In the 1st law of thermodynamics for a CV, the W cv term includes all forms of power (rate of work) done on or by the CV EXCEPT flow work. True False

Answer:

∠ A = 98°

Step-by-step explanation:

ABCD is a cyclic quadrilateral

its opposite angles sum to 180° , that is

∠ A + ∠ C = 180°

∠ A + 82° = 180° ( subtract 82° from both sides )

∠ A = 98°

The integral ∫ ln(x^17)/x^2 dx converges, and its value is given by -ln(x^17)/x + 17/x + C.Therefore, the integral converges.

To determine if the integral ∫ ln(x^17)/x^2 dx converges or diverges, we need to use the integral test. The integral test states that if f(x) is a continuous, positive, and decreasing function on the interval [a, infinity), then the infinite series ∑f(n) and the improper integral ∫f(x) dx both converge or both diverge.

In this case, we have f(x) = ln(x^17)/x^2, which is continuous, positive, and decreasing on the interval [1, infinity). Therefore, we can use the integral test to determine if the improper integral ∫ ln(x^17)/x^2 dx converges or diverges.

To evaluate the integral, we can use integration by parts with u = ln(x^17) and dv = 1/x^2 dx, which gives us:

∫ ln(x^17)/x^2 dx = -ln(x^17)/x - ∫ -17/x^3 dx
= -ln(x^17)/x + 17/x + C

where C is the constant of integration.

Now, to determine if the integral converges or diverges, we need to evaluate the limit as x approaches infinity of the integrand. We have:

lim x→∞ [ln(x^17)/x^2] = lim x→∞ [(17/x) ln(x)]/x
= lim x→∞ (17/x) * lim x→∞ ln(x)/x

Since lim x→∞ ln(x)/x = 0 (which can be shown using L'Hopital's rule), the limit of the integrand is 0.

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The points where tangents are horizontal are only at point A (0,-40). The answer is only at point A and the word count is 307.

In order to find the points where tangents are horizontal, we need to first understand what a tangent is. A tangent is a straight line that touches a curve at only one point. When we say that a tangent is horizontal, it means that it is parallel to the x-axis or has a slope of 0.
To find the points where tangents are horizontal, we need to differentiate the given function and set it equal to 0. This is because the slope of the tangent at a point is given by the derivative of the function at that point. When the derivative is 0, the slope of the tangent is also 0, which means it is a horizontal line.
Let's differentiate the given function:
f(x) = -5x^4 + 30x^2 - 40
f'(x) = -20x^3 + 60x
Now, we set f'(x) = 0 to find the critical points where the slope is 0:
-20x^3 + 60x = 0
-20x(x^2 - 3) = 0
x = 0, √3, -√3
These are the critical points where the slope is 0. To determine whether the tangents at these points are horizontal, we need to look at the second derivative:
f''(x) = -60x^2 + 60
At x = 0, f''(x) = 60 > 0, which means the point (0,-40) is a minimum point and the tangent is horizontal.
At x = √3 and x = -√3, f''(x) = -60 < 0, which means the points (±√3, -28) are maximum points and the tangents are not horizontal.

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The calculated value of the area of the figure is 89 sq meters

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

Composite figure

The shapes in the composite figure are

This means that

Area = Rectangle - Triangle

Using the area formulas on the dimensions of the individual figures, we have

Area = 13 * 8 - 1/2 * (13 - 8) * 6

Evaluate

Area = 89

Hence, the area of the figure  is 89 sq meters

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The assessed value of the property with a market value of $60,000 and an assessment rate of 45% is $27,000. T

To compute the assessed value of a property, you'll need to consider both the market value and the assessment rate. In this case, the market value is $60,000, and the assessment rate is 45%.

To find the assessed value, you can multiply the market value by the assessment rate. Using the provided values, you can calculate the assessed value as follows:

Assessed Value = Market Value x Assessment Rate
Assessed Value = $60,000 x 0.45

Assessed Value = $27,000

In this case, the assessed value of the property with a market value of $60,000 and an assessment rate of 45% is $27,000. This value represents the taxable amount on which property taxes will be calculated and is an essential factor for both property owners and tax authorities.

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A rose garden is formed by rectangular and semi-circular parts. If the gardener wants to build a fence around the garden, then total 139.39 feet of fence are required.

The perimeter is defined as calculating the outer length of boundaries of shape.

We have a rose garden is formed by joining a rectangle and a semicircle, as present in above figure. We have to determine the feet of fence are required to build a fence around the garden.

From the above figure, length of rectangular part, l = 35 ft

Width of rectangular part, w = 27 ft.

Also, diameter of semi-circular part, d

= 27 ft

Radius of of semi-circular part, r = d/2

= 27/2 ft = 13.5 ft

So, the perimeter of semi-circular part, Pₛ = π × r = π × 13.5 ft

= 42.39 ft.

Here, the fence required for the rectangle shape is three sides that two long sides and one wide side. The fourth side of the width is already covered by the semi-circular part. So, the perimeter formula for the rectangle shape, Pᵣ = 2l + w. Therefore, perimeter of garden

= Pₛ + Pᵣ

= 42.39 ft + 2 × 35 ft + 27 ft

= 70 ft + 27 ft + 42.39 ft

= 139.39 ft.

Hence, required value is 139.39 feet.

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The price-demand equation is [tex]p(x) = 77.88^(x/5)[/tex] rounded to two decimal places.

We know that the marginal price dp/dx at x units of demand per week is proportional to the price p. Let k be the constant of proportionality. Then, we have:

dp/dx = kp

Separating variables and integrating, we get:

∫dp/p = ∫kdx

ln|p| = kx + C

Taking exponential on both sides and using the initial condition p(0) = 100, we get:

p(x) = 100e^(kx)

Using the second condition, p(5) = 77.88, we get:

[tex]77.88 = 100e^(5k)[/tex]

Solving for k, we get:

k = ln(77.88/100)/5

Substituting this value of k in the price-demand equation, we get:

[tex]p(x) = 100e^[(ln(77.88/100)/5)x] = 100(77.88/100)^(x/5)[/tex]

Therefore, the price-demand equation is [tex]p(x) = 77.88^(x/5)[/tex] rounded to two decimal places.

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(a) The contents of the ROM are:

X = A'B'.0 + AB.1 + 0.0 + BC'D.1

(b) The contents of the PLA using dot notation as follows:

Y = A0.B0.Y0 + A1.B1.Y1

/Y = A0.B0./Y0 + A0.B1./Y1 + A1.B0./Y0 + A1.B1./Y1

Z = A0.B0.Z0 + A1.B0.Z0 + A0.B1.Z1 + A1.B1.Z1

(c) The contents of the PLA using dot notation as follows:

Z = A0.B0.Z0 + A1.B0.Z0 + A0.B1.Z1 + A1.B1.Z1

(a) Implementing the function X = AB + BC'D + A'B' using a single 16 x 3 ROM:

We can use the formula for X to determine the output values:

X(00C') = A'B'

X(01C') = AB

X(10C') = 0

X(11C') = BC'D

Therefore, the contents of the ROM can be represented using dot notation as follows:

X = A'B'.0 + AB.1 + 0.0 + BC'D.1

(b) Implementing the function Y = AB + BD using a 4 x 8 x 3 PLA:

We can assign the product terms as follows:

Y0 = AB

Y1 = BD

/Y0 = A'B' + A'D + B'C

/Y1 = A'C' + B'C' + BC

Z0 = A + B

Z1 = C + D

Then, we can assign the connections as follows:

A0 = Y0 + /Y0 + Z0

A1 = Y1 + /Y1 + Z0

B0 = Y0 + /Y0 + Z0

B1 = /Y1 + Z1

C0 = /Y0 + Z1

C1 = /Y1 + Z1

D0 = Z0

D1 = Y1 + /Y1 + Z1

Finally, we can represent the contents of the PLA using dot notation as follows:

Y = A0.B0.Y0 + A1.B1.Y1

/Y = A0.B0./Y0 + A0.B1./Y1 + A1.B0./Y0 + A1.B1./Y1

Z = A0.B0.Z0 + A1.B0.Z0 + A0.B1.Z1 + A1.B1.Z1

(c) Implementing the function Z = A + B + C + D using a 4 x 8 x 3 PLA:

We can assign the product terms as follows:

Z0 = A + B + C + D

Z1 = /Z0

Then, we can assign the connections as follows:

A0 = Z0

A1 = Z1

B0 = Z0

B1 = Z1

C0 = Z0

C1 = Z1

D0 = Z0

D1 = Z1

Finally, we can represent the contents of the PLA using dot notation as follows:

Z = A0.B0.Z0 + A1.B0.Z0 + A0.B1.Z1 + A1.B1.Z1

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24 is the sum of the left leaves (9 and 15) in the binary tree.

Here's the implementation of the sum_of_left_leaves function in Python:

class TreeNode:

def __init__(self, x):

self.val = x

self.left = None

self.right = None

def sum_of_left_leaves(root):

if not root:

return 0

elif root.left and not root.left.left and not root.left.right:

# The current node has a left child that is a leaf node

return root.left.val + sum_of_left_leaves(root.right)

else:

# Recursively sum up the left leaves of the left and right subtrees

return: sum_of_left_leaves(root.left) + sum_of_left_leaves(root.right)

It implementation uses recursion to traverse the binary tree and add up the values of all the left leaves. The base case is when the current node is None, in that case we return 0.

If the current node has a left child which is a leaf node .

We are adding its value to the sum and recursively call the function on the right subtree.

In other word we can say that we recursively call the function on both the right and left subtrees and sum up their results.

For use this function with the example result, you can create the binary tree like this:

# Input: 3 9 20 15 7

root = TreeNode(3)

root.left = TreeNode(9)

root.right = TreeNode(20)

root.right.left = TreeNode(15)

root.right.right = TreeNode(7)

# Output: 24

print(sum_of_left_leaves(root))

This will output 24, which is the sum of the left leaves (9 and 15) in the binary tree.

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Correct question is " Sum of Left Leaves in a Binary Tree Given a non-empty binary tree, return the sum of all left leaves. Example: Input: 3 9 20 15 7 Output: 24 Explanations summing up every Left leaf in the tree gives us: 9 + 15 = 24 -1 -2 -3 -4 class TreeNode: def __init__(self, x): self.val = x self.left = self.right = None 5 def sum_of_left_leaves (root): -6 7 18 19 50 51 2 13 Write your code here :type root: TreeNode :rtype: int 11 001 84 15 > root = input_binary_tree"

The H.C.F of the polynomials x⁸ - y⁸, (x⁴ - y⁴)(x + y) is (x - y)(x + y)

The H.C.F - Highest Common Factor of two or more numbers is the highest of their common factors.

Given the polynomials x⁸ - y⁸, (x⁴ - y⁴)(x + y). To find their highest common factors, we proceed as follows.

Now, the polynomial x⁸ - y⁸ = (x⁴)² - (y⁴)²

= (x⁴ - y⁴)(x⁴ + y⁴) difference of two squares.

Also,

(x⁴ - y⁴) = (x²)² - (y²)²

= (x² - y²)(x² + y²) difference of two squares.

= (x² - y²)(x² + y²)

So,

x⁸ - y⁸ = (x⁴ - y⁴)(x⁴ + y⁴)

= (x² - y²)(x² + y²)(x⁴ + y⁴)

Also,  (x² - y²) = (x - y)(x + y)

So, substituting this into the quation, we have that

x⁸ - y⁸ =  (x² - y²)(x² + y²)(x⁴ + y⁴)

=   (x - y)(x + y)(x² + y²)(x⁴ + y⁴)

Now, the polynomial

(x⁴ - y⁴)(x + y) =  (x² - y²)(x² + y²)(x + y)

= (x - y)(x + y)(x² + y²)(x + y)

So, comparing the factors of x⁸ - y⁸ and (x⁴ - y⁴)(x + y), the H.C.F is (x - y)(x + y)

So, the H.C.F is (x - y)(x + y)

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An expression for 2n-1 is: 2n-1 = (2n+1 - 1)/(2n(2n-1)), for n = 1, 2, ....  The general solution is: [tex]y(x) = c1exp(x^2) + c2exp(-2x^2)[/tex], where c1 and c2 are constants. The power series solution of the initial value problem is:[tex]y(x) = 4 - 6x - 12x^2 + 16x^3 - 16x^4 + 256/15 x^5 - 1024/96 x^6 + 2048/315 x^7 - 32768/460[/tex]

2.1 Using the recurrence relation, we can obtain an expression for 2n-1 as follows:

Cn+2 = (n+1)Cn+1 + 22Cn

For n=0, we have:

C2 = C1 + 22C0

Substituting C1 = 1.3.5 and C0 = c, we get:

C2 = 1.3.5 + 22c

For n=1, we have:

C3 = 2C2 + 22C1

Substituting C2 = 1.3.5 + 22c and C1 = 1, we get:

C3 = 2(1.3.5 + 22c) + 22

Simplifying, we get:

C3 = 1.3.5.7 + 2.2.3.5c + 22

Comparing with the general expression for Cn+2, we get:

2n+1 - 1 = 2.2n(2n-1)cn

Solving for 2n-1, we get:

2n-1 = (2n+1 - 1)/(2n(2n-1))

Hence, an expression for 2n-1 is:

2n-1 = (2n+1 - 1)/(2n(2n-1)), for n = 1, 2, ...

2.2 The general solution of the differential equation y" - 2xy' - 4y = 0 can be written as a linear combination of the two linearly independent solutions:

[tex]y1(x) = c1exp(x^2)\\y2(x) = c2exp(-2x^2)[/tex]

Hence, the general solution is:

[tex]y(x) = c1exp(x^2) + c2exp(-2x^2)[/tex]

where c1 and c2 are constants.

2.3 To find the power series solution of the initial value problem y" - 2xy' - 4y = 0, y(0) = 4, y'(0) = -6, we first need to find the coefficients of the power series solution y(x).

Substituting y = Σn=0∞ anxn into the differential equation, we get:

Σn=0∞ [(n+2)(n+1)an+2 - 2n an - 4an]xn = 0

Equating the coefficients of xn, we get:

(n+2)(n+1)an+2 - 2n an - 4an = 0

Simplifying, we get:

an+2 = (2n/(n+2))an

Using the initial conditions y(0) = 4 and y'(0) = -6, we get:

a0 = 4

a1 = -6

Substituting the recurrence relation, we get:

a2 = -12

a3 = 48/3 = 16

a4 = -128/8 = -16

a5 = 256/15

a6 = -1024/96

a7 = 2048/315

a8 = -32768/4608

Hence, the power series solution of the initial value problem is:

[tex]y(x) = 4 - 6x - 12x^2 + 16x^3 - 16x^4 + 256/15 x^5 - 1024/96 x^6 + 2048/315 x^7 - 32768/460[/tex]

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a. The mean of the binomial distribution is  4.7

b. The variance of the binomial distribution is 1.1

c. The standard deviation of the binomial distribution is 1.0

Probability of success p = 0.78

Number of trials n = 6

(a) Mean of the binomial distribution:

The mean or expected value of a binomial distribution is given by np.

Therefore, the mean of this binomial distribution is:

μ = np = 6 x 0.78 = 4.68

So, the mean is 4.7 (rounded to one decimal place).

(b) Variance of the binomial distribution:

The variance of a binomial distribution is given by np(1 - p).

Therefore, the variance of this binomial distribution is:

σ^2 = np(1 - p) = 6 x 0.78 x 0.22 = 1.0608

So, the variance is 1.1 (rounded to one decimal place).

(c) Standard deviation of the binomial distribution:

The standard deviation of a binomial distribution is the square root of the variance.

Therefore, the standard deviation of this binomial distribution is:

σ = √(np(1 - p)) = √(1.0608) = 1.03

So, the standard deviation is 1.0 (rounded to one decimal place).

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The probability sampling method in which you randomly select one of the first k elements and then select every k element thereafter is known as c. systematic sampling. Therefore, option c. systematic sampling is correct.

Systematic sampling is a probability sampling technique where the sample is chosen by selecting every kth element from the population, where k is a constant. This method is often used when the population is large and the complete list of elements is not easily available.

Stratified random sampling is a technique where the population is divided into strata or subgroups based on certain characteristics and a random sample is chosen from each stratum.

Cluster sampling involves dividing the population into clusters or groups and then selecting a random sample of clusters. The elements within each selected cluster are then included in the sample.

Convenience sampling is a non-probability sampling method where the sample is chosen based on convenience and availability. This method is often used in situations where it is difficult or expensive to obtain a random sample.

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A) The probability is approximately 0.257. B) The expected number of customers is 5, and the standard deviation is approximately 1.71.

A) The number of customers who qualify for the membership follows a binomial distribution with parameters n = 20 and p = 0.1. The probability of having between 2 and 5 (inclusive) customers who qualify for the membership can be calculated by summing the probabilities of having exactly 2, 3, 4, or 5 customers who qualify for the membership.

Using the binomial probability formula or a binomial probability table, we find that this probability is approximately 0.257.

B) The expected number of customers who qualify for the membership in a randomly selected sample of 50 customers can be calculated using the formula E(X) = np, where X is the number of customers who qualify for the membership.

Thus, E(X) = 50 × 0.1 = 5. The variance of X is given by Var(X) = np(1-p), so the standard deviation is given by the square root of the variance, which is approximately 1.71.

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The equation that can be used to find the number of quarters q in any number of rolls of quarters r is: q = 40r

This equation simply states that the total number of quarters is equal to the number of rolls multiplied by 40, since each roll contains 40 quarters.

An equation is a mathematical statement that expresses the equality of two expressions using an equal sign (=). It contains one or more variables (letters or symbols that represent an unknown value) and constants (numbers that have a fixed value), as well as mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Equations are used to solve problems and find solutions for unknown values by manipulating the expressions using algebraic rules and properties.

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for this question the required one is the distance b/n the light house and the boat (x) so in this case we are gonna use tan= opposite / hypotenuse :

- tan 20 = 89/ x (tan 20 is equivalent to 0.3640)

- 0.3640 = 89 / x

-x = 89 / 0.3640

- x = 244.5 ft. and when estimated x= 245 ft.

The value of y -x is 30°

A triangle is a three-sided polygon that consists of three edges and three vertices. The types of triangles include; right triangle, equilateral triangle, isosceles triangle , obtuse triangle e.t.c

A triangular theorem states that the sum of angle In a triangle is 180°

Therefore 3x = 180-90

3x =90

divide both side by 3

x = 90/3 = 30°

Therefore x+y+90 = 180

x+y =180 - 90

x+y = 90

y = 90-x

y = 90-30

y = 60°

Therefore the value of y-x will be ;

y-x = 60-30

= 30°

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The best instrument to determine the predictive validity of a metric scale would be a correlation-coefficient.

This measure assesses the strength of the relationship between two variables, in this case, the metric scale scores and the predicted outcome. A high correlation would indicate that the metric scale is a good predictor of the outcome, whereas a low correlation would indicate that the metric scale is not a reliable predictor.

Cronbach's alpha is a measure of internal consistency and would not be appropriate for determining predictive validity. Fisher's r-to-z test is used to compare the strength of two correlations and is not necessary in this scenario. Therefore, the answer is B, a correlation-coefficient.

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Answer:

Step-by-step explanation:

Add the numbers first. (-10 + 4)

f(x)=x^2-10+4;f(-2)

f(x)=x^2-6;f(-2)

Then re-order the terms so that the constants are on the left. f & (-2)

f(x)=x^2-6;f(-2)

f(x)=x^2-6;-2f

Your answer will then be f(x)=x^2-6;-2f.

Answer:

-10

Step-by-step explanation:

f(x) = f(-2)

(-2) × 2 - 10 + 4

= (-4) - 10 + 4

= (-14) + 4

= (-10)

The throw of the dice or the drawing of a card is known as random behavior.

Random behavior refers to events that occur by chance, without any predictability or pattern. These events cannot be influenced or controlled, and their outcomes are determined by the laws of probability. In games such as dice and card games, the outcomes are determined by random variables, such as the position of the dice or the shuffle of the deck.

Random behavior can also occur in other contexts, such as weather patterns or the stock market, where events are subject to unpredictable fluctuations. Understanding random behavior is important in many fields, including mathematics, statistics, and economics, as it can help us make informed decisions based on the probability of different outcomes.

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This is the alternative hypothesis. It is expressed as

H0 : μ < 250

A restaurant advertises that its burritos weigh 250 g. A consumer advocacy group doubts this claim, and they obtain a random sample of these burritos to test if the mean weight is significantly lower than 250 g. Let u be the mean weight of the burritos at this restaurant and ĉ be the mean weight of the burritos in the sample. Which of the following is an appropriate set of hypotheses for their significance test? Choose 1 answer:

A) H0 : x = 250 , Ha : x < 250

B) H0 : x = 250 , Ha : x > 250

C) H0 : μ = 250 , Ha: μ < 250

C) H0 : μ = 250 , Ha: μ > 250

Solution:

The null hypothesis is the hypothesis that is assumed to be true. The restaurant advertises that its burritos weigh 250. This is the null hypothesis. 250 is the population mean,μ . Thus, the null hypothesis is

H0 : μ = 250

The alternative hypothesis is what the researcher expects or predicts. The consumer advocacy group tests if the mean weight is significantly lower than 250g.

This is the alternative hypothesis. It is expressed as

H0 : μ < 250

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Answer:

20%

Step-by-step explanation:

1200 is 120% of 1000, therefor the profit percentage is 20.

Let's start by assigning variables to the unknowns in the problem. Let x be the number of one-dollar bills,

y be the number of five-dollar bills, and z be the number of ten-dollar bills. From the problem, we know that: x + y + z = 35 (since there are 35 bills in total) 1x + 5y + 10z = 144 (since the total amount of money is $144) z = y + 2

(since there are two more ten-dollar bills than five-dollar bills) Now we can substitute the third equation into the second equation: 1x + 5y + 10(y + 2) = 144 Simplifying: 1x + 15y + 20 = 144 1x + 15y = 124 We have two equations with two variables: x + y + z = 35 x + 15y = 124 Solving for x in the second equation: x = 124 - 15y

Substituting into the first equation: (124 - 15y) + y + (y + 2) = 35 126 - 13y = 35 -13y = -91 y = 7

Now we can find z: z = y + 2 = 9 And finally, we can find x: x = 124 - 15y = 124 - 15(7) = 19 Therefore, there are 19 one-dollar bills, 7 five-dollar bills, and 9 ten-dollar bills.

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The valid reasons to use a sample instead of evaluating a much larger population are contacting the whole population would be only marginally more accurate than a sample. A sample can be chosen to validate a specific idea about the population.

Contacting the entire population would be time consuming.
1. Contacting the whole population would be only marginally more accurate than a sample: This is a valid reason because a well-designed sample can often provide an accurate estimate of the population, while using significantly fewer resources.
2. Sampling is a random process, and therefore more accurate than measuring the whole population: This statement is incorrect. A random sample can provide an unbiased estimate, but it is not necessarily more accurate than measuring the entire population.
3. A sample can be chosen to validate a specific idea about the population: This is a valid reason because sampling can be used to test hypotheses or ideas about the population without having to evaluate every member of the population.
4. Contacting the entire population would be time-consuming: This is a valid reason because sampling can save time and resources compared to attempting to gather data from every individual in a population.

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The real variables among the given options are: Bushels of Wheat and Numbers of bees per hectare used to pollinate farm crops. The nominal values are: Price of a pen and Price of a computer.

Real variables are measurable quantities that can take on any value within a given range. In this case, "Bushels of Wheat" and "Numbers of bees per hectare used to pollinate farm crops" are real variables. These variables can be measured and expressed as continuous quantities, such as the amount of wheat harvested or the number of bees present.

On the other hand, nominal values are categories or labels that cannot be measured on a numerical scale. They represent different groups or classes. The "Price of a pen" and "Price of a computer" are nominal values as they represent different price categories or labels for pens and computers, but they do not have a numerical measurement scale associated with them.

In summary, "Bushels of Wheat" and "Numbers of bees per hectare used to pollinate farm crops" are real variables, while "Price of a pen" and "Price of a computer" are nominal values.

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X = 3. 5y is an equation that shows the number of teaspoons of sugar, y, needed to make x tarts where a recipe 3.5 teaspoons of sugar are added to make a tart.

An equation is a mathematical sentence where we equalize two expressions using an equal sign. An expression refers to a phrase with two or more variables or numbers with any mathematical operation.

The situation given is a recipe 3.5 teaspoons of sugar is required to make a tart. Thus to calculate the number of teaspoons of sugar needed to make tarts, we have to multiply 3.5 by the number of teaspoons of sugar to make tarts

Thus if x is the number of tarts

y is the number of teaspoons of sugar

The equation is given by x = 3.5y

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The equation of the tangent plane to the surface [tex]x^2yz[/tex] = 18 at the point (4, 1, 2) is: (8x-3y-16z) = (84 - 31 - 16*2)

8x - 3y - 16z = -21

The region $E$ is defined by the paraboloid [tex]$z = 24 - 2x^2 - 2y^2$[/tex]and the cone [tex]$z = 2\sqrt{x^2 + y^2}$[/tex]. To find the volume of this region, we integrate over [tex]$E$[/tex]using cylindrical coordinates:

[tex]\iiint_E dV &= \int_{0}^{2\pi} \int_{0}^{\sqrt{2}} \int_{2r^2}^{24-2r^2} r , dz , dr , d\theta \[/tex]

[tex]&= \int_{0}^{2\pi} \int_{0}^{\sqrt{2}} (22r^3 - r^5) , dr , d\theta \&= \frac{64}{3} \pi.\end{align*}[/tex]

Therefore, the volume of the region[tex]$E$ is $\frac{64}{3} \pi$[/tex]

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Check the picture below.

[tex]\cfrac{1}{6}+\cfrac{1}{4}\implies \cfrac{(2)1+(3)1}{\underset{\textit{using this LCD}}{12}}\implies \cfrac{5}{12}~\hfill {\Large \begin{array}{llll} > \end{array}} ~\hfill \cfrac{1}{6}+\cfrac{1}{6}\implies \cfrac{2}{6}\implies \cfrac{1}{3}[/tex]

notice, 1/4 is really larger than 1/6 of the same whole.

There are 432 different meal choices that a customer can make if a meal includes a sandwich, chicken wings, and a drink.

To calculate the total number of meal choices, you can multiply the number of options for each item:

Number of sandwich options = 8

Number of chicken wing options = 9

Number of drink options = 6

Total number of meal choices = number of sandwich options x number of chicken wing options x number of drink options

Total number of meal choices = 8 x 9 x 6 = 432

Therefore, there are 432 different meal choices that a customer can make if a meal includes a sandwich, chicken wings, and a drink.

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