Make an accurate drawing of triangle ABC, in which AB = 8 cm,
AC 7 cm and BC = 4 cm.
You must show all of your construction lines.
Measure the size of angle ACB to the nearest degree.

Answer:

To compare who saves the most of their salary each month among Amy, John, and Emily, we need to calculate the amount of salary each person saves.

Let's assume that the monthly salary of each person is 'S'. Then we can calculate the amount saved by each person as follows:

Amy:

Amount saved = 20% of S = 0.2S

Amount spent = S - 0.2S = 0.8S

John:

Amount spent = 2/5 of S = (2/5)S

Amount saved = S - (2/5)S = (3/5)S

Emily:

Let's assume that Emily saves '5x' and spends '8x' of her monthly salary.

Then, according to the question, we have:

Amount saved = 5x

Amount spent = 8x

We know that the ratio of the amount saved to the amount spent is 5:8, so we can write:

Amount saved / Amount spent = 5/8

Substituting the values of amount saved and amount spent, we get:

5x / 8x = 5/8

5x = (5/8) x 8x

5x = 5x

Therefore, the ratio of amount saved to amount spent is equal to 5:8. This means that Emily saves 5/13 of her monthly salary and spends 8/13 of her monthly salary.

So, the amount saved by each person is:

Amy: 0.2S

John: (3/5)S

Emily: 5/13 of S

Now, we need to compare these amounts to find out who saves the most.

To compare these amounts, we can write them in terms of a common denominator:

Amy: 0.2S

John: (3/5)S = (0.6)S

Emily: (5/13)S = (0.3846)S (approx.)

Therefore, we see that John saves the most of his salary each month, followed by Amy and then Emily.

Working out:

Let's assume that each person earns $1000 per month.

Amy:

Amount saved = 20% of $1000 = $200

Amount spent = $800

John:

Amount spent = 2/5 of $1000 = $400

Amount saved = $1000 - $400 = $600

Emily:

Let's assume that Emily saves $5x and spends $8x of her monthly salary.

Then, we have:

Amount saved = $5x

Amount spent = $8x

We know that the ratio of the amount saved to the amount spent is 5:8, so we can write:

$5x / $8x = 5/8

Solving for x, we get:

x = 8/13

Substituting the value of x, we get:

Amount saved = $5 x (8/13) x $1000 = $384.62 (approx.)

Amount spent = $8 x (8/13) x $1000 = $615.38 (approx.)

Therefore, we see that John saves the most of his salary each month, followed by Amy and then Emily.

Therefore, the coordinates of point P are approximately (4.444, -2.222, 4.444). This is the point on the plane 2x - y + 2z = 20 nearest to the origin.

To find the point on the plane nearest to the origin, we need to minimize the distance between the origin and a point on the plane.

The distance between two points, (x₁, y₁, z₁) and (x₂, y₂, z₂), is given by the formula:

distance = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

In this case, we want to find a point (x, y, z) on the plane 2x - y + 2z = 20 that is closest to the origin (0, 0, 0).

We can set up this problem as an optimization problem by minimizing the distance function:

distance = √((x - 0)² + (y - 0)² + (z - 0)²) = √(x² + y² + z²)

subject to the constraint 2x - y + 2z = 20.

To solve this problem, we can use the method of Lagrange multipliers. We define the Lagrangian function:

L(x, y, z, λ) = x² + y² + z² + λ(2x - y + 2z - 20)

Taking the partial derivatives of L with respect to x, y, z, and λ, and setting them equal to zero, we can solve for x, y, z, and λ. However, this process is quite lengthy and involves solving a system of equations.

Alternatively, we can use geometric intuition to find the point on the plane nearest to the origin. The normal vector to the plane is given by the coefficients of x, y, and z, which is (2, -1, 2). This vector is perpendicular to the plane.

The point on the plane closest to the origin will be the one that lies on the line perpendicular to the plane and passes through the origin. Let's call this point P.

The direction vector of the line passing through the origin and perpendicular to the plane is the same as the normal vector, (2, -1, 2). Therefore, the coordinates of point P can be expressed as (2t, -t, 2t), where t is a scalar parameter.

Substituting these coordinates into the equation of the plane, we get:

2(2t) - (-t) + 2(2t) = 20

4t + t + 4t = 20

9t = 20

t ≈ 2.222

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Hello !

square's area = c * c = c²

c² = 25

c = √25 = 5cm

square's perimeter = c + c + c + c = 4c

4c = 4 * 5cm = 20cm

The matrix A representing the linear transformation T is [5 -2; 0 6].

To find the matrix A that represents a linear transformation T, we need to determine the images of the standard basis vectors under T and use them to form the columns of A. In this case, we are given that T(1,0) = (5,0) and T(0,1) = (-2,6). These correspond to the first and second columns of A, respectively. Therefore, the matrix A is:

A = [5 -2]

[0 6]

To apply T to any vector x, we simply multiply it by A:

T(x) = Ax

So, if we have a vector x = [x1, x2], we can calculate T(x) as follows:

T(x) = [5x1 - 2x2, 6x2]

Thus, A fully characterizes the transformation T and enables the computation of T(x) for any given vector x.

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The volume of the pyramid is 266.67 inches³.

The surface area of the pyramid is 288 inches².

The prism is a square base pyramid. The volume and surface area of the prism can be found as follows:

volume of the square pyramid = 1 / 3 × base area × height

Therefore,

Base area = 10² = 100 inches²

volume of the square pyramid = 1 / 3 × 100  × 8

volume of the square pyramid = 800 / 3

volume of the square pyramid = 266.67 inches³

Surface area of the pyramid = a² + 2al

where

Therefore,

l² = 8² + 5²

l = √64 + 25

l = √89

l = 9.43398113206

l = 9.4

Therefore,

Surface area of the pyramid = 10² + 2 × 10 × 9.4

Surface area of the pyramid = 100 + 188

Surface area of the pyramid = 288 inches²

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Hello !

2 x 5 x 35.8miles = 358 miles

Answer:

716 miles

Step-by-step explanation:

There are two trips in one day. One trip averages 35.8 miles. Two trips can be found simply by doing [tex]35.8[/tex] × [tex]2[/tex] which equals 71.6.

To find out how many miles Mr. Adams drives in one week, we need to know how many times he drives in a week. He only drives 5 times a week, and 2 trips are done in 1 day. In 5 days, 10 trips will be made.

Now simply multiply 71.6 by 10.

[tex]71.6[/tex] × [tex]10[/tex] [tex]= 716[/tex]

The answer is 716 miles.

(this is based on the average amount miles per trip)

The random variable x has a uniform distribution, defined on [7,11], therefore the P is (C) 0.75

For a uniform distribution, the probability of a random variable X falling within a specific interval is calculated by dividing the length of the interval by the total length of the distribution. In this case, X has a uniform distribution defined on [7, 11].
To find P(8 ≤ X ≤ 11), we first determine the length of the interval: 11 - 8 = 3. Next, we find the total length of the distribution: 11 - 7 = 4. Now, we can calculate the probability:
P(8 ≤ X ≤ 11) = (length of interval) / (total length of distribution) = 3 / 4 = 0.75
Thus, the correct answer is C, 0.75.

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The final answers are as follows: Function f(x) is even.

Function g(x) is odd.

Function h(x) is neither even nor odd

To determine if a function is even, we check if f(x) = f(-x) for all x in the domain. Let's evaluate f(x) and f(-x) for the given function:

f(x) = [tex]x^{2}[/tex] + 2[tex]x^{4}[/tex]

f(-x) = [tex]-x^{2}[/tex] + 2[tex]-x^{4}[/tex]

Since f(x) = f(-x), the function is even.

Function: g(x) = [tex]x^{3}[/tex] - x

To determine if a function is odd, we check if f(x) = -f(-x) for all x in the domain. Let's evaluate g(x) and -g(-x) for the given function:

g(x) = [tex]x^{3}[/tex] - x

-g(-x) = -[tex]x^{3}[/tex] - (-x) = -[tex]x^{3}[/tex] + x

Since g(x) = -g(-x), the function is odd.

Function: h(x) = 2x + [tex]x^{2}[/tex]

To determine if a function is even or odd, we need to satisfy the conditions mentioned above. Let's evaluate h(x) and h(-x) for the given function:

h(x) = 2x + [tex]x^{2}[/tex]

h(-x) = 2(-x) + -[tex]x^{2}[/tex] = -2x + [tex]x^{2}[/tex]

Since h(x) is not equal to h(-x) or -h(-x), the function is neither even nor odd.

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The maximum blood pressure occurs at a dosage of 10 cc, resulting in a blood pressure of 0.

To find the maximum blood pressure and the corresponding dosage, we analyze the given quadratic function B(x) = 400x² - 4000x.

The maximum blood pressure is represented by the vertex of the parabolic function.

Using the formula x = -b / (2a), we substitute the values a = 400 and b = -4000 to find x = 10. This implies that the maximum blood pressure occurs at a dosage of 10 cc. By substituting x = 10 into the function, we calculate B(10) = 400(10)² - 4000(10) = 40000 - 40000 = 0.

Therefore, the maximum blood pressure is 0, and it is attained at a dosage of 10 cc.

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a) The cable provider would use a discrete uniform distribution to model the selection of customers in the telephone exchange b) The probability that a randomly selected number from the 452 exchange belongs to a new business that does not subscribe to digital TV is 1

To model the selection of customers in a particular telephone exchange, the cable provider would use a uniform distribution. This is because they are selecting numbers with equal probability from a set of 10,000 possible numbers. In a uniform distribution, each value has an equal chance of being selected, making it suitable for this scenario

Second, we are given that the new business incubator was assigned the 500 numbers between 452-2000 and 452-2499, and these businesses don't subscribe to digital TV. To find the probability of randomly selecting a number from this range that doesn't subscribe to digital TV, we need to determine the proportion of numbers in that range that meet the condition.

In this case, there are 500 numbers in the range 452-2000 to 452-2499, and all of them don't subscribe to digital TV. Since we are selecting from a specific range, the probability of selecting a number that doesn't subscribe to digital TV is 100% or 1.

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The evaluated integral of 9x cos(4x) dx is (9/4) x sin(4x) - (9/64) cos(4x) + C, where C is the constant of integration.

To evaluate the integral of 9x cos(4x) dx, we can use integration by parts, which is a technique based on the product rule of differentiation. The integration by parts formula is given as:

∫u dv = uv - ∫v du

Let's assign u and dv as follows:

u = 9x (differential: du = 9 dx)

dv = cos(4x) dx (v = ∫dv = ∫cos(4x) dx)

To find v, we integrate dv:

∫cos(4x) dx = (1/4) sin(4x)

Now, we can apply the integration by parts formula:

∫9x cos(4x) dx = 9x [(1/4) sin(4x)] - ∫(1/4) sin(4x) du

Simplifying:

= (9/4) x sin(4x) - (1/4) ∫sin(4x) du

= (9/4) x sin(4x) - (1/4) ∫sin(4x) (9 dx)

Integrating ∫sin(4x) (9 dx):

= -(9/4) ∫sin(4x) dx

= -(9/4)(-1/4) cos(4x)

= (9/16) cos(4x)

Now, let's substitute the result back into the original equation:

∫9x cos(4x) dx = (9/4) x sin(4x) - (1/4)(9/16) cos(4x) + C

= (9/4) x sin(4x) - (9/64) cos(4x) + C

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(a) By the technique shown in Examples 1 and 2:

To express the function f(x) = 1/4 + x as a geometric power series centered at 0, we can follow the technique shown in Examples 1 and 2, which involves finding a common ratio and using the formula for the sum of an infinite geometric series.

In this case, we can rewrite the function as:

f(x) = 1/4 + x

     = 1/4 + x(1)

Now, we can identify the common ratio, which is x. We can express the function as:

f(x) = 1/4 + x(1) = [tex]1/4 + x(1)^n[/tex]

The formula for the sum of an infinite geometric series is:

S = a / (1 - r)

where S is the sum of the series, a is the first term, and r is the common ratio.

In our case, the first term is a = 1/4 and the common ratio is r = x.

Therefore, the geometric power series representation of f(x) is:

f(x) = [tex]1/4 + x + x^2 + x^3 + ...[/tex]

(b) By long division (Give the first three terms):

To find the geometric power series representation of f(x) = 1/4 + x using long division, we divide 1 by 1 - x.

          1/4 + x

   -----------------

   1 - x | 1

We divide 1 by 1 - x as follows:

             1/4 + x

   ------------------

   1 - x | 1

         - (1 - x)        (subtracting)

   -------------

          x / (1 - x)     (dividing)

        [tex]- (x - x^2)[/tex]

   -------------

             [tex]x^2 / (1 - x)[/tex]

We can continue this process indefinitely, but let's stop at the third term:

f(x) = [tex]1/4 + x + x^2[/tex]

Therefore, the geometric power series representation of f(x) using long division is:

f(x) =[tex]1/4 + x + x^2 + ...[/tex] (infinite terms)

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(a) The mean swipe rate for New York's MetroCard system is 27.5 inches per second.

(b) The standard deviation of the swipe rate is 12.99 inches per second.

(c) The first quartile (25th percentile) is 16.25 inches per second, and the third quartile (75th percentile) is 38.75 inches per second.

(a) The mean swipe rate can be calculated by taking the average of the minimum and maximum values of the uniform distribution: (5 + 50) / 2 = 27.5 inches per second.

(b) The standard deviation of a uniform distribution can be calculated using the following formula: (b - a) / sqrt(12), where a is the lower limit and b is the upper limit of the distribution. In this case, the standard deviation is (50 - 5) / sqrt(12) ≈ 12.99 inches per second.

(c) The quartiles divide the distribution into four equal parts. Since the distribution is uniform, the first quartile occurs 25% of the way through the range, and the third quartile occurs 75% of the way through the range. Therefore, the first quartile is 25% of the way from 5 to 50, which is 16.25 inches per second, and the third quartile is 75% of the way, which is 38.75 inches per second.

(d) To calculate the percentage of subway riders who must re-swipe the card due to being outside the acceptable range, we calculate the proportion of the uniform distribution that falls below 10 inches per second or above 40 inches per second. The range of the acceptable swipe rates is 40 - 10 = 30 inches per second. The proportion of the distribution outside this range is (50 - 40 + 10 - 5) / (50 - 5) = 0.15, or 15%. Therefore, approximately 15% of subway riders must re-swipe the card because their swipe rate is outside the acceptable range.

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The optimal values of x₁ and x₂ in the given linear programming problem are x₁ = 6 and x₂ = 5.

The optimal values of x₁ and x₂ are determined in a linear programming problem by maximizing or minimizing the objective function while satisfying the given constraints. This can be achieved through various optimization techniques, such as graphical methods, simplex algorithm, or other optimization algorithms.

In this specific problem, the objective is to maximize the objective function z = 3x₁ + 4x₂. The constraints x₁ ≤ 2x₂ ≤ 16

2x₁ + 3x₂ ≤ 18

x₁ ≥ 2 and x₂ ≤ 10 define the feasible region. By graphically plotting the constraints and identifying the corner points of the feasible region, we can determine the optimal values of x₁ and x₂ that maximize the objective function.

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a. The expected value is 14 days, the variance is 196 days^2, and the standard deviation is 14 days.

b. The probability that the number of days ahead a traveler will purchase an airline ticket is between 10 and 16 days is approximately 0.345 or 34.5%.

c. The probability that the number of days the traveler purchases the ticket is more than 9 days is approximately 0.591 or 59.1%.

Since the square root of variance is regarded as the standard deviation for the specified data set, variance and standard deviation have a relationship in statistics. The terms variance and standard deviation are defined here.

(a) To find the expected value, variance, and standard deviation of the number of days ahead of time a traveler will purchase an airline ticket, given that it follows an exponential distribution with λ = 1/14, we can use the following formulas:

Expected value (mean): E(X) = 1/λ

Variance: Var(X) = 1/λ²

Standard deviation: SD(X) = √Var(X)

Given λ = 1/14, we can calculate:

Expected value: E(X) = 1 / (1/14) = 14 days

Variance: Var(X) = 1 / (1/14)² = 196 days²

Standard deviation: SD(X) = √Var(X) = √196 = 14 days

Therefore, the expected value is 14 days, the variance is 196 days^2, and the standard deviation is 14 days.

(b) To find the probability that the number of days ahead a traveler will purchase an airline ticket is between 10 and 16 days, we can use the cumulative distribution function (CDF) of the exponential distribution.

P(10 ≤ X ≤ 16) = F(16) - F(10)

where F(x) is the CDF of the exponential distribution.

Using the formula for the CDF of the exponential distribution, we can calculate:

P(10 ≤ X ≤ 16) = [tex]e^{(-10/14)[/tex] - [tex]e^{(-16/14)[/tex] ≈ 0.345

Therefore, the probability that the number of days ahead a traveler will purchase an airline ticket is between 10 and 16 days is approximately 0.345 or 34.5%.

(c) Given that the number of days ahead of time a traveler will purchase an airline ticket is more than 7 days, we need to find the probability that the number of days the traveler purchases the ticket is more than 9 days.

Using conditional probability notation, we want to find P(X > 9 | X > 7).

P(X > 9 | X > 7) = P(X > 9 and X > 7) / P(X > 7)

Since X follows an exponential distribution, the exponential distribution is memoryless, meaning that P(X > a + b | X > a) = P(X > b) for any a, b > 0.

Therefore, P(X > 9 | X > 7) = P(X > 9) / P(X > 7)

Using the formula for the survival function (1 - CDF) of the exponential distribution, we can calculate:

P(X > 9) = 1 - F(9) = [tex]e^{(-9/14)[/tex]

P(X > 7) = 1 - F(7) = [tex]e^{(-7/14)[/tex]

So, P(X > 9 | X > 7) = [tex](e^{(-9/14))[/tex] / [tex](e^{(-7/14)[/tex]) ≈ 0.591

Therefore, given that the number of days ahead of time a traveler will purchase an airline ticket is more than 7 days, the probability that the number of days the traveler purchases the ticket is more than 9 days is approximately 0.591 or 59.1%.

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The predicted number of popcorn pieces a participant will eat if it has been 20 days since their last diet is approximately 25.78 pieces.

To predict the number of popcorn pieces a participant will eat based on the number of days since their last diet, we can use the given correlation and data. Let's solve each problem separately:

   To predict the number of popcorn pieces if it has been 10 days since the participant's last diet, we will use the formula for a simple linear regression:

Ŷ = a + bX

where Ŷ is the predicted number of popcorn pieces, X is the number of days since the participant's last diet, a is the intercept, and b is the slope of the regression line.

Given the correlation (r = 0.60), we can calculate the slope (b) using the formula:

b = r * (S_y / S_x)

where S_y is the standard deviation of Y (the number of popcorn pieces) and S_x is the standard deviation of X (the number of days since the last diet).

Plugging in the values:

r = 0.60

S_y = 5

S_x = 50

b = 0.60 * (5 / 50) = 0.06

To find the intercept (a), we can use the formula:

a = mean(Y) - b * mean(X)

mean(Y) = 25 (given)

mean(X) = 7 (given)

a = 25 - 0.06 * 7 = 24.58

Now we can substitute X = 10 into the regression equation to predict the number of popcorn pieces:

Ŷ = 24.58 + 0.06 * 10 = 25.18

Therefore, the predicted number of popcorn pieces a participant will eat if it has been 10 days since their last diet is approximately 25.18 pieces.

To compute the standard error of the estimate, we can use the formula:

SE = S_y * sqrt(1 - r^2)

Plugging in the values:

S_y = 5

r = 0.60

SE = 5 * sqrt(1 - 0.60^2) = 3.06

So, the standard error of the estimate is approximately 3.06.

   To predict the number of popcorn pieces if it has been 20 days since the participant's last diet, we can use the same regression equation:

Ŷ = 24.58 + 0.06 * X

Substituting X = 20 into the equation:

Ŷ = 24.58 + 0.06 * 20 = 25.78

Therefore, the predicted number of popcorn pieces a participant will eat if it has been 20 days since their last diet is approximately 25.78 pieces.

Regarding the standard error of the estimate, it will be the same as in the previous question (3.06) because the standard error is a measure of the overall precision of the regression line, and it does not depend on the specific value of X being predicted.

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The area of the plane figure is,

⇒ Area = 263.24 m²

We have to given that;

A trapezoid is shown in figure.

Now, We have to given that;

Upper base = 11.8 cm

Height = 16.1 m

Hence, By Pythagoras theorem, we get;

In side triangle,

⇒ Length of base = √18.5² - 16.1²

⇒ Length of base = √342.3 - 259.2

⇒ Length of base = √83.09

⇒ Length of base = 9.1

Hence, Lower base of trapezoid is,

⇒ (11.8 + 9.1)

⇒ 20.9

So, Area of trapezoid is,

⇒ A = (11.8 + 20.9) × 16.1 / 2

⇒ A = 526.47 / 2

⇒ A = 263.24 m²

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The increase in cost from a production level of 0 bikes per month to 600 bikes per month is $240,000.

To compute the increase in cost going from a production level of 0 bikes per month to 600 bikes per month, we need to integrate the marginal cost function over the interval [0, 600].

The marginal cost function is given as:

C'(x) = 600 - x/3

To find the cost function C(x), we need to integrate C'(x) with respect to x:

C(x) = ∫ (600 - x/3) dx

Evaluating the integral, we get:

C(x) = 600x - (1/6)x^2 + C

Now, to find the increase in cost, we need to evaluate C(600) - C(0):

∆C = C(600) - C(0)

= (600(600) - (1/6)(600^2)) - (600(0) - (1/6)(0^2))

= (360000 - 120000) - (0 - 0)

= 240000

Therefore, the increase in cost from a production level of 0 bikes per month to 600 bikes per month is $240,000.

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False.

As an airplane flies toward you at a constant altitude horizontal to the ground, its angle does not increase. The angle between the airplane and the observer remains the same throughout the flight.

Initially, when the airplane is far away, its position can be represented as (x, h), where h represents the altitude. As the airplane moves closer to the observer, its x-coordinate decreases while the altitude remains constant.

We can visualize this by drawing a right triangle where the hypotenuse represents the line of sight from the observer to the airplane, the base represents the horizontal distance (x-coordinate), and the altitude represents the height (h).

Therefore, as an airplane flies towards you at a constant altitude horizontal to the ground, its angle does not increase.

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(a) To find the partial derivatives, we differentiate the function f(x, y) with respect to x and y while treating the other variable as a constant.

fx(x, y) = d/dx [ (4x^2 + 2xy + 4y^2) / (x^2 + y^2) ]

          = [(8x(x^2 + y^2) - (4x^2 + 2xy + 4y^2)(2x)) / (x^2 + y^2)^2]

          = [(8x^3 + 8xy^2 - 8x^3 - 4x^2y - 8xy^2) / (x^2 + y^2)^2]

          = [-4x^2y / (x^2 + y^2)^2]

fy(x, y) = d/dy [ (4x^2 + 2xy + 4y^2) / (x^2 + y^2) ]

          = [(8y(x^2 + y^2) - (4x^2 + 2xy + 4y^2)(2y)) / (x^2 + y^2)^2]

          = [(8xy^2 + 8y^3 - 8xy^2 - 4x^2y - 8y^3) / (x^2 + y^2)^2]

          = [-4x^2y / (x^2 + y^2)^2]

(b) To find fx(0, 0) and fy(0, 0), we substitute x = 0 and y = 0 into the partial derivative expressions:

fx(0, 0) = -4(0)^2(0) / ((0)^2 + (0)^2)^2 = 0

fy(0, 0) = -4(0)^2(0) / ((0)^2 + (0)^2)^2 = 0

(c) Both partial derivatives, fx(x, y) and fy(x, y), are continuous at every point in the set {(x, y): (x, y) ≠ (0, 0)}.

However, at the point (0, 0), the partial derivatives fx(0, 0) and fy(0, 0) are both 0, indicating that the partial derivatives are continuous at that point as well.

Therefore, the correct answer is (a) Yes, both the partial derivatives are continuous at every point in that set.

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Answer:

the area of Figure 6 is Figure 36

Step-by-step explanation:

using formula,

A=[tex]a^{2}[/tex] (a = sides of the square)

  =[tex]6^{2}[/tex]cm

  =36cm.

The equivalent expression would now be given as [tex]4x^3y^4[/tex]. Option A

Regardless of the particular values given to the variables, equivalent expressions in mathematics have the same value. In other words, they stand for the same quantity or mathematical relationship.

We would now just apply the mathematical principle that we need by taking the cube root of all the terms that we have in the expression as shown.

We have the expression;

[tex]\sqrt[3]{} 64x^9y^{12}[/tex]

This would now be;

[tex]4x^3y^4[/tex]

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The constraint that ensures options (Xo) cannot exceed cash (Xc) and stocks (Xs) combined is  Xo <= Xc + Xs. The correct option is (2).

This constraint states that the amount invested in options (Xo) should be less than or equal to the sum of the amounts invested in cash (Xc) and stocks (Xs).

This ensures that the investment in options does not exceed the combined investment in cash and stocks.

Options are subject to this constraint because the problem statement specifies that options cannot exceed stocks and bonds combined.

By including cash in the constraint, it ensures that options are limited to the remaining amount after accounting for cash and stocks.

To summarize, the constraint Xo <= Xc + Xs ensures that options cannot exceed the combined investment in cash and stocks.

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Answer:

A. The domain is al real numbers

Step-by-step explanation

The domain of a function is the set of all values of the input which yield real and defined values for the function

f(d) has no restrictions on the input d. d can range from -∞ to +∞ which is the set of all real numbers

There are 9 different propositional variable assignments (models) for this logic that satisfy the given sentence.

What is Propositional logic?

The study of propositions and the logical connections between them is the focus of propositional logic, often referred to as sentential logic or propositional calculus. The manipulation and assessment of propositions, which are declarative statements that can either be true or wrong, are the main topics of this study. Using logical operators like conjunction (AND), disjunction (OR), negation (NOT), implication (IF-THEN), and biconditional (IF AND ONLY IF), propositions are combined in propositional logic. These logical operators make it possible to construct intricate logical expressions and analyse the truth values of those expressions depending on the truth values of the propositions that make them up. For inference and reasoning in a variety of fields, including mathematics, computer science, philosophy, and artificial intelligence, propositional logic serves as a solid foundation.

In a logic with four propositional variables (A, B, C, and D), we can take into account all potential assignments of truth values to these variables and assess the sentence for each assignment to get the number of models that meet the phrase (A B) (C D).

Since there are four variables, each one has a true or false truth value that it can take. There are therefore a total of 16 possible assignments, or 24.

We can make a list of all possible assignments and determine which ones meet the criteria

A | B | C | D | (A ∧ B) ∨ (C ∧ D)

[tex]T | T | T | T | TT | T | T | F | T\\T | T | F | T | T\\T | T | F | F | F\\T | F | T | T | T\\T | F | T | F | T\\T | F | F | T | F\\T | F | F | F | F\\F | T | T | T | T\\F | T | T | F | T\\F | T | F | T | F\\F | T | F | F | F\\F | F | T | T | T\\F | F | T | F | T\\F | F | F | T | F\\F | F | F | F | F[/tex]

From the table, we can see that 9 out of the 16 possible assignments satisfy the sentence (A ∧ B) ∨ (C ∧ D).

Therefore, there are 9 different propositional variable assignments (models) for this logic that satisfy the given sentence.

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The graph of the polar equation r = 9 over the interval [0, 2π] is a complete circle with radius 9 centered at the origin.

In polar coordinates, the equation r = 9 represents a circle with a constant radius of 9. The angle θ varies from 0 to 2π, which covers one complete revolution around the origin.

Using a graphing utility,  plot the polar equation r = 9 over the interval [0, 2π]. The resulting graph will show a circular shape centered at the origin with a radius of 9 units. As θ increases from 0 to 2π, the graph completes one full revolution, tracing out the entire circle.

The graph of the polar equation r = 9 can help visualize the circular shape and size of the curve in polar coordinates. It provides a geometric representation of the equation and its corresponding points in the polar plane.

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The closing costs and actual amount financed with the mortgage are $4,500 and $154,500 respectively.

Closing costs are fees and taxes associated with buying and selling a home. They can include things like title insurance, appraisal fees, and origination fees. In this case, the Taylors will be financing the closing costs as part of their mortgage. This means that the total amount they will be borrowing will be $150,000 plus the closing costs.

The total amount financed with the mortgage is $150,000 + $4,500 = $154,500.

The closing costs are $4,500.

The actual amount financed with the mortgage is $154,500.

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The "conservative" degrees of freedom refers to using a larger value for degrees of freedom in order to be more cautious and ensure that the test is not too liberal (i.e. too likely to detect a difference when there isn't one).

Specifically, for the two-sample t statistic with sample sizes n1 and n2, the "conservative" degrees of freedom is calculated as the smaller of n1-1 and n2-1, or 12-1 if that value is larger. This is done to account for the fact that the t distribution becomes more normal (and thus the standard error of the mean becomes more reliable) as the sample size increases, so we can be more confident in the results with larger sample sizes.

However, if the sample sizes are small, using the smaller of n1-1 and n2-1 as the degrees of freedom is still appropriate.

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The solution of the algebraic equation is x = -2.5. It is not extraneous.

An algebraic equation is when two expressions are set equal to each other, and at least one variable is included.

An extraneous solution is a solution that is not true for a particular algebraic equation.

Let's solve:

(6x + 4) / (x +8) = -2

6x + 4 = -2(x+8)

6x + 4 = -2x - 16

6x + 2x = -16 - 4

8x = -20

x = -20/8

x = -2.5

Let check if the solution is extraneous or not by substituting x = -2.5 into the equation.

If we get -2, it is not extraneous. Otherwise, it is extraneous (because it is not a true solution).

(6x + 4) / (x +8) = -2

(6*(-2.5) + 4) / (-2.5 +8) = -11/5.5

                                    = -2

Therefore, the solution is not extraneous.

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[tex]\textit{Logarithm of rationals} \\\\ \log_a\left( \frac{x}{y}\right)\implies \log_a(x)-\log_a(y) \\\\[-0.35em] ~\dotfill\\\\ \log(8)-\log(2)\implies \log\left( \cfrac{8}{2} \right)\implies \log(4)[/tex]