How to add these two exponentials? 33.65 e^{j(377 t-25.5)}+31.39 e^{j(377 t+172.8)}

The given function evaluates to [tex]\frac{x^2(1-x)}{x^2 + 1}[/tex].

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences. The German mathematician Peter Dirichlet initially offered the contemporary definition of function in 1837: A variable y is said to be a function of the independent variable x if there is a relationship between them such that whenever a numerical value is assigned to x, there is a rule that determines a specific value of y.

As per the given data:

The function is [tex]f(x) = \frac{2}{x^2 + 1}[/tex]

To determine the function: [tex]f(\frac{1}{x}) - x^2f(-1)[/tex]

Substituting (1/x) and (-1) respectively:

[tex]= \frac{2}{\frac{1}{x}^2 + 1} - x^2\frac{2}{(-1)^2 + 1}[/tex]

[tex]= \frac{2x^2}{x^2 + 1} - x^2\frac{2}{2}[/tex]

[tex]= \frac{2x^2}{x^2 + 1} - x^2[/tex]

Taking the LCM:

[tex]= \frac{2x^2 - x^3 - x^2}{x^2 + 1}[/tex]

[tex]= \frac{x^2 - x^3}{x^2 + 1}\\= \frac{x^2(1-x)}{x^2 + 1}[/tex]

Hence, the given function evaluates to [tex]\frac{x^2(1-x)}{x^2 + 1}[/tex]

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The number of child tickets were sold that day are 73.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.

We are given that;

Cost of child admission= $5.30

Cost of adult admission= $8.70

Now,

Let's assume that "c" represents the number of child tickets sold, and "a" represents the number of adult tickets sold. We can create two equations based on the given information:

c + a = 163 (equation 1: the total number of tickets sold)

5.30c + 8.70a = 1169.90 (equation 2: the total sales)

We can use equation 1 to solve for "a" in terms of "c":

a = 163 - c

Substituting this expression for "a" into equation 2, we get:

5.30c + 8.70(163 - c) = 1169.90

Simplifying and solving for "c", we get:

5.30c + 1418.10 - 8.70c = 1169.90

-3.4c = -248.20

c = 73

Therefore, by algebra the answer will be 73.

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Ms. Twinkle walks a total of 6.6 miles around the park.

What is perimeter ?

Perimeter is the total distance around the outside of a two-dimensional shape. It is the sum of the lengths of all the sides of the shape.

To find how far Ms. Twinkle walks, we need to find the perimeter of the rectangle-shaped park, which is the sum of the lengths of all its sides.

The length of the park is 2 miles and the width is 1 3/10 miles. We can write the width as a mixed number of 1 + 3/10 = 13/10 miles.

Therefore, the perimeter of the park is:

Perimeter = 2(length + width)

Perimeter = 2(2 + 13/10)

Perimeter = 2(20/10 + 13/10)

Perimeter = 2(33/10)

Perimeter = 66/10

Perimeter = 6.6 miles

Therefore, Ms. Twinkle walks a total of 6.6 miles around the park.

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a)f(x) = { 0, if x < 0 \\ 4x, if 0 ≤ x < 4 \\ 0, if x ≥ 4 }

b)P(X=-1) = 0

c)P(X ≤ 3) = 4 * 3 = 12

a) The pdf (probability density function) of the variable X is given by
f(x) =
{
   0, if x < 0 \\
   4x, if 0 ≤ x < 4 \\
   0, if x ≥ 4
}

b) P(X=-1) = 0, since x < 0

c) P(X ≤ 3) = 4 * 3 = 12

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The formula for the exponential function that passes through the given points is (a) 3(2ˣ) and (b) g(x) = (2(3)^(1/2))((1/3)^(x/6)).

To find the formula for an exponential function that passes through two points, we can use the formula:
f(x) = abˣ
where a and b are constants. We can use the two points to create a system of equations and solve for the constants.

For part a:
3/4 = ab⁻²
12 = ab²

Dividing the second equation by the first equation gives us:
16 = b⁴

Taking the fourth root of both sides gives us:
b = 2

Plugging this value back into the first equation gives us:
3/4 = a2⁻²
3/4 = a(1/4)
a = 3

So the formula for the exponential function is:
f(x) = 3(2ˣ)

For part b:
6 = ab⁻³
2 = ab³

Dividing the second equation by the first equation gives us:
1/3 = b⁶

Taking the sixth root of both sides gives us:
b = (1/3)^(1/6)

Plugging this value back into the first equation gives us:
6 = a(1/3)^(-1/2)
6 = a(3)^(1/2)
a = 6 / (3)^(1/2)
a = 2(3)^(1/2)

So the formula for the exponential function is:
g(x) = (2(3)^(1/2))((1/3)^(x/6))

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Answer:

<RHF = 68°

Step-by-step explanation:

<RHG and <FHY are vertical opposite angle which mean they equal to each other.

<RHG = <FHY

(6m+88) = (7m+84)

88 - 84 = 7m - 6m

4 = m

Plug m=4 into the <RHG to solve for the measurement.

<RHG = 6m+88

<RHG = 6*4+88

<RHG = 24+88

<RHG = 112°

<RHG and <RHF are supplementary angle which mean they both linear adding up to 180°

<RHF + <RHG = 180°

<RHF + 112° = 180°

<RHF = 68°

The total number of months he made deposits and multiply that by the monthly deposit amount.

Chase started an IRA at the age of 26 and deposited $400 each month until he retired at the age of 65. To determine the amount of money Chase deposited over the length of the investment, we need to calculate the total number of months he made deposits and multiply that by the monthly deposit amount.

Number of months = (Retirement age - Starting age) * 12
Number of months = (65 - 26) * 12
Number of months = 468

Total deposits = Number of months * Monthly deposit amount
Total deposits = 468 * 400
Total deposits = $187,200

To determine how much Chase made in interest upon retirement, we need to subtract the total deposits from the retirement savings.

Interest = Retirement savings - Total deposits
Interest = $838,879.58 - $187,200
Interest = $651,679.58

Therefore, Chase deposited a total of $187,200 over the length of the investment and made $651,679.58 in interest upon retirement.

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The triangle with measure of the angle is 30°, 60°, and 90° proves that it is right angled triangle.

Let us consider 'x' be the measure of the angles in a triangle.

Ratio of the angles in a triangle is 1 : 2 : 3

Measure of angle 1 = 1x

Measure of angle 2 = 2x

Measure of angle 3 = 3x

Sum of all the interior angles in a triangle = 180°

Substitute the value of the measure of the angles we get,

⇒ 1x + 2x + 3x = 180°

⇒ 6x = 180°

⇒ x = 180° / 6

⇒ x = 30°

Measure of angle 1 = 30°

Measure of angle 2 = 2× 30°

                                 = 60°

Measure of angle 3 = 3× 30°

                                 = 90°

Therefore, as measure of one of the interior angle is 90 degree this implies it is a right angled triangle.

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Answer:

Step-by-step explanation:

The formula to calculate the doubling time for continuously compounded interest is:

t = ln(2) / (r * ln(1 + (r/n)))

where t is the time in years, r is the annual interest rate as a decimal, and n is the number of times the interest is compounded per year (which is infinity for continuous compounding).

For Logan's investment, we have:

r = 0.05

n = infinity

t1 = ln(2) / (0.05 * ln(1 + (0.05/infinity)))

t1 ≈ 13.86 years

For Qasim's investment, we have:

r = 0.06

n = 1

t2 = ln(2) / (0.06 * ln(1 + (0.06/1)))

t2 ≈ 11.55 years

To find the difference in the time it takes for their investments to double, we can subtract t2 from t1:

t1 - t2 ≈ 13.86 - 11.55 ≈ 2.31

So it would take Logan's money approximately 2.31 years longer to double than Qasim's money, to the nearest hundredth of a year.

arctan(4/7) + arctan(7/4) is 90 degrees

What is Trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, and the functions that describe those relationships. It has applications in fields such as engineering, physics, astronomy, and navigation.

We can use the following trigonometric identity to simplify the expression:

arctan(x) + arctan(1/x) = pi/2

where x is a positive number.

In this case, we can set x = 4/7, which is a positive number, and apply the identity to get:

arctan(4/7) + arctan(7/4) = pi/2

Therefore, the sum of the two arctan terms simplifies to pi/2.

So, the simplified expression is:

arctan(4/7) + arctan(7/4) = 90 degrees (rounded to the nearest degree).

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The point of intersection of the two lines is the fixed point or center of the point reflection.

A point reflection is a geometric transformation that maps a point across a fixed point called the center of reflection. The transformation involves reflecting the point across two intersecting lines that pass through the center of reflection. The point reflection can be thought of as a composite of two reflections, one across each line of reflection. The result is a mirror image of the original point that is equidistant from the center of reflection. The concept of point reflection is used in geometry to construct symmetrical figures and in crystallography to describe the symmetry of crystals.

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Answer:its 80 i did the same problem

Step-by-step explanation:

The solution to the equation (-3/4)(-16 + 8x) by applying the distributive property is 12 - 6x

An equation is an expression containing numbers and variables linked together by mathematical operations such as addition, subtraction, division, multiplication and exponents.

The distributive property states that for three variables a, b and c, the following rule apply:

a(b + c) = ab + ac

Given that:

(-3/4)(-16 + 8x)

Applying the distributive property:

= (-3/4)(-16) + (-3/4)(8x)

= 12 - 6x

The solution is 12 - 6x

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B. Lela walked f miles yesterday, and miles fewer today.

What is expression?

One or more variables or numbers are combined with one additional action to form an expression.

The situation that could be described by the expression f + 1/2 is:

B. Lela walked f miles yesterday, and miles fewer today.

The expression f + 1/2 represents the total distance that Lela walked over two days, where "f" represents the distance she walked yesterday, and "1/2" represents the distance she walked today.

Option B describes the situation where Lela walked "f" miles yesterday and "1/2" times fewer miles today (or, equivalently, "1/2" times the distance she walked yesterday).

Therefore, the expression f + 1/2 is a valid way to describe this situation.

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Landon would make a salary of $96,000 in his 23rd year working at this job.

Algebraic expressions are mathematical statements with a minimum of two terms containing variables or numbers.

If Landon starts with a salary of $30,000 and gets a raise of $3,000 every year going forward, then his salary in the 23rd year can be calculated as follows:

Salary in 2nd year = $30,000 + $3,000 = $33,000

Salary in 3rd year = $33,000 + $3,000 = $36,000

Salary in 4th year = $36,000 + $3,000 = $39,000

and so on...

So, we can see that Landon's salary increases by $3,000 every year.

His salary in the 23rd year would be:

Salary in 23rd year = $30,000 + ($3,000 x 22) = $30,000 + $66,000 = $96,000

Therefore, Landon would make a salary of $96,000 in his 23rd year working at this job.

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To input the given commands and find the Z-scores for the different heights, you will need to use the following steps:

1. Convert the heights from feet and inches to inches. For example, 6'4" would be 76 inches, 5'8" would be 68 inches, and 5'2" would be 62 inches.

2. Use the formula Z = (X - mean) / standard deviation to find the Z-scores for each height. For example, for the height of 6'4", you would use the formula Z = (76 - height.xbar) / height.s.

3. Use the pnorm command to find the percentiles for each Z-score. For example, for the Z-score of the height 6'4", you would use the command pnorm(Z).

4. Use the pnorm command with the lower.tail = FALSE argument to find the portion of people taller than a given height. For example, for the height of 6'4", you would use the command pnorm(Z, lower.tail = FALSE).

5. Use the pnorm command with the lower and upper limits to find the portion of people between two heights. For example, for the heights of 5'2" and 5'8", you would use the command pnorm(upper limit) - pnorm(lower limit).

Using these steps, you can input the given commands and find the Z-scores and percentiles for the different heights. Remember to use the variable names height.xbar and height.s for the mean and standard deviation of the height variable, and to use the number, variable, and height terms in your answer.

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The answer response are:

   A duck with wings: Likely

   Duck number 7: Unlikely

   A duck with a number greater than 3: Certain

   Duck number 15: Impossible

   A duck with a number greater than 10: Unlikely

   A duck with a number: Certain

   Duck number 4: Unlikely

   A duck with sunglasses: Impossible

   A duck with a hat: Unlikely

Lastly,  An even-numbered duck: Likely

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See transcribed text below



Date:

Period:

What is the probability you will choose each duck described below? Write the answer as a fraction in lowest terms and place it in the appropriate box (certain, likely, unlikely, impossible).

33

1. A duck with wings

2. duck number 7

3. a duck with a number greater than 3

4. duck number 15

5. a duck with a number greater than 10

6. a duck with a number

7. duck number 4

8. a duck with sunglasses

9. a duck with a hat

10. an even-numbered duck

The ducks are numbered From one through twelve!

Kibb

Certain

Likely

Even

Unlikely Impossible

ว

Mengxi invested $4,200 in the term deposit paying 5% /year and $5,800 in Canada Savings Bonds paying 3.5% /year. The total interest earned is $413.

Let's assume that Mengxi invests x dollars in the term deposit paying 5% /year, and the remaining (10000 - x) dollars in Canada Savings Bonds paying 3.5% /year.

At the end of the year, Mengxi earns a total of $413 in simple interest. The interest earned from the investment in the term deposit is calculated as 0.05x, while the interest earned from the investment in Canada Savings Bonds is 0.035(10000 - x).

Thus, we can write the following equation to represent the total interest earned:

0.05x + 0.035(10000 - x) = 413

Simplifying the equation, we get:

0.015x + 350 = 413

0.015x = 63

x = 4200

Therefore, Mengxi invested $4,200 in the term deposit paying 5% /year, and $5,800 (10000 - 4200) in Canada Savings Bonds paying 3.5% /year.

To check the answer, we can calculate the interest earned from each investment and add them up:

0.05(4200) + 0.035(5800) = 210 + 203 = 413

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Answer:

i need more details i cant answer.

Answer:

Let's call the length of the rectangle "L" and the width "W".

From the problem, we know that:

W = L - 4

And

Area = Length x Width

Substituting the first equation into the second equation, we get:

Area = L x (L - 4)

We also know that the area is 21 square units, so we can set up the following equation:

21 = L x (L - 4)

Expanding the right side of the equation:

21 = L^2 - 4L

Rearranging the terms:

L^2 - 4L - 21 = 0

Now we can solve for L using the quadratic formula:

L = (-b ± sqrt(b^2 - 4ac)) / 2a

Where a = 1, b = -4, and c = -21

L = (-(-4) ± sqrt((-4)^2 - 4(1)(-21))) / 2(1)

L = (4 ± sqrt(100)) / 2

L = (4 ± 10) / 2

L = 7 or L = -3

Since the length cannot be negative, we choose L = 7.

Therefore, the length of the rectangle is 7 units.

Volume is the quantum of space an object occupies. Saddie uses 8 pints of lemon- lime juice rather of orange juice.

This is simply because there are 4 cups of juice in 1 liter. volume is a measure of space possessed by matter.

It is frequently quantified using SI deduced units or colorful US Customary or Imperial units. The description of length refers to volume.

A liter is a measure of commodity liquid, like milk or makeup. A gallon is four liters.

Question:

Sadie is making punch. how numerous further quarts of bomb lime juice will she use than orange juice?

constituents for punch-

8qts Lemon Lime Juice. 4 scoops of vanilla ice cream and 8 mugs of orange juice

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Using the gratuity percentages, we found that the amount of meal is $32.6 and the amount to be paid is $41.89

What is meant by percentage?

A figure or ratio stated as a fraction of 100 is called a percentage. Frequently, it is indicated with the per cent sign, "%". If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The percentage, therefore, refers to a component per hundred. Per 100 is what the word per cent means. As there is no unit of measurement for percentages, they are dimensionless numbers. This is because we divide numbers with the same units in percentage calculation.

Given,

Gratuity guide :

15% would be $4.89

18% would be $5.87

20% would be $6.52

1) The tip percentages are given above.

If the cost of the meal is x.

15% of x = 4.89

18% of x = 5.87

0.15x = 4.89

x = 32.6

0.18x = 5.87

x = 32.6

So the cost of the meal is $32.6.

2)The family choose a 20% tip.

   The tax percentage is 8.5% of 32.6.

Tip = 6.52 ( from gratuity guide)

Tax  =  8.5 % of 32.6

    = 0.085 * 32.6 = $2.77

Total amount to be paid = 32.6 + 6.52 + 2.77 = $41.89

Therefore using the gratuity percentages, we found that the amount of meal is $32.6 and the amount to be paid is $41.89.

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Answer:

system of equation:

(x,y)

(3,-50)

Step-by-step explanation:

equation solving

The simplified form of the given expression by the laws of indices is; x^(-3/10).

As evident in the task content; the given expression is;

1 / ⁵√√x³

Therefore, we have that;

By the laws of indices we have that;

= 1 / ⁵√x^(3/2)

= 1 / x^(3/10)

On this note, the simplified form of the expression as required is; x^(-3/10).

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The zeros of the function c(x) = 6x^3 + 3x^2 - 3x are x = 0, 1/2, and -1, all with a multiplicity of 1.

To find the zeros of the function c(x) = 6x^3 + 3x^2 - 3x, we need to factor the equation and set it equal to zero.

First, we can factor out a common factor of 3x from all of the terms:
c(x) = 3x(2x^2 + x - 1)

Next, we can use the quadratic formula to find the zeros of the remaining quadratic equation:
2x^2 + x - 1 = 0

x = (-1 ± √(1^2 - 4(2)(-1)))/(2(2))
x = (-1 ± √(1 + 8))/4
x = (-1 ± √9)/4
x = (-1 ± 3)/4

So, the two zeros of the quadratic equation are:
x = (-1 + 3)/4 = 1/2
x = (-1 - 3)/4 = -1

Finally, we can combine the zero from the factored term with the zeros from the quadratic equation to find all of the zeros of the function:
x = 0, 1/2, -1

The multiplicities of these zeros are:
x = 0 has a multiplicity of 1
x = 1/2 has a multiplicity of 1
x = -1 has a multiplicity of 1

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The linear equation that shows the interest earned from each investment is (d): "4a + 6b + 5c = 950"

Investment refers to the act of allocating money or other resources to an asset or venture with the expectation of generating a profit or some other form of return over a period of time.  The primary goal of investing is to use the available resources to generate a return that is greater than the initial investment, thus growing wealth over time.

A linear equation is an algebraic equation in which the highest power of the variable is one. In other words, a linear equation is a polynomial of degree one. It represents a straight line when plotted on a graph. The general form of a linear equation with one variable, x, is:

ax + b = 0

In the given question,

Let's assume that Mr. Silverstone invested a dollars in the first annuity with an interest rate of 2%, b dollars in the second annuity with an interest rate of 4%, and c dollars in the bond with an interest rate of 5%. The interest earned from each investment can be calculated as follows:

Interest earned from the first annuity = 0.02a

Interest earned from the second annuity = 0.04b

Interest earned from the bond = 0.05c

The total interest earned from all three investments together is given as $950. Therefore, we can write the equation:

0.02a + 0.04b + 0.05c = 950

This is the same as option (c) given in the question: "0.04a + 0.06b + 0.05c = 950". However, option (c) has some errors in the coefficients, as the interest rates for the annuities were given as 2% and 4%, not 4% and 6%.

Therefore, the linear equation that shows the interest earned from each investment if the total was $950 is:

0.02a + 0.04b + 0.05c = 950

This can also be simplified as:

2a + 4b + 5c = 95000 (by multiplying both sides by 100 to remove the decimals)

So, the correct option is (d): "4a + 6b + 5c = 950".

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The volume is 3x³ + 16x² + 3x - 10, while when the height is reduced by 50%, the new volume is (1/2)(3x³ + 16x² + 3x - 10)

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

The volume of the triangular prism is:

Volume = area of base * height

Substituting:

Base area = (1/2) * (2x + 2) * (x + 5) = x² + 6x + 5; height = 3x - 2

Volume = area of base * height = (x² + 6x + 5)(3x - 2)

Volume = 3x³ + 18x² + 15x - 2x² - 12x - 10 = 3x³ + 16x² + 3x - 10

The volume is 3x³ + 16x² + 3x - 10

If the height is reduced by 50%, new height =  (1/2)(3x - 2)

Volume = area of base * height = (x² + 6x + 5) * (1/2)(3x - 2)

Volume = (1/2)(3x³ + 16x² + 3x - 10)

The new volume is (1/2)(3x³ + 16x² + 3x - 10)

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The simplified functions of f(g(x)) and g(f(x)) are: f(g(x))=x3 and g(f(x))=x3−6x2+12x−6

To find the function f(g(x)), we need to substitute g(x) into f(x). This means we will replace the x in f(x) with the expression for g(x):

f(g(x))=f(x3+2)=(x3+2)−2

Simplifying this expression gives us:

f(g(x))=x3

Similarly, to find the function g(f(x)), we need to substitute f(x) into g(x):

g(f(x))=g(x−2)=(x−2)3+2

Simplifying this expression gives us:

g(f(x))=x3−6x2+12x−6

Therefore, the functions f(g(x)) and g(f(x)) are:

f(g(x))=x3

g(f(x))=x3−6x2+12x−6

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The depth of water is fount to be 2.0 fathom when its was 12 feet.

Given:

Conversion unit:

1 foot = 0.167 fathom

Then,

12 feet = 0.167*12 fathom

12 feet = 2.004 fathom

12 feet = 2.0 fathom

Thus, the depth of the water is fount to be 2.0 fathom when its was 12 feet.

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