cm.

To find the slant height of the triangular pyramid, we can use the formula:

Slant height = sqrt(h^2 + (0.5b)^2)

where h is the height of the triangular pyramid and b is the base length.

First, we need to find the height of the triangular pyramid using the formula for the surface area of a triangular pyramid:

Surface area = 0.5 * Perimeter * Slant height + Base area

where the base area is 0.5 * b * h, and the perimeter is the sum of the lengths of the three sides of the base.

In this case, we know the surface area of the triangular pyramid (532 square cm), the base length (24 cm), and the hypotenuse of the base (25 cm).

The length of the other leg of the base can be found using the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the two legs of the right triangle formed by the base, and c is the hypotenuse.

In this case, we have:

a^2 + 24^2 = 25^2

a^2 = 625 - 576

a^2 = 49

a = 7 cm

Therefore, the perimeter of the base is:

24 + 25 + 7 = 56 cm

Now, we can use the formula for the surface area to find the height:

532 = 0.5 * 56 * Slant height + 0.5 * 24 * h

532 = 28 * Slant height + 12 * h

We need to find the height h in terms of the slant height, so we can isolate h:

h = (532 - 28 * Slant height) / 12

Now, we substitute this expression for h into the formula for the height:

h^2 = 25^2 - (0.5 * 24)^2

h^2 = 625 - 144

h^2 = 481

h = sqrt(481)

h = 21.93 cm

Now, we substitute the expression for h in terms of the slant height into the formula for the slant height:

Slant height = sqrt(h^2 + (0.5b)^2)

Slant height = sqrt((532 - 28 * Slant height)^2 / 144 + 144)

Squaring both sides and simplifying, we get:

756 * Slant height^2 - 149984 * Slant height + 70624 = 0

Using the quadratic formula, we get:

Slant height = (149984 +/- sqrt(149984^2 - 4 * 756 * 70624)) / (2 * 756)

Slant height = (149984 +/- sqrt(141562496)) / 1512

Taking the positive root and simplifying, we get:

Slant height ≈ 19.88 cm

Therefore, the slant height of the triangular pyramid is approximately 19.88 cm.

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Answer:

Step-by-step explanation:

First, let's calculate the volume of the glass sphere:

V_sphere = (4/3) * pi * r^3

V_sphere = (4/3) * 3.14 * 12^3

V_sphere ≈ 7238.23 cm^3

Next, let's find the volume of the cylinder:

V_cylinder = pi * r^2 * h

V_cylinder = 3.14 * 30^2 * 100

V_cylinder ≈ 282,600 cm^3

Since the glass sphere is completely submerged in water, the volume of the water in the cylinder will be the difference between the volume of the cylinder and the volume of the sphere:

V_water = V_cylinder - V_sphere

V_water ≈ 275,362.77 cm^3

Therefore, the approximate volume of the water contained in the cylinder is approximately 275,365 cm^3, which is closest to option D.

The roots of the given equation is real and equal.

Given , 5[tex]x^{2} \\[/tex] - 10x+ k=0

The quadratic equation is b² - 4ac = 0

Here, a= 5, b= -10, c= k

substitute in b² - 4ac = 0

(-10)² - 4 * 5* k =0

100 - 20k =0 , let this be equation (1)

100 = 20k

k = [tex]\frac{100}{20}[/tex]

k = 5.

now, substitute  k= 5 in equation (1)

100 -20k = 0

100 - 20*5 = 0

100 - 100 = 0

Therefore, the given equation is real and equal .

The correct question is 5x² - 10x + k =0

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The expression 2+2+210-(5*2) equals 14.

The expression 2+2+2 x 10-(5 x 2) can be solved using the order of operations, also known as PEDMAS.

In the given expression, we don't have any parentheses or exponents, so we move to the multiplication and division step. We have 2 multiplications, 210 and 52. Since multiplication comes before addition and subtraction, we must perform these multiplications first. Therefore, we get:

2 + 2 + 20 - 10

Now, we have only addition and subtraction left. According to the order of operations, we perform these operations from left to right. Therefore, we get:

4 + 20 - 10

Now, we can perform the addition and subtraction, which gives us the final answer:

14

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Answer:

The Correct answer is A

2x²√7y

The equation of the tangent plane to the surface z=ly2-922 at the point (1,-3,0) is z = -27 - 6ly(x-1).

To find the equation of the tangent plane to the surface z=ly2-922 at the point (1,-3,0), we need to first find the partial derivatives of z with respect to x and y.

∂z/∂x = 0

∂z/∂y = 2ly

Next, we need to evaluate these partial derivatives at the given point (1,-3,0), which gives us:

∂z/∂x(1,-3,0) = 0

∂z/∂y(1,-3,0) = -6l

Using these partial derivatives, we can now write the equation of the tangent plane:

z = f(a,b) + ∂f/∂x(a,b)(x-a) + ∂f/∂y(a,b)(y-b)

where f(a,b) is the value of the function at the point (a,b), and (x-a) and (y-b) are the differences between the given point and the point on the tangent plane.

Plugging in the values we found, we get:

z = -27 - 6ly(x-1)

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a) Cost of n paperbacks = 8n

b) Cost of paperbacks for a Book Club member, including a membership is represented by algebraic equation = 10 + 6n

c) Difference = $14 (96 - 82), Book Club membership saves money.

Cost refers to the amount of money or resources that are required to produce, purchase, or obtain a product or service. It is the value given up in exchange for something.

An algebraic expression is a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division, that can be simplified and evaluated.

According to the given information:

a) The cost of n paperbacks can be represented by the algebraic expression: 8n. This expression means that the cost of n paperbacks is equal to the unit price of $8 multiplied by the number of paperbacks purchased.

b) The cost of paperbacks for a Book Club member, including a membership can be represented by the algebraic expression: 10 + 6n. This expression means that the cost of paperbacks for a Book Club member is equal to the membership fee of $10 plus the discounted unit price of $6 multiplied by the number of paperbacks purchased.

c) Without the Book Club membership, the cost of 12 books would be 12 x 8 = $96. With the Book Club membership, the cost of 12 books would be 10 (membership fee) + 6(12) = $82. Therefore, the difference between the cost of 12 books purchased with and without a membership is $14 (96 - 82). This shows that being a Book Club member results in a significant discount on the cost of paperbacks.

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The values of x where f has a relative maximum are -1 and 3. The answer is D, -1 and 3 only.

To determine the values of x where the function f has a relative maximum, we need to analyze the sign of the derivative in the neighborhood of each critical point.

The critical points of f correspond to the values of x where the derivative is equal to zero or undefined. In this case, the derivative is undefined only at x = -1, where there is a vertical asymptote.

To find the values of x where the derivative is zero, we set f'(x) = 0 and solve for x:

[tex]f'(x) = x(x-3)^2[/tex](x+1) = 0

This equation is satisfied when x = 0 or x = 3, since these are the only values that make one of the factors equal to zero. Therefore, these are the only two critical points of f.

Next, we can use the first derivative test to determine the nature of each critical point. This involves checking the sign of the derivative in the interval to the left and right of each critical point.

For x = -1, the derivative is undefined to the left and right, but we can still examine the sign of f'(x) in the vicinity of -1. For example, if we consider values of x slightly to the left of -1 (say, x = -1.1), we see that f'(x) is negative. Similarly, if we consider values of x slightly to the right of -1 (say, x = -0.9), we see that f'(x) is positive. This means that f has a relative maximum at x = -1.

For x = 0, the derivative is negative to the left and positive to the right. This means that f has a relative minimum at x = 0.

For x = 3, the derivative is positive to the left and negative to the right. This means that f has a relative maximum at x = 3.

Therefore, the values of x where f has a relative maximum are -1 and 3. The answer is D, -1 and 3 only.

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The probability that the sample mean compressive strength exceeds 4985 psi is approximately 1.

We need to find the probability

Calculate the standard error of the sample mean.

Standard Error (SE) = σ / √n = 100 / √9

                               = 100 / 3

                               = 33.33 psi

Calculate the z-score of the sample mean.

z = (sample mean - μ) / SE = (4985 - 5500) / 33.33

                                           = -515 / 33.33

                                           = -15.45

Find the probability using the z-score.

Since the z-score is -15.45, which is very far from the mean in the left tail, the probability of the sample mean

compressive strength exceeding 4985 psi is almost 1.

So, the probability that the sample mean compressive strength exceeds 4985 psi is approximately 1.

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The function f(x) is a quadratic function in factored form, revealing the x-intercepts, vertex, and behavior of the function. X-intercepts are (0,0) and (-4,0), vertex is (-2,8), and y-intercept is (0,0). The graph opens upward.

A function is a relation between two sets of values, such that each input value maps to a unique output value. It describes a mathematical rule or relationship that links inputs to outputs.

According to the given information:

a. The function f(x) is in the form of a quadratic function, which is a second-degree polynomial function. The equation is written in factored form, revealing the x-intercepts (zeros) and the behavior of the function as it approaches the x-axis.

b. To find the x-intercepts, we set f(x) equal to zero and solve for x:

f(x) = (x^2)(x + 4) = 0

x^2 = 0 or x + 4 = 0

x = 0 or x = -4

So the x-intercepts are (0,0) and (-4,0).

To find the vertex, we can use the formula -b/2a to find the x-value of the vertex, where a and b are coefficients in the quadratic equation ax^2 + bx + c. In this case, a = 1 and b = 4, so the x-value of the vertex is -b/2a = -4/2 = -2. To find the y-value, we evaluate f(-2):

f(-2) = (-2)^2(-2+4) = 8

So the vertex is (-2, 8).

To find the y-intercept, we evaluate f(0):

f(0) = (0)^2(0+4) = 0

So the y-intercept is (0,0).

c. Here is a sketch of the graph of f(x):

      |          

      |          

      |          

  ___/ \___      

 /         \      

/           \    

/             \    

---------------  

     |     |      

    -4     4      

The graph has x-intercepts at (0,0) and (-4,0), a vertex at (-2,8), and a y-intercept at (0,0). It opens upward since the leading coefficient (coefficient of x^2) is positive

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Answer:

i want to say A, 9.

Step-by-step explanation:

The answer is (A) 9.

If Sonia removes 1 tile from the pile of 10 tiles, there will be 9 tiles left.If Roberto removes 1 tile, then there will be 8 tiles left. Sonia can remove 2 tiles, leaving 6 tiles for Roberto.

If Roberto removes 1, 2, or 3 tiles, then there will be 5, 4, or 3 tiles left, respectively. Sonia can then remove enough tiles to leave Roberto with a multiple of 6 tiles, ensuring a win on her next turn.For example, if Roberto removes 3 tiles, then there will be 7 tiles left. Sonia can remove 2 tiles, leaving 5 tiles for Roberto. Then, regardless of how many tiles Roberto removes, Sonia can always remove enough tiles to leave Roberto with a multiple of 6 tiles on his turn, ensuring a win on her next turn.

The number of elements in the sample space of the situation is given as follows:

36.

The Fundamental Counting Theorem states that if there are m ways to do one thing and n ways to do another, then there are m x n ways to do both.

This can be extended to more than two events, where the number of ways to do all the events is the product of the number of ways to do each individual event

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

For this problem, we have two events, in which:

Hence the total number of outcomes is given as follows:

6 x 6 = 36.

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In linear regression, unbiasedness of OLS estimator for $\beta$ is guaranteed

True.

In linear regression, unbiasedness of OLS estimator for $\beta$ is guaranteed when the following conditions hold:

The regression model is correctly specified and the true model is linear in the parameters.

The errors are homoscedastic and normally distributed with mean zero and constant variance.

The regressors are not perfectly collinear.

The expected value of the errors given the values of the regressors is zero.

Under these conditions, if the OLS residuals are uncorrelated with the regressors, then the estimate of $\beta$ obtained from OLS is unbiased.

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The expression that can be solved to give a value of  13 is 1/2 * ((12/2) - 2) + 11

From the question, we have the following parameters that can be used in our computation:

1/2 x 12 / 2 - 2 + 11

Next, we solve one after the other to get 13

Using the above as a guide, we have the following expressions:

12 /2 = 6

So, we have

6 - 2 = 4

Next, we have

1/2 * 4  = 2

Lastly, we have

2 + 11 = 13

When the above steps are combined, we have

1/2 * ((12/2) - 2) + 11

Hence, the expression is 1/2 * ((12/2) - 2) + 11

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Answer: 10

Step-by-step explanation: Volume = length*width*depth. 180 = 6*3*depth. 6*3 = 18. We divide 180 by 18, to get 10. Depth = 10

Answer:

20 feet

Step-by-step explanation:

Let's start by using the formula for the volume of a rectangular box:

[tex]\sf\qquad\dashrightarrow V = lwh[/tex]

where:

We know that the box is half filled with grain, so the volume of the grain is:

[tex]\sf:\implies V_{grain} = 0.5V = 0.5lwh[/tex]

We also know that the volume of the grain is 180 cubic feet, so we can set up an equation:

[tex]\sf\qquad\dashrightarrow 0.5lwh = 180[/tex]

We are given that the box is 6 feet wide and 3 feet long, so we can substitute those values in:

[tex]\sf:\implies 0.5(3)(6)h = 180[/tex]

Simplifying:

[tex]\sf:\implies 9h = 180[/tex]

[tex]\sf:\implies \boxed{\bold{\:\:h = 20\:\:}}\:\:\:\green{\checkmark}[/tex]

Therefore, the depth of the box is 20 feet.

Answer:

112 cm²

Step-by-step explanation:

Area of shaded region =

(9 x 16) - (5 x 4) - 2(1/2 x 3 x 4) = 112 cm²

a. At a spending level of $16 million, the percentage of pollution removal is increasing at a rate of 0.97% per million dollars spent.

b. For spending levels below $150 million, the percentage of pollution removal is decreasing, while for spending levels above $150 million, the percentage of pollution removal is increasing.

a. To answer the first part of the question, we need to find the rate at which the percentage of pollution removal is changing when $16 million is spent. This is the derivative of the function P(x) evaluated at x = 16. Taking the derivative, we get P'(x) = 0.06x + 9/100.

Evaluating at x = 16, we get P'(16) = 0.06(16) + 9/100 = 0.969, or approximately 0.97%.

b. To determine for what values of x the function P(x) is increasing or decreasing, we need to find the sign of the derivative P'(x). Taking the derivative of P(x), we get P'(x) = 0.06x + 9/100. This derivative is positive for x > -9/0.06, or x > -150, and negative for x < -150.

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Answer: 35, 92

Step-by-step explanation: We see that the third number is the result of adding the first two numbers (9+13=22). Continuing with this pattering, we know that the fourth spot is 22 + x = 57. 57-22 is 35, so the fourth spot is 35. The sixth spot will be 57 + 35, or, 92.

Answer:

The next numbers in the sequence are 35, and 92.

Step-by-step explanation:

The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding numbers. So, the next numbers in the Fibonacci sequence would be:

Therefore, the next numbers in the sequence are 35, and 92, if the sequence is continued using the Fibonacci sequence.

a) z-scores for Mississauga-Erindale: Z = (143,361 - 102,639.28) / 21,855.384 ≈ 1.865 and for Parkdale-High Park: Z = (102,142 - 102,639.28) / 21,855.384 ≈ -0.023
b) The citizens might feel underrepresented in the House of Commons as their votes have less weight compared to those in smaller ridings.

a) To compare the z-scores for these ridings, we need to use the formula:

z-score = (x - μ) / σ

where x is the population of the riding, μ is the mean population of all ridings, and σ is the standard deviation of all ridings.

For Mississauga-Erindale:

z-score = (143,361 - 102,639.28) / 21,855.384 = 1.87

For Parkdale-High Park:

z-score = (102,142 - 102,639.28) / 21,855.384 = -0.02

Therefore, Mississauga-Erindale has a z-score of 1.87, which means its population is above the mean population of all ridings by 1.87 standard deviations. Parkdale-High Park has a z-score of -0.02, which means its population is almost exactly at the mean population of all ridings.

b) The citizens of Mississauga-Erindale could argue that their riding is overrepresented in the House of Commons. This is because their population is above the mean population of all ridings by almost 2 standard deviations, which means they have more political influence per person compared to other ridings. However, it's important to note that the electoral district boundaries are redrawn every 10 years based on population changes, so the population of each riding may change over time.
a) To compare the z-scores for Mississauga-Erindale and Parkdale-High Park, we need to calculate the z-scores for each riding using the given mean and standard deviation. The formula for calculating z-scores is:

Z = (X - μ) / σ

For Mississauga-Erindale:
Z = (143,361 - 102,639.28) / 21,855.384 ≈ 1.865

For Parkdale-High Park:
Z = (102,142 - 102,639.28) / 21,855.384 ≈ -0.023

b) The citizens of Mississauga-Erindale could argue that their representation in the House of Commons is unfair because their riding has a significantly larger population compared to the average riding size. The z-score of 1.865 indicates that the population of Mississauga-Erindale is approximately 1.865 standard deviations above the mean, meaning it is larger than a majority of other ridings. Consequently, the citizens might feel underrepresented in the House of Commons as their votes have less weight compared to those in smaller ridings.

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This is an exercise in Newton's second law is one of the fundamental laws of physics and is used to describe the relationship between force, mass, and acceleration of an object. This law can be expressed mathematically as F = ma, where F is the net force applied to an object, m is its mass, and a is the acceleration produced. The law states that the acceleration experienced by an object is directly proportional to the net force acting on it and inversely proportional to its mass.

In other words, if the force acting on an object increases, its acceleration will also increase, and if the object's mass increases, its acceleration will decrease. This mathematical relationship is very useful in understanding how objects move and how forces affect their motion. Newton's second law is especially important for dynamics, the branch of physics that deals with the study of the movement of objects and its causes.

The law of force can be intuitively explained by observing how an object moves when a force is applied to it. If a force is applied to an object, such as pushing a box, the box will start to move in the direction of the applied force. If a larger force is applied, the box will move faster, and if a smaller force is applied, the box will move more slowly. If the box is heavier, it will take more force to move it at the same speed as a lighter box.

Newton's second law also states that the direction of the acceleration produced is the same as the direction of the applied force. For example, if a box is pushed to the right, the box will move to the right. If the box is pushed up, the box will move up. This relationship between the direction of force and the direction of acceleration is important in understanding how objects move in different situations.

In addition, Newton's second law is also important in understanding how forces are applied in different situations. For example, if a force is applied to a box at an angle, the box will move in a different direction than the applied force due to the breakdown of the force into horizontal and vertical components. The law of force can be used to calculate the components of force and determine how the object will move.

Newton's second law can also be used to understand the relationship between force and motion in nature. The law of force applies to all objects in the universe and is essential in understanding how planets, stars, and other celestial bodies move. For example, the gravitational force acting between two objects depends on the objects' mass and the distance between them, and this force determines how the objects move in space.

It tells us a model rocket has a mass of 0.1 kg, is pushed by the engine, and has an acceleration of 35 m/s².

We apply the formula F = m × a. We do not clear because it asks us to calculate the force, where:

F = Calculated force in Newton (N).

m = calculated mass in Kilograms (kg).

a = acceleration calculated in meters per second squared (m/s^2).

F = m  × a

F = 0.1 kg × 35 m/s²

F = 3.5 N

The amount of force that the motor exerts on the rocket is 3.5 Newtons.

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The given series 3n!/2ⁿ is divergent.

To determine if the sequence a_n = 3n!/2ⁿ converges or diverges, we can use the ratio test.

Taking the limit of a_(n+1)/a_n as n approaches infinity, we get:

lim [(3(n+1)!/2ⁿ⁺¹) / (3n!/2ⁿ)]
= lim [3(n+1)!/2ⁿ⁺¹ * 2ⁿ/3n!]
= lim [3(n+1)/2]
= infinity

Since the limit is greater than 1, the sequence diverges.

The ratio test is a way to determine the convergence or divergence of a series by taking the limit of the ratio of consecutive terms. If the limit is less than 1, the series converges.

If the limit is greater than 1, the series diverges. In this case, we applied the ratio test to the sequence a_n = 3n!/2ⁿ and found that the limit is infinity, indicating that the sequence diverges. This means that the terms of the sequence do not approach a finite limit as n approaches infinity, but instead grow without bound.

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The median of this distribution is 4.375.

To find the median of a distribution, we need to first arrange the values in order from smallest to largest along with their corresponding frequencies.

VALUE.....1....2...3...4....5

Freq.........4....3...2...6....8

There are a total of 23 observations in this distribution (4 + 3 + 2 + 6 + 8 = 23). Since the total frequency is an odd number, the median will be the value that is exactly in the middle when the observations are arranged in order.

To find this middle value, we need to first find the cumulative frequency for each value. The cumulative frequency is the sum of the frequencies up to and including that value.

VALUE.....1....2...3...4....5

Freq.........4....3...2...6....8

Cumulative Freq...4...7...9..15..23

The median will be the value that has a cumulative frequency of (23 + 1)/2 = 12. This value falls in the interval 4-5 since the cumulative frequency of 5 is greater than 12 and the cumulative frequency of 4 is less than 12.

To find the exact median value, we need to interpolate between the two values in the interval 4-5. We can use the following formula to find the median:

Median = L + [(n/2 - CF) / f] * w

where L is the lower limit of the interval, n is the total frequency, CF is the cumulative frequency up to the lower limit, f is the frequency of the interval, and w is the width of the interval.

For the interval 4-5, L = 4, n = 23, CF = 9, f = 8, and w = 1. The median can be calculated as:

Median = 4 + [(12 - 9) / 8] * 1 = 4.375

Therefore, the median of this distribution is 4.375.

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The least value of k for which the alternating series error bound guarantees that [S-Pk}= 3/100 is k = 34. To get the least value of k for which the alternating series error bound guarantees that [S-Pk}= 3/100,

We need to use the formula for the alternating series error bound: |S-Pk| ≤ a(k+1)
where a is the absolute value of the term following the last one included in Pk. In this case, the last term included in Pk is the kth term, which is (-1)^(k+1) * 2/k. Therefore, a = 2/(k+1).
We want to find the least value of k such that a(k+1) ≤ 3/100. Substituting a and simplifying, we get: 2(k+1)/(k+2) ≤ 3/100
Multiplying both sides by (k+2) and simplifying, we get: 200k + 400 ≤ 3k^2 + 6k
Rearranging and simplifying, we get: 3k^2 - 194k - 400 ≥ 0
Solving this quadratic inequality using the quadratic formula, we get: k ≤ (194 + sqrt(4*3*400+194^2))/6 or k ≥ (194 - sqrt(4*3*400+194^2))/6
k ≤ 33.053 or k ≥ 63.947
Therefore, the least value of k for which the alternating series error bound guarantees that [S-Pk}= 3/100 is k = 34.

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If the perimeter and two-sides of the triangle are 18c, 4cm and 5cm, then the unknown third-side is 9cm.

The "Peri-meter" of a triangle is defined as the sum of lengths of all 3 sides.

Let the unknown third side of the triangle be denoted as "x".

We know that, "first-side" of triangle is = 4 cm,

"Side-2" of triangle = 5 cm,

"Peri-meter" of the triangle is given to be 18 cm,

The Perimeter equation is written as:

⇒ Side 1 + Side 2 + Side 3 = Perimeter,

⇒ 5 + 4 + x = 18,

⇒ 9 + x = 18,

⇒ x = 18 - 9,

⇒ x = 9.

Therefore, the measure of the unknown third side of triangle is 9 cm.

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The given question is incomplete, the complete question is

The two sides of the triangle are 4cm and 5cm , the perimeter of the triangle is 18cm, find the measure of the unknown third side.

No, Alvin's claim is incorrect when she claimed that when two negative rational numbers are multiplied, the result is greater than both of the numbers

No, it does not. When two negative rational numbers are multiplied, the result is positive, not greater than both of the numbers.

For example:

= -1/2 *  -1/3

= 1/6

The 1/6 is positive and less than both -1/2 and -1/3.

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The value of RE for the given triangle is 3 units.

A key idea in geometry known as the Pythagorean theorem outlines the relationship between the sides of a right triangle. It asserts that the square of the length of the longest side, known as the hypotenuse, is equal to the sum of the squares of the two shorter sides of a right triangle. In other words, if a and b are the measurements of a right triangle's two shorter sides and c is the measurement of the hypotenuse,

Triangle DRF is an right angle triangle.

Using Pythagoras Theorem we have:

c² = a² + b²

DF² = 27² + 9²

DF² = 729 + 81

DF² = 810

DF = √(810)

DF = 9 √(10)

Now, the value of DF = 9 sqrt(10).

In triangle DEF we have:

DE² = (9 √(10))² + (3 √(10))

DE² = 81 * 10 + 9 * 10

DE² = 900

DE = 30

Now, DE = DR + RE

Given, DR = 27 substituting the value:

Thus,

30 = 27 + RE

RE = 3

Hence, the value of RE for the given triangle is 3 units.

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The slope of the tangent line to the curve is equal to 122 when x is approximately 24.17.

To find the value of x where the slope of the tangent line to the curve y = 3x^2 - 23x + 10 is equal to 122, we first need to find the derivative of y with respect to x. The derivative represents the slope of the tangent line at any given point.

Using the power rule, the derivative of y with respect to x (denoted as y') is:
y' = 6x - 23

Now we need to set y' equal to 122 and solve for x:
6x - 23 = 122

Add 23 to both sides:
6x = 145

Divide by 6:
x ≈ 24.17 (rounded to two decimal places)

So, the slope of the tangent line to the curve is equal to 122 when x is approximately 24.17.

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Based on the given information, we can determine that the confidence level used was 89.4%.

To determine the confidence level used, we first need to calculate the margin of error.
Margin of Error (ME) = (Upper Limit - Lower Limit) / 2
ME = (5.15 - 3.37) / 2
ME = 0.89

Now, we can use the z-score formula to find the confidence level:
Z = ME / (σ / √n)
Z = 0.89 / (2.75 / √25)
Z = 0.89 / (2.75 / 5)
Z = 0.89 / 0.55
Z ≈ 1.618

Now we will look up this z-score in a standard normal (z) table or use a calculator with an inverse cumulative distribution function to find the corresponding percentage, which is approximately 89.4%.

So, the confidence level used is 89.4%.

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The probability of having 63 or more successes in the sample is approximately 0.0266, or 2.66%.

To use the normal approximation to find the probability P(X ≥ 63) for a sample size of n = 90 and population proportion of successes p = 0.6, follow these steps:

Step 1: Calculate the mean (μ) and standard deviation (σ) for the binomial distribution.
[tex]μ = n * p[/tex] = 90 * 0.6 = 54
[tex]σ = \sqrt{(n * p * (1 - p))} = \sqrt{(90 * 0.6 * 0.4) }[/tex]= √21.6 ≈ 4.65

Step 2: Use the normal approximation.
To find P(X ≥ 63), first convert X to a z-score:
z = [tex](X - μ) / σ[/tex] = (63 - 54) / 4.65 ≈ 1.93

Step 3: Find the probability using a z-table or calculator.
Using a z-table or calculator, find the probability of a z-score less than 1.93 (since we want P(X ≥ 63), we need to find the area to the right of the z-score):
P(Z ≤ 1.93) ≈ 0.9734

Step 4: Calculate the complement probability.
Since we want P(X ≥ 63), we need to find the complement probability (1 - P(Z ≤ 1.93)):
P(X ≥ 63) = 1 - 0.9734 = 0.0266

So, the probability of having 63 or more successes in the sample is approximately 0.0266, or 2.66%.

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