The y-intercept of a function always occurs where y is equal to zero true or false

Angle PBC is 126 degrees.

Let's start by drawing the triangle ABC and marking the point P on AC such that APB = ABP.

Since APB = ABP, we can conclude that the triangle ABP is an isosceles triangle, which means that angles ABP and BAP are equal.

Let's call angle ABP and angle BAP x, then we have:

angle ABC = 2x (since triangle ABP is isosceles)

angle ACB = 2x - 39 (from the given information)

Since the sum of angles in a triangle is 180 degrees, we can write:

angle PBC + angle ABC + angle ACB = 180

Substituting the values we found for angle ABC and angle ACB, we get:

angle PBC + 2x + (2x - 39) = 180

Simplifying the equation, we get:

angle PBC = 219 - 4x

We still need to find the value of x to calculate angle PBC. To do that, let's use the fact that the sum of angles in a triangle is 180 degrees for triangle ABP:

angle ABP + angle BAP + angle APB = 180

Substituting x for angle ABP and BAP, we get:

2x + angle APB = 180

But we also know that angle APB = ABP (from the given information), so we can substitute:

2x + ABP = 180

Solving for ABP, we get:

ABP = 90 - x

Now we can substitute this value for ABP in our equation for angle PBC:

angle PBC = 219 - 4x

= 219 - 4(90 - ABP)

= 219 - 4(90 - (90 - x))

= 219 - 4x

Simplifying, we get:

angle PBC = 3x - 9

So to find the value of angle PBC, we need to find the value of x:

2x + ABP = 180

2x + (90 - x) = 180

x = 45

Now we can substitute x = 45 in our equation for angle PBC:

angle PBC = 3x - 9

= 3(45) - 9

= 126

Therefore, angle PBC is 126 degrees.

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Full Question ;

Point P is on side AC of the triangle ABC such that APB = ABP and ABC - ACB = 39 find PBC in degrees.

How much does it take to cover the cylinder? is different from the rest, as it refers to the surface area of the cylinder, not its volume or capacity.

The term "cover" usually means to place something over the top or on the surface of something else, such as a lid covering a container.

In the context of a cylinder, "covering" would typically refer to finding the surface area of the cylinder, which includes both the top and bottom circles as well as the curved lateral surface.

In contrast, the other statements are related to the volume or capacity of the cylinder, which refers to how much space is contained inside the cylinder.

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The weight of the ice after 10 hours is 1217 lb

A 523 lb mass of ice melts at 4.3 hours

The first step is to calculate the weight after one hour

523= 4.3

x= 1

cross multiply both sides

4.3x= 523

x= 523/4.3

x= 121.7

The weight after 10 hours can be calculated as follows

121.7= 1

y= 10

y= 121.7 × 10

y= 1217

Hence the weight after 10 hours is 1217 lb

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(a) The probability that all 4 missiles will fire is 0.2098 and (b) The probability that at most 2 will not fire is 0.0099.

Here we need to use the concept of combinations to get our required answer.

Here we have been given that 4 out of 14 missiles are defective.

Hence 10 missiles are working.

A sample of 4 missiles was chosen

Hence the simple can be chosen in ¹⁴C₄ ways

a)

The probability that all 4 missiles will fire is

¹⁰C₄/¹⁴C₄

= 0.2098

b)

The probability that at most 2 will not fire is

1 - the probability of including all three defective missiles

= 10³C₃/¹⁴C₄

= 0.0099

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(a) Probability all 4 fire: 0.3297. (b) Probability at most 2 not fire: 0.9591.

(a) To track down the chance that all of the 4 rockets will fire, we actually need to do not forget the quantity of methods we can select 4 operating rockets out of the eleven that are not unsuitable, isolated by the all out range of approaches we will select four rockets out of the whole component.

The amount of ways of selecting four operating rockets out of the eleven that aren't improper is given by way of the combination recipe:

C(eleven,four) = 330

The absolute quantity of methods of selecting four rockets out of the entire part is given via:

C(14,4) = 1001

Subsequently, the chance that each one of the four rockets will hearth is:

P(all four hearth) = C(11,four)/C(14,4) = 330/1001 ≈ zero.3297

(adjusted to 4 decimal spots)

(b) To find the likelihood that at most 2 rockets might not fire, we need to remember every one of the potential conditions in which 0, 1, or 2 poor rockets are selected, and afterward song down the likelihood of every case and upload them up.

Case 1: No defective rockets are selected

The amount of ways of selecting four working rockets out of the eleven that are not faulty is given through the mixture recipe:

C(11,four) = 330

Subsequently, the chance of choosing no defective rockets is:

P(0 inadequate) = C(eleven,4)/C(14,four) ≈ 0.3297

Case 2: One defective rocket is chosen

The quantity of ways of selecting three operating rockets out of the eleven that are not poor is given with the aid of the combo equation:

C(eleven,three) = 165

The amount of ways of selecting 1 damaged rocket out of the three that are available is given through the mix equation:

C(three,1) = 3

Subsequently, the all out number of approaches of selecting 1 insufficient and three running rockets is:

C(eleven,three) × C(3,1) = 495

Thusly, the probability of choosing one deficient rocket and three running rockets is:

P(1 deficient) = C(eleven,three) × C(three,1)/C(14,4) ≈ 0.4655

Case three: Two poor rockets are selected

The amount of approaches of choosing 2 poor rockets out of the three which might be handy is given through the combination recipe:

C(3,2) = 3

The amount of ways of choosing 2 running rockets out of the 11 that aren't faulty is given via the mix recipe:

C(11,2) = 55

In this way, the all out number of methods of selecting 2 broken rockets and 2 working rockets is:

C(3,2) × C(eleven,2) = one hundred sixty five

Subsequently, the chance of choosing two incorrect rockets and two operating rockets is:

P(2 insufficient) = C(3,2) × C(11,2)/C(14,four) ≈ 0.1638

Subsequently, the likelihood that at maximum 2 rockets might not fireplace is:

P(at maximum 2 don't fire) = P(zero imperfect) + P(1 blemished) + P(2 deficient) ≈ 0.9591

(adjusted to 4 decimal spots).

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The missing elements are: Radius = 19 inches, Circumference = 119.32 inches and Area = 1133.54 square inches

The radius of a circle is half the diameter of the same circle.

The diameter is 38 inches.

This means that the radius is 19 inches

The circumference is calculated as:

C = 2πr

C = 2 × 3.14 × 19

C = 119.32

Area will be:

= πr²

= 3.14 × 19²

= 1133.54

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You need to assign a 3-digit identifier to each person to randomly select 900 students from a population of 5 million using a random number table.

To randomly select 900 students from a population of 5 million using a random number table, you need to assign a unique identifier to each person. This identifier should be a sequence of digits that allows you to distinguish between individuals and assign a random number to each.

To determine the number of digits required for this identifier, you can use the formula:

n = log10(N)

where N is the population size and n is the number of digits required. In this case:

N = 5,000,000

n = log10(5,000,000) = 6.7

This means that you need 7 digits to assign a unique identifier to each individual. However, since you are only selecting 900 students, you can truncate the identifier to 3 digits, which will still be sufficient to identify each individual in the sample. Therefore, you need to assign a 3-digit identifier to each person to randomly select 900 students from a population of 5 million using a random number table.

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The first-order and the second-order Taylor formula for f(x, y) = 17e(x+y) at (0,0) is f(x,y) = 17 + 17x + 17y + (17/2)x² + 17xy + (17/2)y²

The first-order Taylor formula for f(x,y) = 17[tex]e^{(x+y)}[/tex] at (0,0) is:

f(x,y) ≈ f(0,0) + ∇f(0,0) · (x,y)

≈ 17[tex]e^{(0+0)}[/tex] + (∂f/∂x, ∂f/∂y)(0,0) · (x,y)

≈ 17 + (17,17) · (x,y)

≈ 17 + 17x + 17y

The second-order Taylor formula for f(x,y) = 17[tex]e^{(x+y)}[/tex] at (0,0) is:

f(x,y) ≈ f(0,0) + ∇f(0,0) · (x,y) + (1/2)(x,y) · Hf(0,0) · (x,y)

≈ 17 + (17,17) · (x,y) + (1/2)(x,y) · ( ∂²f/∂x² ∂²f/∂x∂y ; ∂²f/∂y∂x ∂²f/∂y² ) (0,0) · (x,y)

≈ 17 + 17x + 17y + (1/2)(x,y) · (17 17 ; 17 17) · (x,y)

≈ 17 + 17x + 17y + (1/2)(17x² + 34xy + 17y²)

≈ 17 + 17x + 17y + (17/2)x² + 17xy + (17/2)y²

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Answer:

what

Step-by-step explanation:

The equation of the tangent plane to the given surface at the specified point (3, -2, 18) is z - 18 = 8(x - 3) - 12(y + 2).

To find the equation of the tangent plane to the given surface at the specified point (3,-2,18), we first need to find the partial derivatives of z with respect to x and y:

∂z/∂x = 4(x-1)
∂z/∂y = 12(y+3)

Then, we can evaluate these partial derivatives at the given point (3,-2,18):

∂z/∂x = 4(3-1) = 8
∂z/∂y = 12(-2+3) = -12

Next, we can use these partial derivatives and the point (3,-2,18) to write the equation of the tangent plane in point-normal form:

z - z0 = ∂z/∂x(x - x0) + ∂z/∂y(y - y0)

Plugging in the values we found:

z - 18 = 8(x - 3) - 12(y + 2)

Simplifying:

8x - 12y - z = -22

Therefore, the equation of the tangent plane to the given surface at the point (3,-2,18) is 8x - 12y - z = -22.
To find an equation of the tangent plane to the given surface z = 2(x - 1)^2 + 6(y + 3)^2 + 4 at the specified point (3, -2, 18), follow these steps:

1. Calculate the partial derivatives of the function with respect to x and y:
∂z/∂x = 4(x - 1)
∂z/∂y = 12(y + 3)

2. Evaluate the partial derivatives at the specified point (3, -2, 18):
∂z/∂x(3, -2) = 4(3 - 1) = 8
∂z/∂y(3, -2) = 12(-2 + 3) = -12

3. Use the tangent plane equation to find the tangent plane at the specified point:
z - z0 = ∂z/∂x(x - x0) + ∂z/∂y(y - y0)
where (x0, y0, z0) = (3, -2, 18)

4. Plug in the values and simplify the equation:
z - 18 = 8(x - 3) - 12(y + 2)

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The probabilities and the expected values are calculated below

Next cereal box wll contain blue or yellow

Here, we have

Yellow or blue = 15 + 5 = 20

Total = 50

So, we have

P(Yellow or blue) = 20/50

P(Yellow or blue) = 2/5

Arrival time before 7:30 am

Here, we have

Arrival time before 7:30 am = 7

Total time = 20

So, we have

P(Arrival time before 7:30 am) = 7/20

Team least likely

Convert the probabilities to decimal

So, we have

Nets = 0.67

Rockets = 0.5

Bucks = 0.8

Warriors = 0.375

This means that the team least likely to play in the championship game is 0.375

Section 7 in the game

Here, we have

P(7) = 35/250

In 150 times, we have

n(7) = 35/250 * 150

n(7) = 21

So, the number of times is 21

Expected students to eat chicken nuggets

Here, we have

P(Chicken) = 14/40

In 840 students, we have

n(Chicken) = 14/40 * 840

n(Chicken) = 294

Hence, the expected number of students to eat chicken nuggets is 294

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The correct answer is option b.

To find the median, we need to first put the observations in order:

5, 7, 10, 10, 10, 14, 14, 15, 18, x

Since there are 10 observations, the median is the average of the 5th and 6th observations, which are both 10. Therefore, the median is 10.

To find the mode, we need to find the observation that appears most frequently. Here, both 10 and 14 appear three times each, so the data has two modes: 10 and 14.

To find the standard deviation, we need to first find the mean of the data. We know that the sum of the observations is:

5 + 7 + 10 + 10 + 10 + 14 + 14 + 15 + 18 + x

= 103 + x

Since we know that X = 11.6, we can substitute to get:

Sum of observations = 103 + 11.6 = 114.6

The mean is then:

Mean = (Sum of observations) / (Number of observations)

Mean = 114.6 / 10 = 11.46

To find the standard deviation, we need to calculate the deviation of each observation from the mean, square each deviation, find the average of the squared deviations, and then take the square root.

Deviation of 5 = 11.46 - 5 = 6.46

Deviation of 7 = 11.46 - 7 = 4.46

Deviation of 10 = 11.46 - 10 = 1.46

Deviation of 10 = 11.46 - 10 = 1.46

Deviation of 10 = 11.46 - 10 = 1.46

Deviation of 14 = 11.46 - 14 = -2.54

Deviation of 14 = 11.46 - 14 = -2.54

Deviation of 15 = 11.46 - 15 = -3.54

Deviation of 18 = 11.46 - 18 = -6.54

Deviation of x = 11.46 - x

To find the standard deviation, we need to find the average of the squared deviations.

Average of squared deviations = [(6.46)^2 + (4.46)^2 + (1.46)^2 + (1.46)^2 + (1.46)^2 + (-2.54)^2 + (-2.54)^2 + (-3.54)^2 + (-6.54)^2 + (11.46 - x)^2] / 10

= (41.7316 + 19.8916 + 2.1316 + 2.1316 + 2.1316 + 6.4516 + 6.4516 + 12.5316 + 42.8916 + (11.46 - x)^2) / 10

= (136.786) / 10

= 13.6786

Finally, we take the square root of the average of the squared deviations to find the standard deviation:

Standard deviation = sqrt(13.6786) = 3.8356

Therefore, the correct answer is option b.

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The end behavior of f(x)=(2+x2)(x2−36) is determined by the highest degree terms in the numerator and denominator. In this case, the highest degree terms are both x^4.

The numerator (2+x^2) will approach positive infinity as x approaches positive or negative  infinity because the x^2 term dominates.

The denominator (x^2-36) will approach positive infinity as x approaches positive or negative infinity because the x^2 term dominates.

Therefore, as x approaches positive or negative infinity, f(x) will approach positive infinity.

This is because the highest degree term in the function is x^4, which will dominate the function as x approaches infinity or negative infinity. Since the coefficient of x^4 is positive, the function will approach 0 from both sides as x becomes large or very negative.

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The solution is, the solutions using the Zero Product Property: is x =0 and -5.

The expression to be solved is:

x(x + 5) = 0

we know that,

The zero product property states that the solution to this equation is the values of each term equals to 0.

now, we have,

x (x + 5) = 0

i.e. we get,

(x) × (x + 5) = 0

so, using the Zero Product Property:

we get,

(x) = 0

or,

(x + 5) = 0

so, we have,

x = 0 or, x = -5

The answers are 0 and -5.

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The value of the probability P(3) is 1/4.

We have,

There are 4 outcomes.

i.e

3, 4, 5, and 6.

Now,

P(3)

This means,

The probability of getting 3 as the outcome when spun.

So,

P(3) = 1/4

Thus,

The value of the probability P(3) is 1/4.

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Total overhead costs for the purchasing department: $500,000

Total dollar value of merchandise purchased: $10,000,000

Total number of purchase orders: 5,000

Total number of suppliers: 1,000

Using the traditional method of allocating costs based on the dollar value of merchandise purchased, the cost per dollar of merchandise purchased would be:

$500,000 / $10,000,000 = $0.05 per dollar of merchandise purchased

To calculate the cost per purchase order and per supplier using activity-based costing, we first need to calculate the cost driver rates for each activity. The cost driver rate is the total cost of an activity divided by the total number of units of the cost driver for that activity. In this case, the cost driver for the purchase order activity is the number of purchase orders, and the cost driver for the supplier activity is the number of suppliers.

The cost driver rate for the purchase order activity is:

$500,000 / 5,000 = $100 per purchase order

The cost driver rate for the supplier activity is:

$500,000 / 1,000 = $500 per supplier

Using these cost driver rates, we can allocate the purchasing department costs to products based on the number of purchase orders and suppliers for each product. For example, if a product had 10 purchase orders and was purchased from 3 different suppliers, its total purchasing department costs would be:

(10 purchase orders x $100 per purchase order) + (3 suppliers x $500 per supplier) = $1,500

By using activity-based costing, True Fit can allocate its purchasing department costs more accurately and can identify the cost drivers that are most important for its purchasing department. This information can help True Fit make better purchasing decisions and manage its costs more effectively.

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The equation that represents the number of milliliters y of water in the bowl after x seconds is: y = 16x

The constant of proportionality for this relationship is the slope of the linear function that passes through any two points on the line. We can use the points (20, 320) and (60, 960) from the table to find the slope:

slope = (960 - 320) / (60 - 20) = 640 / 40 = 16

Therefore, the equation that represents the number of milliliters y of water in the bowl after x seconds is: y = 16x

The constant of proportionality is the ratio that relates two given values in what is known as a proportional relationship. Other names for the constant of proportionality include the constant ratio, constant rate, unit rate, constant variation, or even the rate of change.

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The force that is between two 0. 004 c charges that are 0. 35 m apart in air is 1, 174.2 N.

The force between these charges can be found by Coulomb's Law which states that the electric force linking two charged particles is proportional to both their individual quantitative charge and inversely proportionate to the square of their separation distance.

Given charges of 0. 004 c and 0. 35 m apart, the formula shows :

F = (8. 99 x 10 ⁹ N m ² /C² x | 0. 004 C x 0. 004 C| ) / ( 0. 35 m ) ²

F = 143.84  / 0.1225

F = 1, 174.2 N

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The producer’s risk is 0.99 and the consumer’s risks for each sampling plan are 0.347, 0.049, and 0.176 for acceptance criteria 0, acceptance criteria 1, and acceptance criteria 2 respectively.

a. The operating characteristic curve (OC curve) shows the probability of accepting a lot with a given quality level, based on the sample size and acceptance criteria. Here are the OC curves for acceptance criteria of 0, 1, and 2, assuming a binomial distribution:

Acceptance Criteria = 0:
Sample size: 20
Probability of acceptance (p): 0.01
Probability of rejection (1-p): 0.99
OC Curve:
Quality Level (proportion defective) | Probability of acceptance
0% | 0.994
1% | 0.988
2% | 0.977
3% | 0.958
4% | 0.928
5% | 0.883
6% | 0.821
7% | 0.743
8% | 0.653
9% | 0.556
10% | 0.458

Acceptance Criteria = 1:
Sample size: 20
Probability of acceptance (p): 0.92
Probability of rejection (1-p): 0.08
OC Curve:
Quality Level (proportion defective) | Probability of acceptance
0% | 1.000
1% | 1.000
2% | 1.000
3% | 1.000
4% | 1.000
5% | 1.000
6% | 0.999
7% | 0.998
8% | 0.993
9% | 0.981
10% | 0.951

Acceptance Criteria = 2:
Sample size: 20
Probability of acceptance (p): 0.83
Probability of rejection (1-p): 0.17
OC Curve:
Quality Level (proportion defective) | Probability of acceptance
0% | 1.000
1% | 1.000
2% | 1.000
3% | 1.000
4% | 0.999
5% | 0.998
6% | 0.992
7% | 0.978
8% | 0.949
9% | 0.898
10% | 0.824

b. The producer's risk (Type I error) is the probability of rejecting a good lot, while the consumer's risk (Type II error) is the probability of accepting a bad lot. Here are the calculations for each sampling plan:

Acceptance Criteria = 0:
Producer's risk = α = 1 - p0 = 0.99
Consumer's risk = β = 1 - OC at p1 = 1 - 0.653 = 0.347

Acceptance Criteria = 1:
Producer's risk = α = 1 - p0 = 0.99
Consumer's risk = β = 1 - OC at p1 = 1 - 0.951 = 0.049

Acceptance Criteria = 2:

Producer's risk = α = 1 - p0 = 0.99
Consumer's risk = β = 1 - OC at p1 = 1 - 0.824 = 0.176

Note that the producer's risk is the same for all three sampling plans since it is based on the specified probability of a defective unit in the lot. The consumer's risk, however, varies depending on the acceptance criteria and sample size. Generally, a more lenient acceptance criterion (higher p-value) or a larger sample size will result in lower consumer risk.

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if you add up 2 interior degrees the answer is the exterior degree

so the answer is 145

Answer:

23 ft^3

Step-by-step explanation:

The volume would be the little box plus the big rectangle. The volume of the little box is 2*2*2 which equals 8. The volume of the second box is 1*3*5 which equals 15. The volume of the whole shape is 8+15=23ft^3.

The probability of rolling a sum of 12 with these dice is  1/36.

The likelihood of rolling an entirety of 12 with two standard dice can be found utilizing the equation:

P(D1 + D2 = 12) = number of ways to induce an entirety of 12 / total possible results

There's as it were one way to roll a whole of 12: rolling a 6 on both dice.

The whole conceivable results can be found by noticing that there are 6 conceivable results for each dice roll, since each kick the bucket has 6 sides. In this manner, the whole number of conceivable results is:

6 x 6 = 36

So the likelihood of rolling a whole of 12 with two standard dice is:

P(D1 + D2 = 12) = 1/6 * 1/6 = 1/36

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The required support wire makes an angle of approximately 67.54° degrees with the ground.

We can use trigonometry to find the angle x. In the diagram, we can see that the support wire, the height of the tower, and the ground form a right triangle. The length of the support wire is the hypotenuse of this triangle, and the distance from the base of the tower to the point where the wire touches the ground is the adjacent side. We can use the tangent function to find the angle x:

tan(x) = opposite / adjacent

In this case, the opposite side is the height of the tower, which we don't know yet. However, we can use the Pythagorean theorem to find it:

height² = support wire² - adjacent²

height² = 100² - 40²

height ≈ 96.8

Now we can plug in the values to find x:

tan(x) = opposite / adjacent

tan(x) = 96.8 / 40

x ≈ 67.54°

Therefore, the support wire makes an angle of approximately 67.54° degrees with the ground.

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The mean of the distribution of sample means would be 27 hours, which is the same as the average life of the candles.

This is because the sample means would be expected to be centered around the population mean.

The standard deviation of the distribution of sample means (also known as the standard error of the mean) can be calculated using the formula:

standard deviation of sample means = standard deviation of population / square root of sample size

In this case, the standard deviation of sample means would be:

6 / square root of 4 = 3

So the answer is 3.

So, the mean of the distribution of the sample means would be 3. Your answer: 3.

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Therefore, if the person works 54 hours, they would make $1,163.04. Rounded to 2 decimal places, the answer is $1,163.00.

The decimal system employs ten decimal digits, a decimal mark, and a minus sign ("-") for negative quantities when writing numbers. The decimal digits are 0 through 9, with the dot (".") serving as the decimal separator in many (mainly English-speaking) nations and the comma (",") in others.

The fractional portion of the number is represented by the place value that follows the decimal. The number 0.56, for instance, is composed of 5 tenths and 6 hundredths.

We can use proportionality to solve this problem. If the person works 25 hours and makes $539, then their hourly rate is:

$539 ÷ 25 hours = $21.56 per hour

They would make if they work 54 hours, we can multiply their hourly rate by the number of hours worked:

$21.56 per hour × 54 hours = $1,163.04

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The hypothesis that the variance in the number of patients seen per day is equal to 10 cannot be rejected  based on the given data and using a level of significance of 0.05.

To determine if the variance in the number of patients seen per day is equal to a specific value, we can conduct a hypothesis test. Let's assume the null hypothesis is that the variance is equal to the specified value, and the alternative hypothesis is that the variance is not equal to the specified value.

We can use a chi-square test to test this hypothesis, where the test statistic is calculated as (n-1)*s²/σ², where n is the sample size, s² is the sample variance, and σ² is the hypothesized population variance. This test statistic follows a chi-square distribution with n-1 degrees of freedom.

Using a level of significance of 0.05, with 9 degrees of freedom (since there were 10 observations), the critical value for the chi-square distribution is 16.92.

Calculating the sample variance from the given data, we get s^2 = 4.44. Assuming the hypothesized population variance is 10, the test statistic is (9)*4.44/10 = 4.00.

Since the test statistic (4.00) is less than the critical value (16.92), we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that the variance in the number of patients seen per day is not equal to 10.

In conclusion, based on the given data and using a level of significance of 0.05, we cannot reject the hypothesis that the variance in the number of patients seen per day is equal to 10.

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To find the percentage of 84 out of 113, we need to first calculate what percentage of the total score 84 represents.
If a score of 113 is 40%, then we can set up a proportion:
113 / 100 = 40 / x
where x represents the percentage we are trying to find.

Cross-multiplying, we get:
113x = 4000
Dividing both sides by 113, we get:
x = 35.4
So, 84 represents approximately 35.4% of the total score of 113.

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By the Comparison Test, we can conclude that Σ (4sin(n))/([tex]n^2[/tex] - 1) converges.

We can use the Limit Comparison Test to determine the convergence or divergence of the series:

∑(4 sin(n))/n

To do this, we need to find a series whose convergence is known and which can be compared to the given series. Let's choose the series ∑(1/n) since we know that it diverges.

We take the limit of the ratio of the nth term of each series as n approaches infinity:

lim n→∞ (4 sin(n)/n)/(1/n) = lim n→∞ 4 sin(n) = DNE

Since the limit does not exist, we cannot apply the Limit Comparison Test. Therefore, Therefore, by the Comparison Test, we can conclude that Σ (4sin(n))/([tex]n^2[/tex] - 1) converges.

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The 7th derivative of f(x)=cos(3x³) - 1/x⁵ at x=0 is 3240.

To find the 7th derivative of f(x) at x=0, we need to build a Maclaurin series for f(x) from the series for cos(x). The Maclaurin series for cos(x) is:

cos(x) = 1 - x²/²! + x⁴/⁴! - x⁶/⁶! + ...

Using this, we can build a Maclaurin series for f(x) as follows:

f(x) = cos(3x³) - 1/x⁵

= (1 - (3x³)²/²! + (3x³)⁴/⁴! - (3x³)⁶/⁶! + ...) - 1/x⁵

= 1 - 9x⁶/²! + 81x¹²/⁴! - 729x¹⁸/⁶! + ... - 1/x⁵

= 1 - 9x⁶/²! + 81x¹²/⁴! - 729x¹⁸/⁶! + ... - x⁻⁵

Taking the 7th derivative of this expression and evaluating at x=0 gives:

f⁷ * ⁰ = 7! * (-9)/2!

= 3240

Therefore, the 7th derivative of f(x)=cos(3x³) - 1/x⁵ at x=0 is 3240.

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The work done on lifting the steel beam by crane upwards is 3.3×10⁵ joules, based on given data.

Since the crane is lifting the steel beam upwards, the total force will be sum of mass and force due to acceleration due to gravity.

So, the formula will be-

W = F × d

W = m(g + a) × d

W = 425 (10 + 1.8) × 66

Performing addition in the parenthesis first

W = 425 × 11.8 × 66

Multiplying all the digits on Right Hand Side of the equation

W = 3,30,990 Joules

Hence, the work done is 3.3 × 10⁵ Joules

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