Find the Product.
5(cos pi/4 + i sin pi/4) x 3(cos 5pi/3 + i sin 5pi/3)

There are 6 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice matters and 3 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice dοes nοt matter.

The prοduct οf all pοsitive integers less than οr equal tο a given pοsitive integer and denοted by that integer and an exclamatiοn pοint. Thus, factοrial seven is written 7!, meaning 1 × 2 × 3 × 4 × 5 × 6 × 7. Factοrial zerο is defined as equal tο 1.

If the chοice matters (i.e., the οrder οf the chοsen οbjects matters), then there are 3 pοssible chοices fοr the first οbject and 2 pοssible chοices fοr the secοnd οbject.

Thus, there are 3 * 2 = 6 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice matters.

If the chοice dοes nοt matter (i.e., the οrder οf the chοsen οbjects dοes nοt matter), then there are 3 chοοse 2 = 3!/((3-2)!*2!) = 3 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice dοes nοt matter.

Therefοre,  there are 6 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice matters and 3 ways tο chοοse 2 οbjects withοut replacement frοm 3 οbjects if the chοice dοes nοt matter.

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According to the question the solution to the given IVP is [tex](ln(9/8))(e^{2t} - te^{2t} + 1).[/tex]

IVP stands for Initial Value Problem. It is a mathematical problem where an initial value is given at a certain point in time, and the goal is to find the solution for the function at all other points in time. An IVP is important in the study of differential equations, which are used to model a wide range of phenomena in science, engineering, and economics.

The given IVP is a second order linear differential equation with constant coefficients, so the general solution can be written as
[tex]y(t) = c_1e^{2t} + c_2te^{2t} + d[/tex]
where c_1, c_2, and d are constants to be determined.
Substituting the initial conditions into the general solution yields
[tex]y(-ln2) = c_1(-ln2)^2 + c_2(-ln2) + d = 0[/tex]
and
[tex]y'(-ln2) = 2c_1(-ln2) + c_2e^{2(-ln2)} = ln(9/8).[/tex]
Solving these two equations for c_1 and c_2 yields
[tex]c_1 = -(ln(9/8) + d)/2(-ln2)^2[/tex]
and
[tex]c_2 = (ln(9/8) + 2d)/2(-ln2)[/tex]
Substituting these values into the general solution and solving for d yields
d = ln(9/8).
Therefore, the solution to the given IVP is
[tex]y(t) = -(ln(9/8) + ln(9/8))e^{2t}/2(-ln2)^2 + (ln(9/8) + 2ln(9/8))te^{2t}/2(-ln2) + ln(9/8)[/tex]
= [tex](ln(9/8))(e^{2t} - te^{2t} + 1).[/tex]

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Answer:kwbebrhdjejhehe

Step-by-step explanation:

Both fireworks will explode at time 0.55 seconds after Firework B launches.

Distance is a unit of measurement for the separation between two places. This scalar number is typically expressed in terms of metres, feet, or miles. Depending on the situation, other distance formulae can be used, such as the geometry distance formula or the physics average speed formula. Displacement, a vector quantity that considers the direction of motion, is distinct from distance. Distance is the overall length of an object's route, whereas displacement is the change in an object's location from its beginning position to its end position.

Let us suppose time = t.

Given that, Firework B is launched 0.25 s before Firework A.

Time taken for Firework A to explode after Firework B launches is t - 0.25.

The distance traveled by each firework is given by:

Distance of Firework A = 320t

Distance of Firework B = 220(t - 0.25) = 220t - 55

Since both fireworks explode at the same height, their distances traveled must be equal. Therefore:

320t = 220t - 55

100t = 55

t = 0.55 seconds

Hence, both fireworks will explode 0.55 seconds after Firework B launches.

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The correct statements are A and E.

A sample is a subset of a larger population that is selected for research or analysis. The sample should be representative of the population to ensure accurate and unbiased conclusions.

A. Amy should have included students from one other science class because the sample should be representative of the entire middle school population, and including students from only one class may not provide a representative sample.

B. To get a good estimate, Amy does not necessarily need all the students at the school to answer the question. However, a larger sample size would increase the accuracy and precision of the estimate.

C. Mixing up the data from the science class would not be necessary if the sample is random and representative of the middle school population.

D. Amy's results can be used to estimate the average number of hours of sleep a typical student at her middle school gets on Saturday nights, provided the sample is random and representative of the middle school population.

E. The sample Amy used may not be a true random sample if it was only taken from one science class. The sample should be randomly selected from the entire middle school population to be representative.

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Some of the general reason why questionnaire are often not suitable for a study includes:

If the questions in the questionnaire are unclear or open to interpretation, then the results may not be accurate or useful. Respondents may answer questions differently based on their own interpretation, leading to inconsistent or unreliable data.

Also, if the questions in the questionnaire are biased, then, the results may be skewed towards a particular outcome or viewpoint. Respondents may feel pressured or influenced to answer questions in a certain way, rather than providing their honest opinion and this can lead to inaccurate or unrepresentative data.

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Answer:

see the work below

Step-by-step explanation:

20.)    x = √32²+32² = √2048 = 45.25

y = 32(sin60) = 27.7

z = 32(cos60) = 16

22.)   x = 7√3 / cos60 = 24.25 = 14√3

y = 14√3 / tan30 = 42

sin30 = 14√3 / (z+7√3)

z + 7√3 = 14√3 / sin30

z = 14√3/√sin30 - 7√3 = 36.37 = 21√3

24.) z = 18(tan30) = 10.4

h = √10.4² + 18² = √432 = 20.785

x = y

2x² = 20.785²

x = √20.785²/2 = 14.7

y = 14.7

Answer:

Question 20:

Question 22:

Question 24:

Step-by-step explanation:

A 45-45-90 triangle is a special right triangle where the measures of its sides are in the ratio 1 : 1 : √2.  Therefore, the formula for the ratio of the sides is b : b : b√2 where:

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio  1 : √3 : 2.  Therefore, the formula for the ratio of the sides is c: c√3 : 2c where:

Side x is the hypotenuse of a 45-45-90 triangle with congruent legs measuring 32 units. Therefore b = 32.

[tex]\implies x=b\sqrt{2}=32\sqrt{2}\; \sf units[/tex]

Side y is the side opposite the 60° angle in a 30-60-90 triangle with a hypotenuse of 32 units. Therefore, 2c = 32 so c = 16.

[tex]\implies y = c\sqrt{3}=16\sqrt{3}\; \sf units[/tex]

Side y is the side opposite the 30° angle in the same 30-60-90 triangle.

[tex]\implies z=c = 16\; \sf units[/tex]

Side x is the hypotenuse of a 30-60-90 triangle with the leg opposite the 30° angle measuring 7√3 units. Therefore c = 7√3.

[tex]\implies x=2c=2 \cdot 7 \sqrt{3}=14\sqrt{3}\; \sf units[/tex]

Therefore, other leg of the same triangle (opposite the 60° angle) measures c√3 = 7√3 · √3 = 21 units.

Side y is the hypotenuse of a 30-60-90 triangle with the leg opposite the 30° angle measuring 21 units. Therefore c = 21.

[tex]\implies y=2c=2 \cdot 21 = 42\; \sf units[/tex]

Side z is the leg of the same 30-60-90 triangle opposite the 60° angle.

[tex]\implies z=c\sqrt{3}=21\sqrt{3}\; \sf units[/tex]

Side z is the side opposite the 30° angle in a 30-60-90 triangle with the other leg (opposite the 60° angle) measuring 18 units. Therefore, c√3 = 18, so c = 18/√3 = 6√3 units.

[tex]\implies z=c = 6\sqrt{3}\; \sf units[/tex]

Therefore, the hypotenuse of the same triangle measures 2c = 12√3 units.

Sides x and y are the congruent legs of a 45-45-90 triangle with hypotenuse measuring 12√3 units. Therefore b√2 = 12√3, so b = 6√6.

[tex]\implies x=6 \sqrt{6}\; \sf units[/tex]

[tex]\implies y=6 \sqrt{6}\; \sf units[/tex]

Therefore , the solution of the given problem of probability comes out to be rounded to four decimal places, is approximately 0.1041.

The assessment of the probability that a claim is true or that a particular event will happen is the main objective of the constructions inside style known as criteria. Chance can be symbolised by any number between zero and 1, for which 1 typically denotes certainty range and 0 typically denotes possibility. A probability diagram illustrates the likelihood that a particular occurrence will take place. Decimal digits 0, 1, rationals with just 0% and roughly 100%.

Here,

A typical six-sided die has a 1/6 chance of rolling a 1 on any particular roll.

We can describe this scenario using a binomial distribution with

=> n = 15 and p = 1/6 since we are rolling the die 15 times.

The binomial probability formula can be used to calculate the likelihood of rolling precisely four ones:

=> P(X = 4) = (15 choose 4) (15 choose 4) * (1/6)^4 * (5/6)^11

Calculating the answer, we obtain:

=> P(X = 4) ≈ 0.1041

The likelihood of rolling precisely four one-sided dice, rounded to four decimal places, is approximately 0.1041.

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Answer:

1) 4x -28 - 2x = 0

2) 2x = 28

3) x = 14

Step-by-step explanation:

1) 4 * (-7) = -28

2) 4x - 2x = 28

   2x = 28

3) x = 28/2

   x = 14

Answer: To solve this expression, we need to simplify it using the order of operations (PEMDAS) and perform the indicated operations:

2.3x^2 + 69 / 465

First, we need to simplify the fraction 69/465. Both the numerator and denominator have a common factor of 3, so we can simplify the fraction by dividing both by 3:

69 / 465 = 23 / 155

Now we can substitute this simplified fraction back into the original expression:

2.3x^2 + 23 / 155

The expression is now in simplified form, and there are no more operations we can perform. So this is the final answer.

Alternatively, if the expression was meant to be solved for x, we would need an equation (e.g., 2.3x^2 + 69 / 465 = some value) in order to solve for x. As it stands, we cannot solve for x with the given expression.

Step-by-step explanation:

[tex]\cfrac{cos^2(a)}{sin(a)-1}\implies \cfrac{1-sin^2(a)}{sin(a)-1}\implies \cfrac{\stackrel{ \textit{difference of squares} }{1^2-sin^2(a)}}{sin(a)-1} \\\\\\ \cfrac{[1-sin(a)][1+sin(a)]}{sin(a)-1}\implies \cfrac{~~\begin{matrix} [1-sin(a)] \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~[1+sin(a)]}{-[~~\begin{matrix} 1-sin(a) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~]}\implies -[1+sin(a)][/tex]

Answer:-1

Step-by-step explanation:

[tex]\frac{cos^{2}a}{sina-1} \\cos^{2}a=\frac{1+cos2a}{2}\\\frac{\frac{1+cos2a}{2}}{sina-1}=\frac{1+cos2a}{2(sina-1)}=\frac{1+cos2a}{2sina-2}=\frac{1+cos2a}{-(2-2sina)}\\1-2sina=cos2a\\\frac{1+cos2a}{-(1+(1-2sina))}=\frac{1+cos2a}{-(1+cos2a)}=-1[/tex]

Answer:

m<v = m<w

Step-by-step explanation:

I do not know how to explain it. CD are parallel with those to lines. They ahve the same degree as the other two angles.

<y = <z are opposite angle.

<w + <x = 180 are suplementary angle which mean they add up to 180.

Answer:

Step-by-step explanation:

it will take 4 days to pay of the loan

Answer: Z=-5

Step-by-step explanation:

we want to group Z on only one side of the equation

-18=Z-3+2Z

-18-Z-2Z=-3 (subtract Z and 2Z from both sides, to remove Z from one side and to move it to the other side)

-3Z=-3+18 (regroup Z and add 18 on both sides, to have Z on only one side)

-3Z=15 (add -3+18)

3Z=-15 (multiply both sides by -1 to get a positive Z)

Z=-5 (divide both sides by 3)

Answer:

the least is 40  the greatest is 49

Step-by-step explanation:

Answer:

40  the greatest is 49

Step-by-step explanation:

Answer:

We can use the following formula to calculate the amount of direct materials purchased during the month:

Direct Materials Purchased = Ending Direct Materials Inventory Balance - Beginning Direct Materials Inventory Balance + Direct Materials Issued to Production

Substituting the given values, we get:

Direct Materials Purchased = $38,500 - $31,000 + $185,000

Direct Materials Purchased = $192,500

Therefore, the amount of direct materials purchased during the month was $192,500.

Answer: The total number of dogs at the event will depend on the number of playpens in each tent and the number of tents.

In each tent, there are x puppy playpens and y senior dog playpens, and there are 4 dogs in each playpen.

To find the total number of dogs at the event, we need to multiply the number of playpens in each tent by the number of dogs per playpen, and then multiply that by the number of tents.

So the expressions that represent the total number of dogs at the event are:

5xy(4) (multiplying the number of playpens in each tent by the number of dogs per playpen, and then by the number of tents).

20xy (combining the multiplication).

4(5xy) (multiplying the number of dogs per playpen by the number of playpens in each tent, and then by the number of tents).

20xy (combining the multiplication).

Therefore, the expressions that represent the total number of dogs at the event are 20xy and 4(5xy).

Step-by-step explanation:

The number of puppies at the event is represented by the expression 20x, and the number of senior dogs at the event is represented by the expression 20y.

The total number of dogs at the event can be calculated by adding the number of puppies and senior dogs. Since there are x puppy playpens and y senior dog playpens in each tent, and there are 5 tents, we would have 5 * x playpens for puppies and 5 * y playpens for senior dogs. Each playpen holds 4 dogs, so the number of puppies would be 4 * 5 * x, and the number of senior dogs would be 4 * 5 * y. Therefore, the expressions that represent the total number of dogs at the event are 20x (for the puppies) and 20y  (for the senior dogs).

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Answer:

a = 45 b = 35.8 c = 57.503

Step-by-step explanation:

there

Answer:

15 + 2j

Step-by-step explanation:

Sum, which means addition, is used to say that 15 will be added to something.

That "something" is twice Jose's score, which is 2 * j, since j is the score.

15 + 2 * j, or just 15 + 2j

Answer:1/9 i gotchu

Step-by-step explanation:

Answer:

4/8

Step-by-step explanation:

one third of 33 is 11

one fifth of 45 is 9.

The critical points are A = -5 and B = 7, and the function is increasing on the interval (-∞, -5] ∪ [7, ∞), and decreasing on the interval [-5, 7].

To find A and B, we need to find the critical points of the function f(x) and determine the intervals of increase and decrease.

First, we find the derivative of f(x) as:

f'(x) = 6x^2 - 12x - 210

Next, we find the critical points by setting f'(x) = 0:

6x^2 - 12x - 210 = 0

Simplifying, we get:

x^2 - 2x - 35 = 0

Factoring, we get:

(x - 7)(x + 5) = 0

So the critical points are x = 7 and x = -5.

Now we can determine the intervals of increase and decrease. We create a sign chart based on the sign of f'(x) in each interval:

Interval | f'(x) | f(x) increasing/decreasing

(-∞, -5) | - | decreasing

(-5, 7) | + | increasing

(7, ∞) | - | decreasing

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Answer:

7.2

Step-by-step explanation:

468/65 = 7.2

Answer:

348 in²

Step-by-step explanation:

[tex] \frac{ad \times (cd + ab)}{2} [/tex]

[tex] \frac{19.2(16 + 24)}{2} [/tex]

384 in²

After answering the presented question, we can conclude that As a triangle result, the figure has a surface area of 24.5 square millimeters.

Because it has four or more pieces, a triangle is a polygon. It has a straightforward rectangular shape. A triangle ABC is a rectangle with A, B, and C edges. Euclidean geometry provides a single plane and cube when the sides are not collinear. A polygon is a triangle with three components and three angles. The corners of a triangle are the spots where the three edges meet. A triangle's sides add up to 180 degrees.

To calculate the area of the figure, we must first calculate the areas of each shape and then add them together.

c² = a² + b²5² = 4

_g² +b²25 = 16+ b²_9_ = 3

As a result, the base of the second triangle is 3mm. We can now pinpoint its location:

A = 1/2 * 3mm * 3mm * 3mm = 4.5mm2

A is equal to (5mm + 5mm) * 4mm / 2 = 20mm2.

We can now add the regions of all three shapes:

Total surface area = 4.5mm2 + 20mm2 = 24.5mm2

As a result, the figure has a surface area of 24.5 square millimeters.

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Answer:

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

The scale factor k affects the points with the rule:

Apply the rule to the given points, given k = 3:

The required correct answer is (D) (-22, -1).

By graphing y=-f(x), we can reflect the graph of any function f about the x-axis, and by plotting y=f, we can reflect the graph about the y-axis.(-x). By graphing y=-f, we can even mirror it about both axes.(-x).

When a point is reflected across the x-axis, the y-coordinate changes sign while the x-coordinate remains the same.

Therefore, to find the coordinates of point A after it is reflected across the x-axis, we change the sign of the y-coordinate of A.

So, if the original coordinates of point A are (-22, 1), the coordinates of A after it is reflected across the x-axis are (-22, -1).

Therefore, the correct answer is (D) (-22, -1).

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Answer:

Step-by-step explanation:To prove that the roots of the equation x² + (1-k)x + k-3 = 0 are real for all real values of k, we need to show that the discriminant of the equation is non-negative for all values of k.

The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac. If the discriminant is positive, then the equation has two distinct real roots; if it is zero, then the equation has one real root (a repeated root); and if it is negative, then the equation has no real roots.

So, in this case, the discriminant of the equation is:

(1-k)² - 4(1)(k-3)

= 1 - 2k + k² - 4k + 12

= k² - 6k + 13

We need to show that k² - 6k + 13 ≥ 0 for all real values of k.

To do this, we can complete the square:

k² - 6k + 13

= (k - 3)² + 4

Since the square of any real number is non-negative, we have (k-3)² ≥ 0 for all k, which means that (k-3)² + 4 ≥ 4.

Therefore, k² - 6k + 13 ≥ 4 for all real values of k, which means that the discriminant of the quadratic equation x² + (1-k)x + k-3 = 0 is non-negative for all real values of k. Hence, the roots of the equation are real for all real values of k.

Answer:  we move (5, 1/3) to (5-3, 1/3*3) = (2, 1)

Step-by-step explanation:

The requried, we would expect 8 of the next 16 muffins sold to be bran muffins.

Out of a total of 50 muffins sold, 25 were bran muffins. That means that the proportion of bran muffins to total muffins sold is:

25 / 50 = 1/2

To find how many of the next 16 muffins sold would be bran muffins, we can multiply the proportion of bran muffins by the total number of muffins:

(1/2) * 16 = 8

Therefore, we would expect 8 of the next 16 muffins sold to be bran muffins.

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