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Answer:   \(\sqrt{\frac{3+2\sqrt{2}}{6}}\)

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Explanation

"180° is greater than or equal to x ≥ 90°" is the same as writing \(180 \ge \text{x} \ge 90\) which is the same as \(90 \le \text{x} \le 180\)

The angle x is in quadrant Q2 where cosine is negative and sine is positive

sin(x) = 5/15 = 1/3

\(\cos^2(\text{x})+\sin^2(\text{x}) = 1\\\\\cos^2(\text{x}) = 1-\sin^2(\text{x})\\\\\cos^2(\text{x}) = 1-\left(\frac{1}{3}\right)^2\\\\\cos^2(\text{x}) = \frac{8}{9}\\\\\cos(\text{x}) = -\sqrt{\frac{8}{9}} \ \ \text{.... cos is negative in Q2}\\\\\cos(\text{x}) = -\frac{\sqrt{8}}{\sqrt{9}}\\\\\cos(\text{x}) = -\frac{2\sqrt{2}}{3}\\\\\)

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\(90 \le \text{x} \le 180\) leads to \(45 \le \frac{\text{x}}{2} \le 90\) after dividing all sides by 2.

The angle x/2 is between 45 and 90, which places it in quadrant Q1 or on the positive y axis. If \(45 \le \frac{\text{x}}{2} \le 90\) then \(\cos\left(\frac{\text{x}}{2}\right) \ge 0\)

We'll use a trig identity to compute cos(x/2).

\(\cos\left(\frac{\text{x}}{2}\right) = \sqrt{\frac{1+\cos(\text{x})}{2}}\\\\\cos\left(\frac{\text{x}}{2}\right) = \sqrt{\frac{1+\frac{2\sqrt{2}}{3}}{2}}\\\\\cos\left(\frac{\text{x}}{2}\right) = \sqrt{\frac{\frac{3+2\sqrt{2}}{3}}{2}}\\\\\cos\left(\frac{\text{x}}{2}\right) = \sqrt{\frac{3+2\sqrt{2}}{3}*\frac{1}{2}}\\\\\cos\left(\frac{\text{x}}{2}\right) = \sqrt{\frac{3+2\sqrt{2}}{6}}\\\\\)

Answer:

understand

Step-by-step explanation:

The fixed amount Geet earns for each television is $60.

Answer:

the answers are 12.5, 5, 2, 0.8 and 0.32

Step-by-step explanation:

2(0.4)^-2

=2(6.25)

= 12.5

2(0.4)^-1

= 2(2.5)

=5

2(0.4)^0

=2(1)

=2

2(0.4)^1

=2(0.4)

=0.8

2(0.4)^2

=2(0.16)

=0.32

The calculated value of the slope of the graph is -2

from the question, we have the following parameters that can be used in our computation:

The graph

Where, we have

(0, 5) and (4, -3)

The rate of the graph is calculated as

Rate = Change in y/x

using the above as a guide, we have the following:

Rate = (-3 - 5)/(4 - 0)

Evaluate

Rate = -2

Hence, the slope is -2

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These expressions back into the original differential equation yields:

(4Ae^(2x)sin(2x) + 4Be^(2x)cos(2x) + 4Ae^(2x)cos

We can use D-operator methods to find the general solutions of these differential equations.

(D^2 - 5D + 6)y = e^-2x + sin 2x

To solve this equation, we first find the roots of the characteristic equation:

r^2 - 5r + 6 = 0

This equation factors as (r - 2)(r - 3) = 0, so the roots are r = 2 and r = 3. Therefore, the homogeneous solution is:

y_h = c1e^(2x) + c2e^(3x)

Next, we find a particular solution for the non-homogeneous part of the equation. Since the right-hand side contains both exponential and trigonometric terms, we first try a guess of the form:

y_p = Ae^(-2x) + Bsin(2x) + Ccos(2x)

Taking the first and second derivatives of y_p gives:

y'_p = -2Ae^(-2x) + 2Bcos(2x) - 2Csin(2x)

y"_p = 4Ae^(-2x) - 4Bsin(2x) - 4Ccos(2x)

Substituting these expressions back into the original differential equation yields:

(4A-2Bcos(2x)+2Csin(2x)-5(-2Ae^(-2x)+2Bcos(2x)-2Csin(2x))+6(Ae^(-2x)+Bsin(2x)+Ccos(2x))) = e^-2x + sin(2x)

Simplifying this expression and matching coefficients of like terms gives:

(10A + 2Bcos(2x) - 2Csin(2x))e^(-2x) + (4B - 4C + 6A)sin(2x) + (6C + 6A)e^(2x) = e^-2x + sin(2x)

Equating the coefficients of each term on both sides gives a system of linear equations:

10A = 1

4B - 4C + 6A = 1

6C + 6A = 0

Solving this system yields A = 1/10, B = -1/8, and C = -3/40. Therefore, the particular solution is:

y_p = (1/10)e^(-2x) - (1/8)sin(2x) - (3/40)cos(2x)

The general solution is then:

y = y_h + y_p = c1e^(2x) + c2e^(3x) + (1/10)e^(-2x) - (1/8)sin(2x) - (3/40)cos(2x)

(D² + 2D + 4)y = e^(2x)sin(2x)

To solve this equation, we first find the roots of the characteristic equation:

r^2 + 2r + 4 = 0

This equation has complex roots, which are given by:

r = (-2 ± sqrt(-4))/2 = -1 ± i√3

Therefore, the homogeneous solution is:

y_h = c1e^(-x)cos(√3x) + c2e^(-x)sin(√3x)

Next, we find a particular solution for the non-homogeneous part of the equation. Since the right-hand side contains both exponential and trigonometric terms, we first try a guess of the form:

y_p = Ae^(2x)sin(2x) + Be^(2x)cos(2x)

Taking the first and second derivatives of y_p gives:

y'_p = 2Ae^(2x)sin(2x) + 2Be^(2x)cos(2x) + 2Ae^(2x)cos(2x) - 2Be^(2x)sin(2x)

y"_p = 4Ae^(2x)sin(2x) + 4Be^(2x)cos(2x) + 4Ae^(2x)cos(2x) - 4Be^(2x)sin(2x) + 4Ae^(2x)cos(2x) + 4Be^(2x)sin(2x)

Substituting these expressions back into the original differential equation yields:

(4Ae^(2x)sin(2x) + 4Be^(2x)cos(2x) + 4Ae^(2x)cos

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The smallest integer 'a' such that for all 'n' greater than or equal to 'a', exactly 'n' cents can be found is 4. To understand why 4 is the smallest integer satisfying the given condition, let's consider the possibilities for smaller values of 'a'.

If 'a' is 1, it means we only have pennies (1 cent coins), and we cannot represent 2 cents. If 'a' is 2, we have pennies and nickels (5 cent coins), but we still cannot represent 4 cents. If 'a' is 3, we have pennies, nickels, and dimes (10 cent coins), but we still cannot represent 6 cents. However, when 'a' is 4, we have pennies, nickels, dimes, and quarters (25 cent coins), allowing us to represent any number of cents from that point onwards. Therefore, 'a' equals 4 is the smallest integer that satisfies the condition.

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The increasing and decreasing behavior of the function f(x) on the given intervals is as follows:

On the interval (-∞, -3), f(x) is increasing.

On the interval (-3, 0), f(x) is decreasing.

On the interval (0, 3), f(x) is increasing.

On the interval (3, ∞), f(x) is decreasing.

We have,

To determine the increasing and decreasing behavior of the function f(x) = x / (x² - 9) on the given intervals, we can evaluate the sign of the derivative of the function.

Taking the derivative of f(x) with respect to x and simplifying, we have:

\(f'(x) = (-x^2 + 9 - x(2x)) / (x^2 - 9)^2\\= (-x^2 + 9 - 2x^2) / (x^2 - 9)^2\\= (-3x^2 + 9) / (x^2 - 9)^2\)

To identify the increasing and decreasing behavior, we need to examine the sign of f'(x) on each interval.

For the interval (-∞, -3):

Plugging in a value less than -3, such as -4, into f'(x) yields a positive result.

Plugging in a value between -3 and 0, such as -2, into f'(x) gives a negative result.

Therefore, f'(x) is positive for x < -3 and negative for -3 < x < 0, indicating that f(x) is increasing on the interval (-∞, -3) and decreasing on the interval (-3, 0).

For the interval (-3, 3):

Plugging in a value between -3 and 0, such as -2, into f'(x) yields a negative result.

Plugging in a value between 0 and 3, such as 1, into f'(x) gives a positive result.

Therefore, f'(x) is negative for -3 < x < 0 and positive for 0 < x < 3, indicating that f(x) is decreasing on the interval (-3, 0) and increasing on the interval (0, 3).

For the interval (3, ∞):

Plugging in a value between 3 and 4, such as 3.5, into f'(x) yields a positive result.

Plugging in a value greater than 4, such as 5, into f'(x) gives a negative result.

Therefore, f'(x) is positive for 3 < x < 4 and negative for x > 4, indicating that f(x) is increasing on the interval (3, 4) and decreasing on the interval (4, ∞).

Thus,

The increasing and decreasing behavior of the function f(x) on the given intervals is as follows:

On the interval (-∞, -3), f(x) is increasing.

On the interval (-3, 0), f(x) is decreasing.

On the interval (0, 3), f(x) is increasing.

On the interval (3, ∞), f(x) is decreasing.

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Answer:

Step-by-step explanation:

Let the length of the shorter piece be x

Then the longer piece is x + 7

Short piece = 19 in

Long piece = 26 in

The solution to this question is as the function could be horizontally translated to the left as follows: Therefore,g(x) = 3x +6

The term "function" refers to the relationship between a collection of inputs and outputs with one each. Simply described, a function is an association of inputs where each input is coupled with exactly one output. Every function has an associated domain, codomain, or range. The most common way to represent a function is as f(x), where x represents the input..

Here,

f(x)=3x+4

As we have to find g(x) a horizontal translation 2 units to the left.

Therefore

g(x)=f(x)+2

g(x)=(3x+2)+4

g(x)=3x+6

Therefore ,solution is g(x) = 3x +6

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Answer:

Yes, he will.

Step-by-step explanation:

Since he needs 9 pieces of 1/3 ft, yo would start by dividing 9 by 3 (9 x 1/3). This would give you 3, meaning that you would need 3 feet of would in all, and since he is given 3 and 1/3 feet, he has enough.

Hope this helps!

Answer:

6715

Step-by-step explanation:

I hope this helps, :)

t = 0

t = 5

Step-by-step explanation:

(-10t^2+50t)/(t^2+4t+3)

I'm making t into x so its not confusing with the plus sign

(-10x^2+50x)/(x^2+4x+3)

with (x^2+4x+3) :

what 2 numbers when multiplied make 3 and when added make 4 ?

they are 1 and 3

(x+1)(x+3)

(-10x^2+50x)/(x+1)(x+3)

(-10x(x-5))/(x+1)(x+3) =0

(-10t(t-5))/(t+1)(t+3)=0

t = 0

t = 5

The angle of depression that each chain makes with the ceiling are approximately 67° and 56°.

To solve the problem, use trigonometry. Let's call the angle of depression that the first chain makes with the ceiling x, and the angle of depression that the second chain makes with the ceiling y. Then:

In triangle ABC, where A is one of the hooks, B is the other hook, and C is the point where the chains meet:

AC = 1.9 m

BC = 2.2 m

angle BAC = 86°

Find angle BCA, which is the angle of depression that chain AC makes with the ceiling. Use the law of sines to find this angle:

sin BCA / AC = sin BAC / BC

sin BCA = AC sin BAC / BC

sin BCA = 1.9 sin 86° / 2.2

sin BCA ≈ 0.925

BCA ≈ 67.3°

So the angle of depression that chain AC makes with the ceiling is approximately 67 degrees.

Similarly, find the angle of depression that chain BD makes with the ceiling by considering triangle ABD:

AD = 2.8 m

BD = 2.2 m

angle ADB = 86°

Using the law of sines:

sin ADB / BD = sin BAD / AD

sin ADB = BD sin BAD / AD

sin ADB = 2.2 sin 86° / 2.8

sin ADB ≈ 0.836

ADB ≈ 56.4°

So the angle of depression that chain BD makes with the ceiling is approximately 56 degrees.

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Answer:

Dyscalculia is a learning disorder that affects a person's ability to understand number-based information and math

Explaining a bar graph

A bar graph graphically displays information. It uses bars that extend to various heights to symbolize value. Bar graphs can be created using any of the following: vertical bars, horizontal bars, grouped bars (several bars that compare values in a category), and stacked bars (bars containing multiple types of information).

Two sketches are shown in Histogram I for Box A.

20 drawings from Box B are represented in Histogram II.

20 drawings from Box A are represented in Histogram III.

Two draws are shown in Histogram IV for Box B.

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The matrix exponential is defined as: eAt = I + At + (At)^2/2! + (At)^3/3! + ...

To find the general solution of the given system, we can start by writing the system in the form dX/dt = AX, where X is the column vector [x y]T and A is the matrix:

[ 4 -1 ]

[ 16 -4 ]

We can then use the matrix exponential to find the general solution:

X(t) = eAt X(0)

The matrix exponential is defined as:

eAt = I + At + (At)^2/2! + (At)^3/3! + ...

To find eAt, we can compute the matrix powers of A and sum them up using the formula above.

Once we have found the general solution, we can use the initial conditions X(0) to find the specific solution for the given system.

Therefore, The matrix exponential is defined as:

eAt = I + At + (At)^2/2! + (At)^3/3! + ...

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Answer:

(1,5)

Step-by-step explanation:

when x = 1

\(y \geq -4 + 3 (1)\\y \geq -1\\\)

when x = 5

\(y \geq -4 + 3(5)\\y \geq 11\\\)

Only (1,5) pair is a solution.

Answer:

Below

Step-by-step explanation:

the  exterior angle measurement  is equal to half the difference of the measure of intercepted arcs.

? = (152-50)° / 2 = 51°

49.7 temperature if you round it up it would be 50.

1.7 times 4 is 6.8 and 56.5 minus 6.8 is 49.7 and of you round that up it would be 50.

Answer:

Step-by-step explanation:

The picture below shows the graph of the  inequality of x² ≤ 4

Option C is correct.

In mathematics, an inequality is described as a relation which makes a non-equal comparison between two numbers or other mathematical expressions which is used most often to compare two numbers on the number line by their size.

A graph of inequality denotes that the variables is the set of points that represents all solutions to the inequality.

A linear inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality.

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Answer: 29,692.5 gallons of water

Step-by-step explanation: first i multiplied 7.5 and 107 and got 802.5 and since it wanted to know about 37 days i multiplied 802.5 by 37 and got 29,692.5

Answer: 1/4 times 5

Step-by-step explanation: gtg sorry don’t have time

Answer:

100% of 36 = \(\frac{100}{100}\)×36= 36

25% of36 = \(\frac{25}{100}\)×36= 9

125% of 36 = \(\frac{125}{100}\)×36= 45

Answer:I believe the answer is b (62.8 cubic units)

Step-by-step explanation:

Answer:

62.8

Step-by-step explanation:

Answer: $0.95 each

Step-by-step explanation:

The price of the oranges needs to be subtracted from the total cost before tax of $22.02.

The result will then be divided by 20 to find out the cost per juice box.

Equation is:

jj = (22.02 - 3.02) / 20

Solving it:

= (22.02 - 3.02) / 20

= 19 / 20

= $0.95 each

Answe Jamie is Right

Step-by-step explanation:

If Christian wants to divide the pies into 4 people, everyone would have 3 pieces. It also Dylan wouldn't make since because there are 12 pies.

The diameter of a new ball is 4.0ft

According to the statement

we have given that the the company wants to spend a maximum of $1 each.

And we have to find a diameter of a new ball.

Remember that for a sphere of diameter D, the surface area is

A = 4*pi*(D/2)^2

In this case, the cost is $0.02 per square foot, and the company wants to expend (at maximum) $1 per ball, so first we need to solve:

$0.02*A = $1

A = $1/$0.02 = 50

So the surface of the ball must be 50 square feet.

Then we solve:

50ft^2 = 4*3.14*(D/2)^2

D = 2*√(50 ft^2/(4*3.14)) = 4.0 ft

here the diameter of a ball is 4.0ft.

So, The diameter of a new ball is 4.0ft

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Answer:

D) 2

Step-by-step explanation:

Use the following slope formula:

m (slope) = (y₂ - y₁)/(x₂ - x₁)

Let:

(x₁ , y₁) = (4 , 1)

(x₂ , y₂) = (1 , -5)

Plug in the corresponding numbers to the corresponding variables:

m = (-5 - 1)/(1 - 4)

Simplify. Remember to follow PEMDAS. First subtract, then divide:

m = (-5 - 1)/(1 - 4)

m = (-6)/(-3)

m = 2

D) 2 is your answer.

~

Answer:

Both rectangles and squares have four sides of equal length.

Step-by-step explanation:

Rectangle and square are the geometrical figures having four sides. They are the quadrilaterals with four sides. A rectangle is the quadrilateral in which opposite sides are equal. In a square all the sides are of equal length. The sides of both the rectangle and square are parallel. All angles form the right angles.