Need help, thanks!
Last question

Need Help, Thanks!Last Question

Answers

Answer 1

The maximum total deduction that can be claimed is OE. $17,100.

What is deduction ?

Deduction refers to the process of subtracting or taking away something. In the context of taxes, a deduction is a specific amount of money that a taxpayer is allowed to subtract from their income before calculating their tax liability. Deductions are one way to reduce the amount of taxes owed and can help to lower a person's overall tax burden.

There are two main types of deductions in the U.S. tax system: standard deductions and itemized deductions. The standard deduction is a fixed amount that varies depending on the taxpayer's filing status, while itemized deductions are specific expenses that a taxpayer can deduct from their taxable income. Some common examples of itemized deductions include mortgage interest, charitable contributions, medical expenses, and state and local taxes.

It's important to note that not all deductions are available to all taxpayers, and there are specific rules and limits that apply to each type of deduction. It's also important to keep accurate records and documentation to support any deductions that are claimed on a tax return.

The maximum total deduction that can be claimed is OE. $17,100, because that is the sum of the deductible expenses and the standard deduction.

The question is asking for the maximum total deduction that can be claimed, given certain deductible expenses and a standard deduction.

The standard deduction for the tax year is $6,100, and the deductible expenses are $6,000 in mortgage interest and $5,000 in charitable contributions.

Since the deductible expenses ($6,000 + $5,000 = $11,000) are less than the standard deduction of $6,100, it makes sense to use the standard deduction instead of itemizing deductions.

Therefore, the maximum total deduction would be the sum of the deductible expenses and the standard deduction:

$11,000 + $6,100 = $17,100

So, the maximum total deduction that can be claimed is OE. $17,100.

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Related Questions

my son is having trouble with this problem he (6) can you tell him the answer 9+10

Answers

Answer:

19

Step-by-step explanation:

111111111

+

1111111111

count how many 1ns are there.

ans is 19

What is the amplitude of this function f(x) ? f(x)=3cos(2x)+5 enter your answer in the box.

Answers

Answer:

3

Step-by-step explanation:

Amplitude is the absolute value of the number in front

3cos(2x) + 5

What is the periodic interest rate for an account that is billed monthly with an APR of 21.99% round your answer to the nearest hundredth

Answers

Answer:

To calculate the periodic interest rate for an account billed monthly with an APR of 21.99%, we need to divide the APR by the number of billing periods in a year.


Since there are 12 months in a year for a monthly billing cycle, we can divide 21.99% by 12 to get the monthly periodic interest rate.


Periodic Interest Rate = APR / Number of Billing Periods in a Year


Periodic Interest Rate = 21.99% / 12


Periodic Interest Rate = 1.8325%


Rounding this answer to the nearest hundredth, we get a periodic interest rate of 1.83%.

A sequence $\{a_n\}$ satisfies $a_1 = 1$ and
\[a_n = \frac{a_{n - 1}}{1 + a_{n - 1}}\]for all $n \ge 2.$ Find $a_{10}.$

Answers

The sum of the given series is -

6 ∑{1/(2n + 1)² - 1}.

What is integration?

Integration is a process of adding up small elemental volumes to find the larger volume.

Given is the series as -

6/(3² - 1) + 6/(5² - 1) + 6/(7² - 1) + ....

The given series is -

S = 6/(3² - 1) + 6/(5² - 1) + 6/(7² - 1) + ....

S = 6{1/(3² - 1) + 1/(5² - 1) + 1/(7² - 1) + ....}

S = 6 ∑ {1/(2n + 1)² - 1}

Therefore, the sum of the given series is -

6 ∑{1/(2n + 1)² - 1}.

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Kristina paid mortgage interest of $6,320.00 and Medical Expenses of $$19,500.00 during 2022. Her GROSS INCOME was $32,500.
19. How much did she save by
itemizing her deductions?

Answers

Answer:

$448.25

Step-by-step explanation:

For her medical expenses, it has to be more than 7.5% of her gross income of $32,500 => 7.5% of $32,500 = $2437.5.

so her deduction for medical expenses is $19,500 - $2437.50 = $17,062.5

so her total itemized deduction is $6,320.00 + $17,062.50 = $23,382.50

Given Kristina will file a single so her standard deduction for 2022 is $12,950.

If she uses standard deduction, her taxable income is 32,500 - 19,500 = 13,000

For the first 10,000 taxed at 10% so it is 10,000 x 10% = $1,000

The next $3000 taxed at 12% = $360

Her total tax is $1,000 + $360 = $1,360

If she uses itemized deduction, her taxable income is 32,500 - 23,382.50 = 9117.5

For the first 10,000 taxed at 10% so it is 9117.5 x 10% = $911.75

The tax saved is $1,360 - $911.75 = $448.25

Hope this helps.

grogg draws the following figure. it so happens that $\angle sap$ is $174^\circ$ less than 4 times $\angle sop$. find the degree measure of $\angle sap.$

Answers

The degree measure of [tex]$\angle sap$ is $726^\circ - 4 \times \angle sop$.[/tex]

What is angle?

Angel is a geometric figure that can be described as the amount of rotation between two straight line or planes and angles is measured in degree with the full circle being 360°.

We can use the following formula to calculate the degree measure of $\angle sap$:

[tex]$\angle sap = 4 \times \angle sop - 174^\circ$[/tex]

Therefore, [tex]$\angle sap = 4 \times \angle sop - 174^\circ = 4 \times (180^\circ - \angle sop) - 174^\circ = 726^\circ - 4 \times \angle sop$[/tex]

Therefore, the degree measure of  [tex]$\angle sap$ is $726^\circ - 4 \times \angle sop$.[/tex]

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Evaluate the function f(x)=46x+2 at x=13.8 .

Answers

The value of f(13.8) = 636.8

What is a function?

function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

For example f(y) = 5y+7

Similar the function given ;

f(x)=46x+2

when x = 13.8

f(13.8) = 46(13.8) + 2

= 636.8

therefore the value of the function f(x)=46x+2 at x = 13.8 is 636.8

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The value of the function f(x) = 46x + 2 at x = 13.8 is 636.8.

What is function?

Function is a rule that assigns a unique output value to each input value. A function is typically denoted by a letter such as f(x), where x is the input variable and f(x) is the output value.

To evaluate the function f(x) = 46x + 2 at x = 13.8, we simply substitute x = 13.8 into the expression for f(x):

f(13.8) = 46(13.8) + 2

Simplifying this expression, we get:

f(13.8) = 634.8 + 2

f(13.8) = 636.8

Therefore, the value of the function f(x) = 46x + 2 at x = 13.8 is 636.8.

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Please help will get brainliest and 40 coins! <33

Answers

A. The height of the cuboid is 2.5 cm.

B. The suggested dimensions of the cuboid is 25 x 5 x 1.

What is the volume of a figure?

The volume of a given three dimensional figure is the total quantity of matter that it can contained. Thus various figures has different methods of determining their volume.

volume of a cube = length x length x length

                             = (length)^3

volume of a cuboid = length x width x height

Considering the given question, since the volumes of the cube and cuboid are the same, then we have:

volume of a cube = volume of a cuboid

(length)^3 = length x width x height

5^3 = 10 x 5 x height

125 = 50 x height

height = 125/ 50

           = 2.5

The height of the cuboid is 2.5 cm.

B. volume of the cuboid = length x width x height

                          = 10 x 5 x 2.5

                          = 125 cm^3

Thus possible dimensions for the cuboid 25 x 5 x 1.

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what is the slope of this graph​

Answers

The slope of this graph​ is equal to 0.375 or 3/8.

How to calculate the slope of a line?

In Mathematics, the slope of any straight line can be determined by using this mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = rise/run

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Substituting the given data points into the slope formula, we have the following;

Slope, m = (35 - 20)/(40 - 0)

Slope, m = 15/40

Slope, m = 0.375 or 3/8.

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Guys help!!

Slope of one of the dotted lines:

Slope of the line of reflection:

Answers

Dotted lines: 1-2
Line of Reflection: -2

Generate an equivalent expression using the Distributive Property.
question one: 6(x + 4)
question two: 2(5 + r)
question three: 11(n + 3)

Answers

Answer:

6(x + 4) can be distributed as 6x + 24.2(5 + r) can be distributed as 10 + 2r.11(n + 3) can be distributed as 11n + 33.

Step-by-step explanation:

The Distributive Property states that multiplying a sum by a number is the same as multiplying each addend in the sum by the number and then adding the products. In each of the given expressions, we can use the Distributive Property to expand the expression by multiplying the coefficient outside the parentheses to each term inside the parentheses, and then adding the resulting products. This results in an equivalent expression that is simplified but has the same value as the original expression.

Answer:

Step-by-step explanation:

6(x+4)

Distribute 6 between the terms x and 4 as:

6(x)+6(4)

=6x+24

2(5+r)

Distribute 2 between the terms 5 and r as:

2(5)+2(r)

=10+2r

11(n+3)

Distribute 11 between the terms n and 3 as:

11(n)+11(3)

=11n+33

A 16ft pipe has been cut into two parts, one 3 times as long as the other. How long (in ft) is each part?

Answers

Answer:

12 feet

Step-by-step explanation:

Let x be the length of the shorter part of the pipe.

According to the problem, the longer part is three times as long as the shorter part. Therefore, the length of the longer part is 3x.

We also know that the sum of the lengths of the two parts is equal to the original length of the pipe, which is 16 feet.

So we can write an equation:

x + 3x = 16

Simplifying this equation, we get:

4x = 16

Dividing both sides by 4, we get:

x = 4

So the length of the shorter part is 4 feet, and the length of the longer part is 3 times as long, or 3 × 4 = 12 feet.

16x2=32 32x3=64 answer is 64

Find the Area of the figure below, composed of a rectangle and a semicircle. Round to the nearest tenths place. PLEASE EXPLAIN WELL THANK YOU

Answers

Answer:

68.1 square units

Step-by-step explanation:

l = Length of rectangle

 = 9 units

w = Width of rectangle

   = 6 units


The dashed lines represent the diameter of the circle. Semi-circle is exact half of a circle. This diameter is parallel to the width of the rectangle and is equal in measurement:

r = Radius of semi-circle

 = [tex]\frac{Diameter}{2}[/tex]

 = [tex]\frac{6}{2}[/tex] units

 = 3 units

∴Area of composite solid = Area of rectangle + Area of semi-circle

                                           = [(l × w) + [tex]\frac{1}{2}(\pi r^{2})[/tex]] [tex]units^{2}[/tex]

                                          = {[9 × 6] + [tex]\frac{1}{2}[\pi (3)^{2}][/tex]} [tex]units^{2}[/tex]

                                          = [54 + [tex]\frac{9}{2}\pi[/tex]] [tex]units^{2}[/tex]

                                          = 68.137 [tex]units^{2}[/tex]

                                         = 68.1 square units (Rounded to the nearest tenths place)

             

Evaluate the expression 1/3x^2 for x=6
Please help

Answers

1/3x6=
1/3 x6/1 = 6/3 =2
2^2=4

-95 times a number -14 is equal to -70 one less thing the number

Answers

The equation for the given statement can be written as:

-95x - 14 = -70 - x

To solve for x, we can rearrange the equation and combine like terms:

-95x + x = -70 + 14

-94x = -56

Next, we can divide both sides by -94 to isolate x:

x = -56/-94

Finally, we can simplify the fraction to get our answer:

x = 28/47

Therefore, the solution to the equation is x = 28/47.

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Solve the system of equations. Write the solution as an ordered pair. 4x + 3y = 36 and y = -1/3x + 6

Answers

Substitute the expression for y in the second equation into the first equation and solve for x:

4x + 3(-1/3x + 6) = 36

4x - x + 18 = 36

3x = 18

x = 6

Now substitute x = 6 into the second equation to solve for y:

y = (-1/3)(6) + 6

y = 4

Therefore, the solution to the system of equations is the ordered pair (6, 4).

Sorry about the bad quality of the pic, can i get an explanation with answer, thanks. :)

Answers

Answer:

3. Initial cost = 99 dollars.  Each hour (h) is 19 dollars. The total value will be initial cost + hours*(cost per hour)=> t=99+19h.

4. She will be using the car for 3 days. Each day costs 29.99. There is no initial cost, so her total cost will be 29.99*3=89.97 dollars.

A baker produces 500 loaves a day. On Monday, 50 loaves did not meet quality standards. The baker generates a random sample to simulate 10 loaves to inspect on Tuesday. The integers 1 to 50 represent sub-standard loaves.
351 207 148 428 272
121 47 205 56 4
Based on this sample, how many loaves will not meet quality standards on Tuesday?

What is the difference between the number of sub-standard loaves produced on Monday and the number predicted to be sub-standard on Tuesday?

Answers

Answer:

To determine how many loaves will not meet quality standards on Tuesday, we need to count the number of integers in the sample that are between 1 and 50, inclusive. From the given sample, we can see that there are 2 integers that fall between 1 and 50: 47 and 4. Therefore, we can predict that 2 loaves will not meet quality standards on Tuesday.

To calculate the difference between the number of sub-standard loaves produced on Monday and the number predicted to be sub-standard on Tuesday, we need to subtract the number of sub-standard loaves on Monday from the number predicted for Tuesday.

Number of sub-standard loaves on Monday = 50

Number predicted to be sub-standard on Tuesday = 2

Difference = 2 - 50 = -48

The difference is -48, which means that there were 48 fewer sub-standard loaves predicted on Tuesday than the number produced on Monday.

A triangle has two sides of length 7 and 10. What is the largest possible whole-number length for the third side?

Answers

Answer:

16 units.

------------------

According to triangle inequality theorem, any side of a triangle is less than the sum of the other two.

Let the missing side is largest, then it must be less than:

7 + 10 = 17

The closest whole number less than 17 is 16.

The figure shows the graph of the function.
f(x)=

Answers

The domain are the possible input while the range are the possible output of a function.

(a) The domain = [-√2, √2], the range = [0, 2]

(b) The domain = [-1, 1], the range = [0, 1]

(c) The domain = [-1, 1], the range = [0, -1]

(d) The domain = [0, 2], the range = [0, 1]

(e) The domain = [-(2 + √2), (√2 - 2)], the range = [0, 2]

Reasons:

The given functions can be expressed by the equation; (-x + 1)·(x + 1) = -x² + 1

Therefore, we have;

(a) y = f(x) + 1 = -x² + 1 + 1 = -x² + 2

The x-intercept of the above function are, x = √2, and x = -√2

Which gives;

The domain = [-√2, √2]

The range = [0, 2]

(b) y = 3·f(x) = 3 × (-x² + 1) = -3·x² + 3

At the x–intercepts, we have;

-3·x² + 3 = 0

x = ±1

The domain = [-1, 1]

The maximum value of y is given at x = 0, therefore;

= -3 × 0² + 3 = 3

The range = [0, 1]

(c) y = -f(x) = -(-x² + 1) = x² - 1

At the x–intercepts, x² - 1 = 0

x = ± 1

The domain = [-1, 1]

The minimum value of y is given at x = 0, which is y = -1

The range = [0, -1]

(d) y = f(x - 1) = -(x - 1)² + 1 = -x² + 2·x

At the x–intercepts, we have;  -x² + 2·x = 0, which gives;

(-x + 2)·x = 0

Which gives, x = 0, or x = 2

The domain = [0, 2]

The maximum value of y is given when x = -b/(2·a) = -2/(2×(-1)) = 1

y = f(1) =  -1² + 2×1 = 1

Therefore;

The range = [0, 1]

(e) y = f(x + 2) + 1 = (-(x + 2)² + 1) + 1 = -x² - 4·x - 2

At the x–intercepts, we have;  -x² - 4·x - 2 = 0, which gives;

x = -(2 + √2) or x = x = √2 - 2

The domain = [-(2 + √2), (√2 - 2)]

The maximum value of y is given when x = -4/(2)) = -2

Which gives;

-(-2)² - 4·(-2) - 2 = 2

The range = [0, 2]

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find the volume of a sphere vd(r) of radius r in d dimensions. in order to accomplish this task, first find the surface area ad of a sphere of unit radius in d dimensions by considering the integral

Answers

The formula for the volume of a d-dimensional sphere of radius r is:

V_d(r) = (π^(d/2) / Γ(d/2 + 1)) * r^d

How to explain the volume

It should be noted that in the formula, Γ is the gamma function.

Let's break down the formula:

π^(d/2) is pi raised to the power of d/2.

Γ(d/2 + 1) is the gamma function evaluated at d/2 + 1.

r^d is the radius raised to the power of d.

Putting it all together, we get the formula above.

For a 2-dimensional sphere (a circle) of radius r, we have d=2, so:

V_2(r) = (π^(2/2) / Γ(2/2 + 1)) * r^2

= (π / Γ(3/2)) * r^2

= (π / (1/2) * √π) * r^2

= 2πr^2

For a 3-dimensional sphere of radius r, we have d=3, so:

V_3(r) = (π^(3/2) / Γ(3/2 + 1)) * r^3

= (4/3) * π * r^3

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Find the value of each variable, help pls??

Answers

Answer:

x = 145

Step-by-step explanation:

We are given a polygon that has 7 sides. We can use the formula [tex]S=(n-2) * 180[/tex] which allows us to write an algebraic expression to solve for x where n = number of sides.

[tex]S=(7-2) * 180[/tex]

Evaluate

[tex]S=5*180[/tex]

[tex]S=900[/tex]

Now that we have our number, lets write an algebraic expression that will help us find the value of x.

[tex]x+116+129+120+130+135+125=900[/tex]

Add like terms

[tex]x+755=900[/tex]

Subtract 755 on both sides

[tex]x=145[/tex]

We can check our work by adding all the angle measures together

[tex]145+116+129+120+130+135+125=900[/tex]

Thus we know 145 is the correct answer.

Using the digits 0-9 once, create accurate number line

Answers

The digits are arranged in increasing order from left to right, and that each digit is used exactly once.

What is a number line?

A number line is a visual representation of numbers arranged in a straight line from left to right or from right to left. It is a tool commonly used in mathematics to help visualize numerical relationships, such as addition, subtraction, multiplication, and division.

A typical number line starts at zero in the middle and extends infinitely in both directions, with positive numbers increasing to the right and negative numbers decreasing to the left. The distance between any two points on the number line represents the absolute difference between the corresponding numbers.

Here's a number line using the digits 0-9 exactly once:

-9 --- -8 --- -7 --- -6 --- -5 --- -4 --- -3 --- -2 --- -1 --- 0 --- 1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9

Note that the digits are arranged in increasing order from left to right, and that each digit is used exactly once.

Therefore, the digits are arranged in increasing order from left to right, and that each digit is used exactly once.

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read in the values for a tic tac toe game and evaluate whether x or o won the game. the first number in the files represents the number of data sets to follow. each data set will contain a 9 letter string. each 9 letter string contains a complete tic tac toe game.

Answers

To evaluate whether x or o won the tic tac toe game, we need to check for the three possible winning conditions:
- A horizontal row of three x's or o's
- A vertical column of three x's or o's
- A diagonal of three x's or o's

Here's the step-by-step process:

1. Read in the first number from the file, which represents the number of data sets to follow.
2. For each data set, read in the 9 letter string representing the tic tac toe game.
3. Check for the three winning conditions by comparing the values in the string.
4. If any of the winning conditions are met, return the winning player (x or o).
5. If none of the winning conditions are met, return "No winner".

Here's the code in Python:

```
# Read in the first number from the file
num_data_sets = int(input())

# Loop through each data set
for i in range(num_data_sets):
 # Read in the 9 letter string
 game = input()

 # Check for the three winning conditions
 if (game[0] == game[1] == game[2]) or (game[3] == game[4] == game[5]) or (game[6] == game[7] == game[8]) or (game[0] == game[3] == game[6]) or (game[1] == game[4] == game[7]) or (game[2] == game[5] == game[8]) or (game[0] == game[4] == game[8]) or (game[2] == game[4] == game[6]):
   # If any of the winning conditions are met, return the winning player
   print(game[0])
 else:
   # If none of the winning conditions are met, return "No winner"
   print("No winner")

```

This code will read in the values for a tic tac toe game and evaluate whether x or o won the game.

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A certain company creates handmade ceramic decorative pieces. Because they are handmade, each piece is unique. The following data represents the number pieces created by a certain craftsman on eight days, \( x \), and the total weight of the pieces in ounces, y. A table of the number of crafts, \( x \), and the number of ounces, \( y \) Part 1: Complete the following table (The last column is for your sums): A table of the number of rrafte \( x \) and the numher of aunroe is
Part 1: Complete the following table (The last column is for your sums): A table of the number of crafts. \( x \) a and the number of ounces. \( v \) Part 2: Use the sums from the table in Part 1 along with the Correlation Coefficient formula to find the Correlation Coefficient. (Make sure you have read the quiz directions)

Answers

The solution for the table has been provided below. It has been obtained by using arithmetic operations.

What are arithmetic operations?

Mathematicians claim that all real numbers may be described using the four basic operations, also referred to as "arithmetic operations." The four mathematical operations that produce quotient, product, sum, and difference, respectively, are divide, multiply, add, and subtract.

We are given a table of the number of crafts x and the number of ounces y.

x        4          9         7            6            5           5            4           8

y        19        46       30          31          23         28          21         42

xy     76       414      210        186        115         140        84         336

      16       81         49         36         25          25         16          64

y²       361     2116     900      961        529        784       441        1764

Hence, the table has been completed.

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The function h(x)
f(x) is defined as:
f(x)=
=
1
x-3
<
>
can be expressed in the form f(g(x)) where g(x) = (x − 3) and
-

Answers

Answer:

[tex]f(x)=\frac{1}{x}[/tex]

Step-by-step explanation:

You are looking for a function that when you plug g(x) into f(x), you get the result of h(x).

If we define f(x) as 1/x then that would mean when we plug in g(x) we would get h(x) = 1 / (x-3), which is what we're looking for.

can you help me with this exercise please

Answers

The solution of the system of equations is

x = 1/2 and y = -2

What are system of equations?

A system of equations are a pair of equations.

How to solve the system of equations?

Given the system of equations

4/x - 2/y = 12 (1) and

5/x + 1/y = 8 (2)

We proceed to solve as follows.

Since we have

4/x - 2/y = 12 (1) and

5/x + 1/y = 8 (2)

Multiplying equation (1) by 1 and equation (2) by 2, we have that

4/x - 2/y = 12 (1) × 1 and

5/x + 1/y = 8 (2) × 2

4/x - 2/y = 12 (4)  and

10/x + 2/y = 16 (5)

Now, adding equations (4) and (5), we have that

4/x - 2/y = 12 (4)  

+

10/x + 2/y = 16 (5)

14/x = 28

14 = 28x

x = 14/28

x = 1/2

Substituting x = 1/2 into equation (1), we have that

4/x - 2/y = 12 (1)

4/(1/2) - 2/y = 12 (1)

4 × 2 - 2y = 12

8 - 2y = 12

-2y = 12 - 8

-2y = 4

y = 4/-2

y = -2

So,

x = 1/2 and y = -2

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Need help solving problem

Answers

The equations that determine the cost of each hamburger and each fries is 4x + 5y = 20 and 10x + 7y = 41.20

What is an equation?

An equation is an expression that shows the relationship between two numbers and variables using mathematical operations like addition, subtraction, multiplication, exponents

Let x represent the cost of each hamburger and y represent the cost of each fries.

She purchased 4 hamburgers and 5 fries for $20.00, hence:

4x + 5y = 20    (1)

She purchased 10 hamburgers and 7 fries for $41.20, hence:

10x + 7y = 41.20   (2)

From both equations, solving simultaneously:

x = 3; y = 1.6

The cost of each hamburger is $3 while the cost of each fries is $1.6

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What is the height of the statue?
SEE ATTACHED.

Answers

Answer:

x ≈ 2.7 m

Step-by-step explanation:

since the 2 triangles are similar then the ratios of corresponding sides are in proportion, that is

[tex]\frac{x}{1.6}[/tex] = [tex]\frac{4}{2.4}[/tex] ( cross- multiply )

2.4x = 4 × 1.6 = 6.4 ( divide both sides by 2.4 )

x = [tex]\frac{6.4}{2.4}[/tex] = 2.6666...

height of statue x ≈ 2.7 m ( to 1 decimal place )

show how the damping ratio for a mass-spring-damper system is defined by the ode for a mass-spring-damper system in terms of mass, m, stiffness, k, and damping, b?

Answers

The equation that shows  the damping ratio for a mass-spring-damper system which is defined in terms of the mass, m, stiffness, k, and damping, b is  ζ = b / (2sqrt(mk)).

What is damping ratio?

The damping ratio ζ is defined as the ratio of the damping coefficient b to the critical damping coefficient b_c, which is the minimum damping required to prevent the system from oscillating.

The damping ratio is a dimensionless quantity that represents the amount of damping in the system relative to the critical damping needed to prevent oscillations. A higher damping ratio indicates a more heavily damped system, while a lower damping ratio indicates a less damped system that may oscillate more.

The motion of a mass-spring-damper system can be described by a second-order linear ordinary differential equation (ODE) of the form:

md²x/dt² + bdx/dt + kx = F(t)

where m is the mass of the system, b is the damping coefficient, k is the spring constant, x is the displacement of the mass from its equilibrium position, and  the external force applied to the system is F(t).

The critical equation will be -

b_c = 2√(mk)

Therefore, the damping ratio can be expressed as:

ζ = b / b_c

Substituting the expression for b_c into the above equation, we get:

ζ = b / (2√(mk))

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