If a=3 and b=-2i, then find the value of the ab^{2} in fully simplified form.

Answer:

[tex]\theta =\dfrac{2}{3} \pi ,\;\; \theta =\dfrac{4}{3} \pi[/tex]

Step-by-step explanation:

Given:

[tex]\cos 2\theta=-5 \cos \theta-3, \quad \quad 0 \leq \theta < 2 \pi[/tex]

[tex]\boxed{\begin{minipage}{6.5 cm}\underline{Cos Double Angle Identity}\\\\$\cos (A \pm B)=\cos A \cos B \mp \sin A \sin B$\\\end{minipage}}[/tex]

Use the cos double angle identity to create a quadratic:

[tex]\begin{aligned}\cos 2\theta &=-5 \cos \theta-3\\\cos^2\theta - \sin^2 \theta &=-5 \cos \theta-3\\\cos^2\theta - (1-\cos^2 \theta) &=-5 \cos \theta-3\\2\cos^2\theta - 1 &=-5 \cos \theta-3\\2\cos^2\theta+5 \cos \theta +2 &=0\end{aligned}[/tex]

Factor the quadratic:

[tex]\begin{aligned}2\cos^2\theta+5 \cos \theta +2 &=0\\2\cos^2\theta+4 \cos \theta +\cos \theta +2 &=0\\2\cos \theta(\cos \theta +2)+1(\cos \theta +2) &=0\\(2\cos \theta+1)(\cos \theta +2)&=0\end{aligned}[/tex]

Apply the zero-product property:

[tex]2 \cos \theta + 1 = 0 \implies \cos \theta =-\dfrac{1}{2}[/tex]

[tex]\cos \theta + 2= 0 \implies \cos \theta =-2[/tex]

As  -1 ≤ cos θ ≤ 1,  cos θ = -2 is undefined.

Therefore, the only valid solution is  cos θ = -¹/₂

Therefore:

[tex]\implies \cos \theta =-\dfrac{1}{2}[/tex]

[tex]\implies \theta =\cos^{-1}\left(-\dfrac{1}{2}\right)[/tex]

[tex]\implies \theta =\dfrac{2}{3} \pi +2\pi n,\;\; \dfrac{4}{3} \pi+2\pi n[/tex]

Solutions in the given interval 0 ≤ θ < 2π :

[tex]\theta =\dfrac{2}{3} \pi ,\;\; \theta =\dfrac{4}{3} \pi[/tex]

Using the graph given, the rate of change from hour 4 to hour 7 is: -20.

The rate of change of a relationship that is represented on a graph is determined using the formula below:

Rate of change = f(b) - f(a) / b - a.

Given the graph above, we are asked to find the rate of change between the interval x = 4 to x = 7:

a = 4

f(a) = f(4) = 120

b = 7

f(b) = f(7) = 60

Plug in the values:

Rate of change = f(b) - f(a) / b - a = (60 - 120) / (7 - 4)

= -60/3

Rate of change = -20.

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Answer:

see the attached image.

Step-by-step explanation:

First, divide by 35 on both sides to isolate 1 multiple of z.

[tex]35z > 34\\\overline{\ 35\ } \ \ \ \: \overline{35}[/tex]

[tex]z > \dfrac{34}{35}[/tex]

See the attached image for a graph.

The values of z stop at an open circle at the point 34/35.

Answer:

40 cc of A and 60 cc of B

Step-by-step explanation:

A+B = 100

0.5A + 0.8B = 100 * 0.68 = 68

0.5A + 0.8 ( 100 - A) = 68

0.5A + 80 - 0.8A = 68

-0.3A = -12

3A = 120

A=40

40 cc of A and 60 cc of B

The required inequality for number of outfit is n ≤  6.3  and Christopher can purchase 6 biking outfits.

Inequality shows relation between two expression which are not equal to each others.

Given that,

Maximum amount that Christopher can spend = $780.

The price of new bicycle = $446.45.

Also, Christopher buys 2 bicycle reflectors for $16.82 each.

And a pair of bike gloves for $24.60.

Let Christopher buys n outfits.

The price of 1 outfit is $43.70.

The price of n outfits = $43.70n

Since,  Price of 1 bicycle reflectors = $16.82

Price of 2 reflectors = $16.82 x 2 = $33.64

The inequality to determine the number of outfits can be written as,

43.70n + 446.45 + 24.60 + 33.64 ≤ 780.

43.70n + 504.69 ≤ 780.

43.70n ≤ 780 - 504.69

43.70n  ≤ 275.31

n ≤  6.3

The maximum number of outfit Christopher can purchase is 6.

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Effectively what you're asking is to solve this equation

x^2 - 12x + 48 = 0

Compare that to this form

ax^2 + bx + c = 0

to find that

Let's compute the discriminant (aka the stuff under the square root in the quadratic formula).

d = b^2 - 4ac

d = (-12)^2 - 4(1)(48)

d = 144 - 192

d = -48

The discriminant is negative, which tells us that there aren't any real numbered solutions to x^2 - 12x + 48 = 0

Therefore, we have no way to factor x^2 - 12x + 48 over the real numbers. Going back at the problem at hand: there aren't any real numbers that (a) multiply to 48 and (b) add to -12.

The two mystery numbers would be complex numbers in the form a+bi, where i = sqrt(-1). If your teacher hasn't covered these types of numbers yet, then you simply would say "no solution".

Answer:

-13/2x

Step-by-step explanation:

1/2x +(−7x)

Combine Like Terms:

12x +(−7x)

(1/2x +−7x)

−13/2x

The inequality represented by the graph is |2x − 1| ≥ 9

The number line represents the given parameter

On the number line, we have the following endpoints

x = -4 and x = 5

So, we test the above solutions in the list of options (representing them as an equation)

So, we have

|2x + 1| = 7

When x = -4

|2(-4) + 1| = 7 ⇒ 7 = 7 --- true

When x = 5

|2(5) + 1| = 7 ⇒ 11 = 7 --- false

2|x + 1| ≤ 8

When x = -4

2|(-4) + 1| = 8 ⇒ 6 = 8 --- false

|2x − 1| ≥ 9

When x = -4

|2(-4) - 1| = 9 ⇒ 9 = 9 --- true

When x = 5

|2(5) - 1| = 9 ⇒ 9 = 9 --- true

Hence, the inequality is (c) |2x − 1| ≥ 9

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Answer: She can make 3 full loaves

Step-by-step explanation: 3 x 3 = 9   3 x 4 = 12 you dont have 12 you only have 11

technically you can make 3 [tex]\frac{1}{3}[/tex] loaves but if you are looking for full loaves you can only make 3

The triangle has a perimeter of 48 units.

Triangles are figures with three sides, the perimeter is the sum of the lengths of the three sides. The length of each side is determined by Pythagorean theorem:

l = √(x² + y²)

Where:

First, determine the horizontal and vertical length of each side:

Side 1

x = 18, y = 0

Sides 2, 3

x = 9, y = 12

Second, determine the side lengths:

Side 1:

l = 18

Sides 2, 3

l = √(9² + 12²)

l = 15

Third, calculate the perimeter of the triangle:

p = 18 + 15 + 15

p = 48

The perimeter of the triangle is equal to 48 units.

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Answer:

Step-by-step explanation:

x= 80 degrees

I think that's the answer

113-33= 80

Answer:

133+33+x=180

X=180-133-33

X=14

Answer:

Step-by-step explanation:

1. x is inversely proportional to y

x=k/y

k is the constant of proportionality

x =60/y

2. x=60/y

x=60/12

x=5

Answer:

the answer is 20x -35

Step-by-step explanation:

5 x 4x = 20x

5 x -7 = -35

The answer options for exercise 5(4x-7) are:

A) 20x-35

B) 5x-5

C) 35-20x

D) 4x-35

To solve this exercise 5(4x-7), we organize the expression with the distributive law.

The distributive law is: a(b+c)=ab+ac.

Therefore, the correct option is alternative "A".

Answer:

30

Step-by-step explanation:

Answer:

51.85g

Step-by-step explanation:

At t = 25 years,

A(t) = 200e^-0.054t

A(25) = 200(e^-0.054(25))

= 200(e^-1.35)

= 200(0.2592)

= 51.85

Answers:

Min cost = $464.76

Dimensions = 11.619 feet by 5.164 feet

Each value mentioned above is approximate.

========================================================

Explanation:

x = horizontal dimension

y = vertical dimension

xy = 60

y = 60/x

The amount of outer fencing needed is 2x+2y feet, aka the perimeter of the largest rectangle.

There are 5 vertical inner fences, so we need 5y feet of inner fencing.

Let's set up the cost function.

Cost = 10*(amount of outer fencing) + 5*(amount of inner fencing)

C = 10*(2x+2y) + 5*(5y)

C = 20x+20y+25y

C = 20x+45y

Plug y = 60/x into the cost function.

C = 20x+45y

C = 20x+45(60/x)

C = 20x + (2700/x)

Then use either calculus or a graphing calculator to find the lowest point on the cost curve is roughly located at (11.619, 464.758)

Use the x value x = 11.619 to find its paired y value

y = 60/x

y = 60/11.619

y = 5.16395558998192

y = 5.164

The entire enclosure should be roughly 11.619 feet by 5.164 feet

The total cost is minimized at approximately  $464.76

Answer:

Step-by-step explanation:

You want the dimensions and cost of a rectangular enclosure with an area of 60 square feet and 5 internal partitions. The cost of outside fence is $10 per foot, and the cost of partition fence is $5 per foot.

Let x represent the length of the enclosure, and y the width (and the partition length). Then we have ...

  xy = 60

  10(2x) = cost fencing for length

  10(2y) +5(5y) = cost of fencing for width

We know the cost will be minimized when the cost of fence for the length is equal to the cost of fence for the width.

  20x = 45(60/x)

  x² = 135 . . . . . . . . . . . multiply by x/20

  x = 3√15 ≈ 11.619 . . . .square root

  y = 60/(3√15) = (4/3)√15 ≈ 5.164 . . . . . find y

Then the cost is ...

  2(10(2x)) = 120√15 ≈ 464.76 . . . . minimum cost

The enclosure is 11.619 by 5.164 feet, for a cost of $464.76.

Answer:

h(x) = - x² + 3

Step-by-step explanation:

Given f(x) reflected in the x- axis , then

h(x) = - x²

Given h(x) then h(x) + c is a vertical translation

• If c > 0 then a shift up of c units

• If c < 0 then a shift down of c units

Here h(x) is shifted 3 units up , then

h(x) = - x² + 3


i gotchu

Answer:

Step-by-step explanation:

it is 17 i did the quiz hope it helps

Answer:

-7/13 is equivalent to -2 1/3, so plot it 1/3 of a unit to the left of -2.

14/3 is equivalent to 4 2/3, so plot it 2/3 of a unit to the right of 4.

The required median of the prices of the various cars is $26,000.

When a dataset is ordered, the median is the value that is exactly in the middle. It is a measure of central tendency that distinguishes between the lowest and top half of data.

here,
Arrange the data set in order,
$22,000, $22,000, $26,000, $28,000, $90,000,

Total number = 5 [ odd numbers ]
Median is given as = n + 1 / 2 th term
                             = 5 + 1 / 2 th term
                             = 3 rd term

Thus, as of the given condition, the median of the prices is $26,000.

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The markup amount on the sweater of cost price $15 is $3.

A fraction is written in the form of p/q, where q ≠ 0.

Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.

Given, A clothing store buys a sweater for $15 before marking it up by 1/5.

So, the markup amount can be obtained if we multiply the cost price by the fraction of mark up which is,

= 15×(1/5).

= 3.

The markup price is $3.

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The number of students signing up for neither Math nor Biology is 25.

A set is a collection of well-defined objects.

A subset contains all the elements or a few elements of the given set.

The improper subset is when it contains all the elements of the given set and the proper subset is when it doesn't contain all the elements of the given set.

Given, 120 students are signing up for classes. 80 of them are signing up for a Math class (M) and 50 of them are signing up for a Biology.

We know N(A∪B) = N(A) + N(B) - N(A∩B).

∴ N(M∪B) = 80 + 50 - 35.

N(M∪B) = 95.

So, the number of students signing up for neither Math nor Biology are

(120 - 95) = 25

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The y-intercept indicates that 174 million smartphones were shipped to stores in 2009. This implies that after two years, in 2011, smartphone manufacturers shipped 500 million smartphones to stores worldwide.

Since 2009 the number of smartphones shipped from manufacturers to stores around the world has increased exponentially.

What is the exponential function?

The mathematical expression f(x)=exp or ex denotes the exponential function. The word, unless specifically stated differently, normally refers to the positive-valued function of a real variable, though it can be extended to the complex numbers or adapted to other mathematical objects like matrices or Lie algebras.

The growth from 2009 through 2015 can be modeled using the function G(t) = 174 * 1.67^t where t is the number of years since 2009 and G(t) is measured in millions of smartphones.

The required graph of the exponential function G(t) has been attached below

Now, finding the y-intercept

G(t) = 174· 1.67^t...(i)

Substitute the value of t = 0 in the above function,

G(t) = 174* 1.67⁰

G(t) = 174(1)

G(t) = 174

The meaning of the y-intercept is that 174 million smartphones are being shipped to stores in years 2009.

Substitute the value of G(t) =500 in the exponential function(i),

500 = 174 * 1.67^t

500/174 = 1.67^t

2.8735 = 1.67^t

In(2.8735 ) = t In (1.67)

t = In (2.8735 ) / In (1.67

t = 2.05827 ≈ 2

Hence, This means after 2 years i.e. in 2011, smartphone manufacturers were shipping 500 million smartphones to stores around the world.

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If Kylie's hourly wage at a nearby restaurant is $8.50. She worked 20.88 hours last week, according to her paycheck. Kylie's salary is $177.48.

It is defined as the operation in which we do the addition of numbers, subtraction, multiplication, and division. It has basic four operators that are +, -, ×, and ÷.

It is given that, Kylie makes $8.50 an hour at a local restaurant. Her paycheck shows she worked 20.88 hours last week.

We have to find the money did Kylie make,

We can utilize multiplication for a number of common tasks, such as figuring out how much it will cost to buy several identical things, calculating sales tax, and figuring out area and other geometric measurements.

Suppose the money did Kylie make is x,

x = 8.50 ×  20.88

x=$ 177.48

Thus, if Kylie's hourly wage at a nearby restaurant is $8.50. She worked 20.88 hours last week, according to her paycheck. Kylie's salary is $177.48.

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The number of persons who own a dog but no cat is 24 - 7 = 17, while the total number of people who own a dog is 17 + 4 = 21.

formal a customer of a specific eatery, lodging facility, or other establishment. Only customers may use this parking lot. synonyms and words that are similar. a hotel or guest house stay.

Since there are a total of 41 customers, the number of those without cats is calculated as follows: 41 – 17 = 24.

13 people(20 - 7) do not own dogs but do own cats.

There are 4 persons who own both a dog and a cat (17 - 13).

The people who have a dog, and do not have a cat is  24 - 7 = 17, and the total of the people who have a dog is 17+4 = 21.

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[tex]g(x)=\frac{5+2x}{2+x} \\ \\ \implies g(4x)=\frac{5+2(4x)}{2+4x}=\boxed{\frac{5+8x}{2+4x}}[/tex]