Rewrite the expression in terms of sine and cosine and utilize the Fundamental Pythagorean Identity: sin²(x)+cos²(x)=1
Verify the identity using the Pythagorean Identity:
[tex]csc(x)+cot(x)=\frac{1}{csc(x)-cot(x)}[/tex]

(a) The total amount of the nurse's annual salary that is not taxed is $6644. (b) His annual taxable salary is $22,226.8. (c) The tax he pays annually is $5856.7.

(a) To find the total amount of the nurse's annual salary that is not taxed, we need to add up all of the non-taxable allowances:

$37 x 12 (months) = $444 for national insurance

$3000 for personal allowance

$1800 for his wife

$1400 for his children

Total non-taxable allowances = $444 + $3000 + $1800 + $1400 = $6644

Therefore, the nurse's total non-taxable salary is $6644.

(b) To find the nurse's annual taxable salary, we need to subtract his non-taxable allowances from his gross salary:

Annual taxable salary = $29,460 - 2% x $29,460 - $6644

Annual taxable salary = $29,460 - $589.2 - $6644

Annual taxable salary = $22,226.8

Therefore, the nurse's annual taxable salary is $22,226.8.

(c) To calculate the nurse's annual tax, we need to apply the income tax rates:

Tax on first $4000 at 20% = $800

Tax on remainder at 25% = 0.25 x ($22,226.8 - $4000) = $5056.7

Total annual tax = $800 + $5056.7 = $5856.7

Therefore, the nurse pays an annual tax of $5856.7.

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Complete question is attached below

The area of the polygon is  6. 9m²

From the information given, we have that the regular polygon takes the shape of a triangle.

The formula for calculating the area of a triangle is expressed with the equation;

A = 1/2bh

Such that the parameters in the formula are;

Area of a equilateral triangle is;

= √3/4a²

Then,

= √3/4 (4)²

= √3 × 4

= 6. 9m²

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Answer:

Step-by-step explanation:

Answer:

x=40

Step-by-step explanation:

alternate angles are equal so the one of the triangles angles r 50 then 50 plus 90, in a right angled triangle, adds to 140

180 minus 140 is 40

The slope of the line passing through (0,3) and (1,-1) is -4.

What is slope of the line?

The slope of a line is a measure of its steepness or incline. It describes how much the line rises or falls as it moves horizontally.

To find the slope of the line passing through (0,3) and (1,-1), we can use the slope formula:

m = (y2 - y1)/(x2 - x1)

where (x1,y1) and (x2,y2) are the coordinates of the two points.

Plugging in the given coordinates, we get:

m = (-1 - 3)/(1 - 0)

m = -4/1

m = -4

Therefore, the slope of the line passing through (0,3) and (1,-1) is -4.

The slope of a line can be positive, negative, zero, or undefined (when the line is vertical). A positive slope means that the line is increasing as it moves to the right, while a negative slope means that the line is decreasing as it moves to the right. A slope of zero means that the line is horizontal, and an undefined slope means that the line is vertical.

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Complete question is: Slope of the given line passing through (0,3) and (1,-1) is -4.

Answer:

  about -0.79 mg/hour

Step-by-step explanation:

You want the rate of change of the amount of medication in a patient after 4 hours if the amount after t hours is modeled by y=88/(1+e^(t+0.7)).

The rate of change is found by taking the derivative of the function with respect to time. Let u = 1+e^(t+0.7). Then du/dt = e^(t +0.7).

The function can be written ...

  y = 88/u = 88·u^-1

so its derivative is ...

  dy/dt = (-88·u^-2)·du/dt

  dy/dt = -88(e^(t +0.7))/(1 +e^(t +0.7))^2

The value of the rate of change at t=4 is ...

  dy/dt = -88(e^(4 +0.7))/(1 +e^(4 +0.7))^2 ≈ -88(109.95)/(1 +109.95)^2

  dy/dt ≈ -0.78602 ≈ -0.79 . . . . . . mg/h

The rate of change in the amount of medication is about -0.79 mg/h.

We must raise the $1,190 starting rent by 1.7% in order to determine the monthly rent after a year.

We can begin by figuring out how much the rise will be:

Amount of increase: 1.7% of $1,190 multiplied by 0.017 times equals $20.23. (rounded to the nearest cent)

We must multiply the initial rent by this sum in order to determine the new monthly rate:

New rate is $1,190 plus $20.23 per month, or $1,210.23. (rounded to the nearest cent)

As a result, the rate will be $1,210 per month after a year.

When the input parameter value is 0, the output of a function is its starting value, or 0. The first number would be the amount of money you had to start with on day 0 if your function, for instance, was tracking the sum of money you earned over a specific period of time.

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The monthly rent after 1 year will be $1,210.

"Amount" is a general term that refers to the quantity, value, or total of something. It can be used in different contexts to refer to different things.

To calculate the monthly rent after 1 year, we need to find the 1.7% increase and add it to the initial rent.

First, we need to calculate the amount of the increase:

Increase = 1.7% of $1,190

Increase = 0.017 × $1,190

Increase = $20.23 (rounded to two decimal places)

Next, we can find the new monthly rent by adding the increase to the initial rent:

New rent = $1,190 + $20.23

New rent = $1,210.23 (rounded to two decimal places)

Therefore, the monthly rent after 1 year will be $1,210.

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The best way to calculate the periodic interest rate for most credit cards is (a) to divide the annual interest rate by 12.

Interest is a fee charged by a lender to a borrower for the use of borrowed money or assets. It is typically expressed as a percentage of the principal amount borrowed or invested, and it is calculated over a specific period of time. Interest can be charged on loans, credit cards, mortgages, bonds, and other financial instruments. Interest can be positive or negative, depending on whether the borrower or lender is paying or receiving the interest. Positive interest is also known as a return or yield, and it is earned by investors who lend money or invest in financial instruments. Negative interest, on the other hand, is a fee charged by lenders to borrowers, and it is typically associated with negative interest rates or deflationary economic conditions.

Here,

Most credit cards compound interest on a monthly basis, which means that the interest is calculated and added to the balance each month. Therefore, to calculate the monthly or periodic interest rate, we need to divide the annual interest rate by 12.

For example, if the annual interest rate on a credit card is 18%, then the monthly or periodic interest rate would be:

Periodic interest rate = 18% / 12

= 1.5%

Therefore, the correct answer is (a) Divide the annual interest rate by 12.

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The measure of DE is 8x. This was found by using the distance formula to calculate the length of the side.

Measurements are the process of comparing an unknown quantity to a known reference value in order to estimate the size, quantity, distance, weight, or other characteristics of that unknown quantity. Measurements are used in science and engineering to quantify the properties of objects, or to compare objects to each other. Measurements are also used in everyday life to help make decisions and assess the impact of changes.

To find the measure of DE, we must first calculate the length of the side. We can do this by using the distance formula. The distance formula is used to calculate the length of a line between two points in a plane. The formula is given as d = √(x2 - x1)2 + (y2 - y1)2.

For the side DE, we can use the coordinates (2x+7, 4(x-3)). The first point is (2x+7, 4(x-3)) and the second point is (2x+7, 4(x+3)). We can substitute these points into the distance formula to calculate the length of DE.

The formula becomes d = √[(2x+7 - 2x+7)2 + (4(x+3) - 4(x-3))2]

Simplifying, we get d = √[(0)2 + (8x)2]

d = √[64x2]
d = 8x

Therefore, the measure of DE is 8x.

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Given Data:

Number of miles covered in 25 minutes = 5

To Find:

Number of miles covered in 30 minutes = ?

Solution:

Number of miles covered per minutes = 5/25             (rate=distance/time)                          

                                                                =0.2 miles/minute

Number of miles covered in 30 minutes = 0.2*30

                                                                   =6 miles

The probability that the next vehicle to be rented will be a minivan from all 80 vehicles for rented wheels is equals to the 7/40.

We have, Rent wheels keeps track of the types of vehicles that are rented each week. The above table shows the results from this week and type of vehicle number.

Number of SUV vehicle = 31

Number of truck vehicle = 8

Number of minivan vehicle = 14

Number of sedan vehicle = 27

So, total number of vehicles or outcomes

= 80

Probability is defined as chances of occurrence of an event. It is calculated by dividing the favourable responses to the total number of possible outcomes or responses. Let E be an event that rented vehicle is minivan. The probability that the next vehicle to be rented will be a minivan, P(E) = 14/80

=> P(E) = 7/40

Hence, the required probability is 7/40.

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Complete question:

The above figure completes the question. rent wheels keeps track of the types of vehicles that are rented each week. the table below shows the results from this week. type of vehicle number rented suv 31 truck 8 minivan 14 sedan 27 based on the data, what is the probability that the next vehicle to be rented will be a minivan?

This method ensures that the side dish selection reflects customer preferences while incorporating an element of randomness.

Hi! To simulate choosing a side dish using 10 index cards, the manager can follow these steps:

1. Assign a number to each side dish based on the survey results.
2. Determine the probability of each side dish being chosen according to customer preferences.
3. Based on the probabilities, allocate the appropriate number of index cards to each side dish (e.g., if a dish has a 30% chance of being chosen, it would be assigned 3 out of the 10 index cards).
4. Write the side dish name or its assigned number on the allocated index cards.
5. Shuffle the index cards and randomly draw one card each week to determine the special's side dish.
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Answer: approximately 7.3 kilometers.

Step-by-step explanation:

Using the Pythagorean theorem, we can write:

x^2 = 7^2 + 2^2

x^2 = 49 + 4

x^2 = 53

Taking the square root of both sides, we get:

x = sqrt(53)

Using a calculator or estimating, we find:

x ≈ 7.3 km

So the straight-line distance between Ryan's house and Emmy's house is approximately 7.3 kilometers.

The expressions when factorized are x(1 + 2yz), 9x^3y^2 and 6(3x^2 − 2y) + (5ab + 6y)

Factorization of X + 2xyz:

We can factor out x from the given expression to get:

x(1 + 2yz)

Therefore, the factorization of X + 2xyz is x(1 + 2yz).

Factorization of 9x3y2:

The given expression, 9x3y2, is already in its simplest form and cannot be factorized further.

Factorization of 18x2 + 5ab − 6y:

We can use the grouping method to factorize the given expression:

(18x2 − 12y) + (5ab + 6y)

6(3x2 − 2y) + (5ab + 6y)

Therefore, the factorization of 18x2 + 5ab − 6y is 6(3x2 − 2y) + (5ab + 6y).

Factorization of 4x4 + 3x2 − x − 4:

Cannot be factorized

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The width of the cooling tower at the base of the structure is 100 meters, rounded to the nearest whole number.

Given that the cooling tower's walls are modeled by where the measurements are in meters and the width of the

cooling tower at the base of the structure is required, we can find the width by using the formula for the area of a

trapezoid. The area of the trapezoid represents the area of the base of the cooling tower.

Area of a trapezoid = 1/2 × (a + b) × h

where a and b are the lengths of the parallel sides of the trapezoid and h is the height or perpendicular distance

between the parallel sides.

In this case, the parallel sides of the trapezoid are not given, but we are given the height of the cooling tower, which is

100 meters. We also know that the area of the base of the cooling tower is 10,000 square meters.

Therefore, we can use the formula for the area of a trapezoid to find the sum of the lengths of the parallel sides of the

trapezoid, which will give us the width of the cooling tower at the base.

Area of a trapezoid = 1/2 × (a + b) × h

10,000 = 1/2 × (a + b) × 100

100 = (a + b)/2

200 = a + b

Width of cooling tower at the base = (a + b)/2

Width of cooling tower at the base = 200/2

Width of cooling tower at the base = 100 meters

Therefore, the width of the cooling tower at the base of the structure is 100 meters, rounded to the nearest whole number.

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After answering the presented question, we can conclude that Pythagorean theorem Therefore, the value of x is 15.

The Pythagorean theorem is a fundamental mathematical principle that describes the relationship between the sides of a right triangle. It states that the sum of the squares of the other two sides' lengths equals the square of the hypotenuse's length (the side opposite the right angle). The mathematical theorem is as follows: c2 = a2 + b2 Where "c" denotes the hypotenuse length and "a" and "b" represent the lengths of the other two sides, known as the legs.

Pythagorean theorem

[tex]x^2 = 12^2 + y^2\\y/x = 3/5\\y = (3/5)x\\x^2 = 12^2 + (3/5)^2 x^2\\(1 - (3/5)^2) x^2 = 12^2\\(1 - 9/25) x^2 = 12^2\\(16/25) x^2 = 12^2\\x^2 = (25/16) * (12^2)\\x = (5/4) * 12 = 15\\[/tex]

Therefore, the value of x is 15.

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When C is the midpoint of AB, D is the midpoint of AC, E is the midpoint of AD, F is the midpoint of ED, G is the midpoint of EF, and H is the midpoint of DB, the length of GH is 77.

How to calculate midpoint for the given problem?

Since D is the midpoint of AC, we know that AD = DC = 56/2 = 28.

Similarly, E is the midpoint of AD, so AE = ED = 28/2 = 14.

Now, F is the midpoint of ED, so EF = ED = 14.

Next, G is the midpoint of EF, so EG = GF = 14/2 = 7.

Finally, H is the midpoint of DB, so DH = HB = AB/2.

Since C is the midpoint of AB, we have AB = 2AC = 2DC = 2*56 = 112.

Therefore, DH = HB = AB/2 = 112/2 = 56.

Now, we can see that GH = GD + DH = (GE + ED) + HB = (7 + 14) + 56 = 77.

Therefore, GH = 77.

The given problem involves a set of line segments and their midpoints. We can use the concept that the midpoint of a line segment divides it into two equal parts. Using this concept, we can determine the lengths of various line segments in terms of the given length DC.

Then we can apply this information to find the length of GH by adding up the appropriate line segments. Specifically, we use the fact that GH is the sum of GD and DH, which in turn are the sum of several mid-segment lengths that can be determined based on the given information. Following this logic, we can calculate that GH is equal to 77.

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Answer:

[tex]f(t) = 4100( {2}^{ \frac{t}{2} }) [/tex]

No, the triangles are not identical

Identical triangles are two triangles that have the exact same

In other words, all corresponding sides and angles are congruent (equal).

Identical triangles can be thought of as the same triangle, just in different positions or orientations in space.

The triangles are similar triangles but not identical triangles as the sides are not equal

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Therefore , the solution of the given problem of inequality comes out to be  150 units per employee, to satisfy the production demand.

Despite the fact that algebra lacks a comparable symbol, its distinction can be expressed by a pair or collection for numbers. Equilibrium is typically followed by equity. The ongoing disparity in norms is what causes inequality. Disparity and fairness are not synonymous. Even though the components are typically not connected or situated closely together, that was our most popular symbol.

Here,

A) Assume Y is the number of units created and X is the number of employees. The following inequalities represent the circumstances in the issue:

=> 100 ≤ X ≤ 200 (the factory must have at least 100 but no more than 200 workers)

=> Y ≥ 30X (the factory must manufacture at least 30 units per worker)

B) The lines 100, 200, and 30X would form the borders of the darkened area.

C) The following three options meet the set of inequalities:

=> X = 100, Y = 3000

=> X = 150, Y = 4500

=> X = 200, Y = 6000

The factory in the second solution employs 150 people and generates 4500 units, or 150 units per employee, to satisfy the production demand.

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1)
a) the mean is µ = 2.5.

b)  the variance is σ² = 1.25.

c) Standard Deviation σ =  1.12

2)

a) the mean is µ = 6.40.

b)  the variance is σ² = 2.12

c) Standard Deviation σ =  1.46.

1) To find the mean (µ), we can multiply each value of x by its corresponding probability (P(x)), and then add up the products: (See attached table I)

Sum | 1 | 2.5


Therefore, the mean is µ = 2.5.

To find the variance (σ²), we need to find the squared differences between each value of x and the mean, multiply each squared difference by its corresponding probability, and then add up the products:

See table II
Sum | 1 | | 5 | 1.25

Therefore, the variance is σ² = 1.25.

Finally, to find the standard deviation (σ), we can take the square root of the variance:

σ = √(σ²) = √(1.25) ≈ 1.12.


2)
To find the mean (µ), we can multiply each value of x by its corresponding probability (P(x)), and then add up the products: See table III...


Sum | 1.00 | 6.40

Therefore, the mean is µ = 6.40/1.00 = 6.40.

To find the variance (σ^2), we need to find the squared differences between each value of x and the mean, multiply each squared difference by its corresponding probability, and then add up the products: See table IV

Sum | 1.00 | | 40.80 | 2.1200

Therefore, the variance is σ² = 2.1200.

Finally, to find the standard deviation (σ), we can take the square root of the variance:

σ = √(σ²) = √(2.1200) ≈ 1.46.

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Answer:

Step-by-step explanation:

The ratio for Purple People Eater can be expressed in fraction form as

✔ 3/5

.

The ratio for Purple Majesty can be expressed in fraction form as

✔ 4/7

.

To compare these fractions, rewrite them with a common denominator of

✔ 35

.

The ratio of red paint to blue paint in Purple People Eater is

✔ greater than

the ratio in Purple Majesty.

Answer:

x = 15.

Therefore, the width of the walkway is 15m.

Step-by-step explanation:

Let's represent the width of the walkway by "x".

We know that the total width/base of the area (including the walkway) is 300m, so we can set up the equation:

length of field + 2(width of field) + 2(width of walkway) = 300

Substituting the given values, we get:

200 + 2w + 2x = 300

Simplifying the equation, we get:

2w + 2x = 100

We also know that the area of the field is 30,000m, so we can set up the equation:

length of field x width of field = 30,000

Substituting the given values, we get:

200w = 30,000

Simplifying the equation, we get:

w = 150

Finally, we know that the total area (including the walkway) is 60,000m, so we can set up the equation:

(length of field + 2x) x (width of field + 2x) = 60,000

Substituting the values we've found so far, we get:

(200 + 2x) x (150 + 2x) = 60,000

Expanding the equation, we get:

300x^2 + 1000x - 45,000 = 0

Solving for x using the quadratic formula, we get:

x = 15 or x = -30/100

Since x can't be negative, we can discard the negative solution and conclude that x = 15.

Therefore, the width of the walkway is 15m.

The sequence of transformation that makes the shapes congruent is rotation then transformation

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

The figures

From the figure, we have the shapes to have different orientation but same size

This means that one of the shapes is rotated to form the other

In this case, the rotation is a 90° clockwise

Followed by the rotation is a translation transformation

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After answering the given question, we can conclude that As a result, the triangle projection's lacking length is 8.

A triangle is a closed, double-symmetrical structure composed of three line segments known as sides that meet at three locations known as vertices. Triangles are distinguished by their edges and angles. Triangles can be equilateral (all groups equal), isosceles, or scalene based on their edges. Triangles are classified as acute (all angles are less than 90 degrees), okay (one angle is identical to 90 degrees), or orbicular (all angles are higher than 90 degrees) (all angles greater than 90 degrees). The area of a triangle can be calculated using the formula A = (1/2)bh, where an is the neighbourhood, b is the triangle's base, and h is the triangle's height.

Because the initial triangular and its projection are comparable, the sides that correspond to them are proportional. Let x be the length of the initial triangle's side equivalent to n. Then, given that the extension of the length x side has length 8, we can write:

x/n = (initial triangle side length)/(projected triangle side length) = x/8

When we solve for n, we get:

n = (8x)/x = 8

As a result, the projection's lacking length is 8.

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Step-by-step explanation:

To find the probability of Mario randomly selecting a red chip or a white chip, we need to first determine the total number of red and white chips he has.

Total number of red and white chips = 45 + 24 = 69

Therefore, the probability of Mario randomly selecting a red chip or a white chip is:

P(red or white) = (number of red chips + number of white chips) / total number of chips

P(red or white) = (45 + 24) / (45 + 12 + 24)

P(red or white) = 69 / 81

P(red or white) = 0.85

So the probability of Mario randomly selecting a red chip or a white chip is 0.85 or 85%.

Answer: 0.8519 ≈ 85%

Step-by-step explanation:

        To find the probability, we will divide wanted outcomes by total possible outcomes. See below.

[tex]\displaystyle \frac{\text{wanted outcomes}}{\text{possible outcomes}} =\frac{\text{number of red or white chip}}{\text{total number of chips}} =\frac{45+24}{45+12+24}= \frac{69}{81}[/tex]

       = 0.8519 ≈ 85%

        The probability that Mario will randomly select a red or white chip is 0.8519 or about 85%.

The perimeter of rectangular painting which is 42 cm wide and 51 cm long is 186 cm.

Perimeter of rectangle = 2×(l + b)

Perimeter = 2 × 93

Perimeter = 186 cm

Perimeter is the total length of the boundary of any closed shape. Let’s try and understand this using an example. For example, you have a huge square-shaped farm. Now, in order to save your farm from street animals, you decide to fence it. If you know the length of one side of the farm, you just have to multiply it by 4 in order to find the total length of the boundary of the farm. There are many such instances where we might be using the concept of finding perimeter without even knowing about it.

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Answer: 25%

Step-by-step explanation:

The key features of the graph are (a) x-int = 5 and y-int= 5, HA; y = 0.5 and VA: x = 0.5 and hole = (-2, 7/5)

The x-intercept of a graph is the point at which the graph intersects the x-axis while the y-intercept of a graph is the point at which the graph intersects the y-axis.

From the given graph, we have the intercepts to be

x-intercept = 5 and y-intercept = 5

The horizontal asymptote of a graph is a horizontal line that the graph approaches as x approaches positive or negative infinity

While the vertical asymptote is a vertical line that the graph approaches as x approaches a certain value.

From the given graph, we have the asymptote to be

Horizontal; y = 0.5 and Vertical: x = 0.5

A hole in a graph refers to a point on the graph where the function is undefined but can be made continuous by removing the point and filling the gap with a single point.

From the given graph, we have the hole to be

Hole = (-2, 7/5)

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Answer:

Original angle = 122*

Supplementary angle = 58*

Step-by-step explanation:

Take the formula and do a little algebra.

x + (x + 64) = 180

Subtract 64 from both sides

x + x = 116

Combine the x's

2x = 116

Divide both side by 2

x = 58

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