Three years ago A's age was double of B's
age. Seven years hence, the sum of their
ages will be 83 years. Find the present
age of A?​

By De Moivre's formula, we have the following four roots: [tex]z_{1} = 1.565 \cdot \left(\cos \frac{\pi}{6} + i\,\sin \frac{\pi}{6} \right)[/tex], [tex]z_{2} = 1.565\cdot \left(\cos \frac{2\pi}{3} + i\,\sin \frac{2\pi}{3} \right)[/tex], [tex]z_{3} = 1.565\cdot \left(\cos \frac{7\pi}{6}+i\,\sin \frac{7\pi}{6} \right)[/tex] and [tex]z_{4} = 1.565\cdot \left(\cos \frac{5\pi}{3} + i\,\sin \frac{5\pi}{3} \right)[/tex].

According to complex analysis, the fourth roots of a complex number can be found by De Moivre's formula:

[tex]\sqrt[n]{z} = \sqrt[n]{r}\cdot \left[\cos \left(\frac{\theta + 2\pi\cdot i}{n} \right)+ i\,\sin \left(\frac{\theta + 2\pi\cdot i}{n} \right)\right][/tex], for [tex]i = \{0, 1, ..., n - 1\}[/tex]     (1)

Where:

First, we find the magnitude and direction of the complex number:

[tex]r = \sqrt{(-3)^{2}+(3\sqrt{3})^{2}}[/tex]

r = 6

[tex]\theta = \tan^{-1} \left(\frac{3\sqrt{3}}{-3} \right)[/tex]

[tex]\theta = \frac{2\pi}{3} \,rad[/tex]

Now we find the four quartic roots of the complex number:

[tex]z_{1} = 1.565 \cdot \left(\cos \frac{\pi}{6} + i\,\sin \frac{\pi}{6} \right)[/tex]

[tex]z_{2} = 1.565\cdot \left(\cos \frac{2\pi}{3} + i\,\sin \frac{2\pi}{3} \right)[/tex]

[tex]z_{3} = 1.565\cdot \left(\cos \frac{7\pi}{6}+i\,\sin \frac{7\pi}{6} \right)[/tex]

[tex]z_{4} = 1.565\cdot \left(\cos \frac{5\pi}{3} + i\,\sin \frac{5\pi}{3} \right)[/tex]

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The answer is no. b.

Step-by-step explanation:

The answer is no b. u just take common

The percent of students that preferred alternative ​music is 23.7%

The music type and the preference are given as:

Music Preferences

Rap            945

Alternative 462

Rock and Roll 275

Country         168

Jazz              15

Other          85

From the above table, we have:

Alternative = 462

Total = 945 + 462 + 275 + 168 + 15 + 85

Total = 1950

The percentage is then calculated as:

Percentage = Alternative/Total * 100%

This gives

Percentage = 462/1950 * 100%

Evaluate

Percentage = 23.7%

Hence, the percent of students that preferred alternative ​music is 23.7%

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The equations that can be used to find the lengths of the legs of the triangle are as follows:

The equation that can be used to find the length of the leg of the right triangle uses Pythagoras theorem,

Therefore,

c²  = a² + b²

where

Therefore,

10²  = (x + 2)² + x²

100 = (x + 2)² + x²

Therefore, using the area of a triangle,

area of a triangle  = 1 / 2 bh

where

Therefore,

area of the triangle = 1 / 2 × (x + 2)(x)

24 = 0.5 × (x + 2)(x)

24 = x² + 2x / 2

cross multiply

48 = x² + 2x

x² + 2x - 48 = 0

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Answer:

d.) $112.50

Step-by-step explanation:

Area of floor:

15 ft × 10 ft = 150 ft²

150 ft² is how many times 12 ft²?

It is 150 ft² / 12 ft² = 12.5.

150 ft² is 12.5 times 12 ft².

12 ft² cost $9.00, so 12.5 times 12 ft² costs 12.5 times $9.00.

12.5 × $9.00 = $112.5

Answer: d.) $112.50

Using an exponential function, the median household income after 5 years was of $42,578.

An increasing exponential function is modeled by:

[tex]A(t) = A(0)(1 + r)^t[/tex]

In which:

For this problem, the parameters are:

A(0) = 42050, t = 5, r = 0.0025[/tex]

Hence:

[tex]A(t) = A(0)(1 + r)^t[/tex]

[tex]A(t) = 42050(1.0025)^t[/tex]

[tex]A(5) = 42050(1.0025)^5 = 42578[/tex]

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The y-intercept of the linear function y = 3x - 2 is -2

The function is given as

y = 3x - 2

The above function is a linear function, and the y-intercept is the point on the graph, where x = 0 i.e. the point (0, y)

As a general rule, linear functions are those functions that have constant rates or slopes

Next, we set x to 0, and calculate y to determine the value of the y-intercept

y = 3(0) - 2

Remove the bracket in the above equation

y = 3 * 0 - 2

Evaluate the product of 3 and 0 i.e. multiply 3 and 0

y = 0 - 2

Evaluate the difference of 0 and -2 i.e. subtract 0 from 2

y = -2

The above means that the value of y when x is 0 is -2

Hence, the y-intercept of the linear function y = 3x - 2 is -2

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Using it's concepts, we have that:

The absolute change between two values is given by the absolute value of the difference between two values.

In this problem, the values are of $40,156 million, which is the final value, and $25,003 million, which is the initial value, hence the absolute change is:

A = 40,156 - 25,003 = $15,153 million.

The relative change is the absolute change multiplied by 100% and divided by the initial value, hence:

(15,153/25,003) x 100% = 60.60%.

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(a) The truth about Karen's sales is that Karen had sales lower in value than about 74% of the salespeople. So, the correct option is D.

(b) The truth about Karen's and Brian's sales is that both Karen and Brian had lower sales in value than the median. So, the correct option is D.

Given that department stores rank their 1,000,000 salespeople by the monetary value of their sales. Karen's sales are at the 26th percentile and brian's sales are at the 33rd percentile.

A percentile is a measure used in statistics that indicates the value below which a certain percentage of observations in a group of observations falls.

(a)  Karen had sales lower in value than about 74% of the salespeople because the 26% percentile represents his relative position among all the employees in terms of sales.

(b) Both Karen and Brian had sales lower in value than the median because the Median represents the 50th percentile. Since Karen's sales are at the 26th percentile and Brian's sales are at the 33rd percentile it implies their sales are less than the median value.

Hence, the truth about Karen's sales is  Karen had sales lower in value than about 74% of the salespeople, and true about Karen's and Brian's sales is both Karen and Brian had lower sales in value than the median.

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The spending variance for salaries and wages cost in February would be closest to; $282 F.

The spending variance as the name implies is the difference between the budgeted cost and actual cost paid for an item. It therefore follows from the definition above that for the month of February;

Spending variance = $7,640 - (2850 + (16×317))

Spending variance = $7,640 - (7922)

Spending variance is therefore; $282 F.

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The price of the chairs and the quantity that maximizes profit is to sell 45 chairs at $75 each.

Given that a furniture store can sell 40 chairs per week at a price of $80 each, and the manager estimates that for each and $5 price reduction, she can sell five more chairs per week, and the chairs cost the store $30 each, to determine the price of the chairs and the quantity that maximize profit, the following calculation must be made:

Therefore, the price of the chairs and the quantity that maximizes profit is to sell 45 chairs at $75 each.

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Answer:

Hello,

Step-by-step explanation:

[tex]log(5-2x)=0\\\\\ \Longleftrightarrow\ 10^{log(5-2x)}=10^0\\\\\\\Longleftrightarrow\ 5-2x=1\\\\\Longleftrightarrow\ 2x=5-1\\\\\\\Longleftrightarrow\ x=\dfrac{4}{2}\\\\\\\Longleftrightarrow\ \boxed{x=2}\\[/tex]

Step-by-step explanation:

3 answer.....

......dkdkdk

Answer:

x = -3

Step-by-step explanation:

Axis of symmetry is the vertical line that divides the parabola into two equal halves and it passes through the vertex of the parabola.

From the graph, Vertex (-3, 4)

Axis of symmetry:  x = -3

The set of all months that contains less than 31 days is; {February, April, June, September, November}.

It follows from the task content that the set whose elements are to be determined is characterized by containing less than 31 days.

On this note, it follows that the elements of the set are; February (28/29 days), April (30 days), September (30 days), June (30 days) and November (30 days).

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The confidence interval expressed in the form p - E is:

(0.104 - 0.039, 0.104 + 0.039).

The mean is the mean of the bounds, hence:

p = (0.065 + 0.143)/2 = 0.104.

The margin of error is the difference between the bounds and the mean, hence:

E = 0.104 - 0.065 = 0.143 - 0.104 = 0.039.

Hence the interval in the desired format is:

(0.104 - 0.039, 0.104 + 0.039).

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The simplification of the equation above in words is = Twelve minus thirteen X.

In mathematics, coefficients are those numbers that multiplies a variable. For example in 13x, the coefficient is 13 which multiplies a variable known as X.

The equation given is,

10 - 13x + 2

Arrange the digits without a coefficient together.

That is ,

10+2 -13x

12 - 13x

Therefore the simplification of the given equation in words is Twelve minus thirteen X.

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Answer:

B.) (1, -5)

Given, equation:

[tex]\sf \dfrac{(x-1)^2}{25} -\dfrac{(y+5)^2}{49} =1[/tex]

Comparing it with standard formula of hyperbola:

[tex]\sf \dfrac{(x-h)^2}{a^2} -\dfrac{(y-k)^2}{b^2} =1[/tex]

where (h, k) is center

The given equation in standard form:

[tex]\sf \dfrac{(x-1)^2}{5^2} -\dfrac{(y-(-5))^2}{7^2} =1[/tex]

Determined:

(h, k) = (1, -5)

a = 5, b = 7

So, here (1, -5) is the center of the given hyperbola.

Answer:

Step-by-step explanation:

Initial volume of mixture is 100 ml with oil to water ratio of 1 : 4.

First, let's find the volume of each component.

If the oil is x then water is 4x according to ratio and their sum is 100 ml:

So, we have 20 ml of oil and 80 ml of water.

Now, let's added volume of oil be y. Then we have the ratio:

So, we have 40 ml of oil and 80 ml of water.

Lastly, let's added water be z. Then we have the ratio:

We must add 40 ml of water.

The attached figure represents the image of A"B"C" after the transformation

The transformation rule is given as:

A"B"C" = Ro90° (T(-4,3)(ABC))

This means that we rotate the triangle 90 degrees clockwise, and then translate the triangle

From the figure, the coordinates of ABC are

A = (-1, 2)

B = (1, 4)

C = (3, -1)

The rule of 90 degrees clockwise rotation is

(x,y) ⇒ (y,-x)

So, we have

A' = (2, 1)

B' = (4, -1)

C' = (-1, -3)

The translation of the triangle by T(-4,3) is

(x,y) ⇒ (x - 4, y + 3)

So, we have

A'' = (-2, 4)

B'' = (0, 2)

C'' = (-5, 0)

See attachment for the image of the transformation

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Answer:

Step-by-step explanation:

eq. of any line perpendicular to 5x+6y=-6 is

5y-6x=c

where c is a constant.

it passes through (-2,-1) so

5(-1)-6(-2)=c

c=-5+12=7

so reqd. eq. is 5y-6x=7

or 6x-5y=-7

or

slope of given line =-5/6

slope of reqd. line=6/5

eq. of line through (-2,-1) with slope 6/5 is

y+1=6/5(x+2)

5y+5=6x+12

5y=6x+12-5

or 5y=6x+7

or

6x-5y=-7

The value of csc theta is [tex]\frac{3. 61}{2}[/tex]

It is important to note that the ratio for cot is given as;

Cot theta = adjacent/opposite

cosec theta = hypotenuse/opposite

If cot theta = 3/2, then

Adjacent = 3

Opposite = 2

Using Pythagoras theorem, we have

Hypotenuse square = opposite + adjacent square

[tex]H = \sqrt{2^2 + 3^2}[/tex]

[tex]H = \sqrt{4 + 9}[/tex]

[tex]H = 3. 61[/tex]

Cosec theta = [tex]\frac{3. 61}{2}[/tex]

Thus, the value of csc theta is [tex]\frac{3. 61}{2}[/tex]

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Answer:

3.16/2

Step-by-step explanation:

Have a great day

Using an exponential function, it is found that:

a) The doubling time of the salary is of approximately 20 years.

b) The salary will be of $160,000.

An increasing exponential function is modeled by:

[tex]A(t) = A(0)(1 + r)^t[/tex]

In which:

The growth rate for this problem is:

r = 0.035.

The doubling time is t for which A(t) = 2A(0), hence:

[tex]A(t) = A(0)(1 + r)^t[/tex]

[tex]2A(0) = A(0)(1.035)^t[/tex]

[tex](1.035)^t = 2[/tex]

[tex]\log{(1.035)^t} = \log{2}[/tex]

[tex]t\log{1.035} = \log{2}[/tex]

[tex]t = \frac{\log{2}}{\log{1.035}}[/tex]

t = 20 years.

You retire in 40 years, which is 2 doubling periods, hence the salary will be of:

40000 x 2 x 2 = $160,000.

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Population density implies the number of an object, items, or people that can be found in a unit area at a particular time. Therefore, the number of fewer people that live in Elmira than in White Plains is 27,653.

Population density implies the total number of objects, items, or people that exist in a unit area or a place at a particular time. So that the number of people that exist in a place, area, city, or country can be referred to as population, in a particular period.

Thus in the given question, it can be deduced that;

The population of people living in White Plains =56,853

The population of people living in Elmira = 29,200

So, the difference in the population between the two cities can be determined as;

                         = 56,853 - 29,200

                         = 27,653

Thus, the number of fewer people living in Elmira than in White Plains is 27,653

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The confidence interval for the population proportion, p is (0.766, 0.833)

The formula for calculating the confidence interval for proportion is expressed as:

CI = p±√pq/n

Given the following

p = 112/140 = 0.8

q = 1-p = 0.2

Substitute

CI = 0.8±√(0.8)(0.2)/140

CI = 0.8±0.0338

CI = (0.766, 0.833)

Hence the confidence interval for the population proportion, p is (0.766, 0.833)

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Answer:

c. [-1, infinity)

Step-by-step explanation:

the domain the interval or set of valid x values (while the range is the same for the valid y values).

what is the smallest value of x we see for the graph ?

for x values we need to look left and right.

and the more left the value, the smaller it is.

so, in other words, what is the left-most x value e can find ?

x = -1

that is the "starting point" of the function. there is no functional value for any smaller x.

and then clearly, the function goes on and on to the right without any boundary in sight. so, it keeps going to the right in all eternity, that means it goes to infinity.

therefore the answer.

keep in mind that "infinity" is not a number, only a concept. so, when writing "infinity" at an interval end, it is not included (and we use a round bracket).

Answer:

Step-by-step explanation:

The statements that correctly compare the sum and product of the function are:

A linear function simply means a function that represents a straight line on the coordinate plane.

In this case, the sum is linear and the product is quadratic while the sum has one y-intercept and the product has two x-intercepts.

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