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Answer:

θ = 36.9°, 90°, 216.9°, or 270°

Step-by-step explanation:

4 sin 2θ − 3 cos 2θ = 3

Use double angle formulas:

4 (2 sin θ cos θ) − 3 (cos²θ − sin²θ) = 3

8 sin θ cos θ − 3 cos²θ + 3 sin²θ = 3

Use Pythagorean identity:

8 sin θ cos θ − 3 cos²θ + 3 sin²θ = 3 (sin²θ + cos²θ)

8 sin θ cos θ − 3 cos²θ + 3 sin²θ = 3 sin²θ + 3 cos²θ

8 sin θ cos θ − 6 cos²θ = 0

2 cos θ (4 sin θ − 3 cos θ) = 0

2 cos θ = 0

θ = 90° or 270°

4 sin θ − 3 cos θ = 0

4 sin θ = 3 cos θ

tan θ = 3/4

θ = 36.9° or 216.9°

In order to find the height of 2 different 3-D shapes, we must identify which dimension represents the height for each individual shape and then add them together.

The first thing is that we must identify which dimension represents the height for each individual shape, for instance, if one shape is a rectangular prism, its height would be the length of one of its sides.

After we identified the height for each shape, you can add them together to get the total height of the composite figure, but, we must understand that this method assumes that the two shapes are stacked vertically on top of each other with their bases aligned.

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Using addition of polynomials, it is found that:

a) The total amount is: T(t) = -0.028t³ - 0.32t² + 1.6t + 59.

b) $55.5 million was spent on buying new and used vehicles in the fifth year.

To add polynomials, we have to combine the like terms, that is, the terms that have t elevated to the same power.

For this problem, the functions are given as follows:

Hence the total amount is:

T(t) = N(t) + U(t)

T(t) = -0.028t³ + 0.06t² + 0.1t + 17 - 0.38t² + 1.5t + 42.

T(t) = -0.028t³ + (0.06 - 0.38)t² + (0.1 + 1.5)t + 17 + 42.

T(t) = -0.028t³ - 0.32t² + 1.6t + 59.

For the 5th year, the amount is given by:

T(5) = -0.028(5)³ - 0.32(5)² + 1.6(5) + 59 = 55.5.

$55.5 million was spent on buying new and used vehicles in the fifth year.

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Answer:

3/8, 5/8, 2/3

5 has x^o and -5 has x^0 LIKE TERMS (x^0 = 1)

-5 has x^0 and -12x has x^1 UNLIKE TERMS

5x has x^1 and -12x has x^1 LIKE TERMS

5x has x^1 and 5y has y^1 UNLIKE TERMS (variables have to have same "letter" and exponent in order to be LIKE TERMS)

The function that could be used to represent the total cost of the shoes by applying the $15 coupon first and the 20% member discount second for original cost, using proportions, is of:

C(x) = 0.8(x - 15).

A proportion represents a fraction relative to a total amount, and this fraction is combined with the basic arithmetic operations, especially division and multiplication, to find the desired amounts in whichever context of the problem.

The value of shoe is given as follows:

x.

The discounts are given as follows:

Hence the function that gives the total cost paid is presented as follows:

C(x) = 0.8(x - 15).

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Answer:

  y = -5/2x +1

Step-by-step explanation:

You want the slope-intercept form equation for the line through the point (-2, 6) that is parallel to 5x +2y = -4.

The equation of a parallel line can be the same as the given equation, except for the constant. The new constant can be found by substituting the given point coordinates:

  5(-2) +2(6) = c

  -10 +12 = c

  2 = c

Now we know the equation of the parallel line can be written as ...

  5x +2y = 2

Solving for y puts this in slope-intercept form:

  2y = -5x +2 . . . . . . . . subtract 5x

  y = -5/2x +1 . . . . . . . . divide by 2

We don't know what your boxes look like, but we can separate the numbers to make it look like this:

  \(\boxed{y=\dfrac{-5}{2}x+1}\)

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Raw materials purchases in July - Payment for raw materials purchases in July = Desired ending raw materials inventory for July

X pounds - 0.35 * X pounds = 9,500 pounds

Solving this equation will give us the value of X, which represents the pounds of raw materials that should be purchased in July.

To determine the number of pounds of raw materials that should be purchased in July, we need to calculate the raw materials production needs for August and then consider the inventory policies given in the information provided.

Each unit of finished goods requires 5 pounds of raw materials. The budgeted unit sales for August are 19,000 units. Therefore, the raw materials production needs for August would be 19,000 units multiplied by 5 pounds per unit, which equals 95,000 pounds.

The ending raw materials inventory equals 10% of the following month’s raw materials production needs. Therefore, the desired ending raw materials inventory for July would be 10% of 95,000 pounds, which is 9,500 pounds.

To calculate the raw materials purchases for July, we need to consider the payment terms provided. Thirty-five percent of raw materials purchases are paid for in the month of purchase and 65% in the following month.

Let's assume the raw materials purchases for July are X pounds. Then the payment for 35% of X pounds will be made in July, and the payment for 65% of X pounds will be made in August.

The payment for raw materials purchases in July (35% of X pounds) will be:

0.35 * X pounds

The payment for raw materials purchases in August (65% of X pounds) will be:

0.65 * X pounds

Since the raw materials purchases for July should cover the desired ending raw materials inventory for July (9,500 pounds), we can set up the following equation:

Raw materials purchases in July - Payment for raw materials purchases in July = Desired ending raw materials inventory for July

X pounds - 0.35 * X pounds = 9,500 pounds

Solving this equation will give us the value of X, which represents the pounds of raw materials that should be purchased in July.

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The angle of elevation θ is equal to 1.13186 in radian rounded to five decimal place.

The angle of elevation is the angle between the horizontal line and the line of sight which is above the horizontal line.

We shall evaluate for the angle of elevation θ in degree, and then convert the result to radian as follows:

θ = tan⁻¹(4255/2000)

θ = 64.82482°

we multiply by π/180 to convert to radian;

θ' = 64.82482° × π/180

θ' = 1.13186 rounded to five decimal place.

Therefore, the angle of elevation θ is equal to 1.13186 in radian rounded to five decimal place.

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The value of x is 1.75 and y is 5. The solution is obtained using properties of congruent triangles.

What is a congruent triangle?

Two triangles that are congruent will have precisely the same three sides and three angles.

The dimensions of the triangles' sides and angles determine whether two or more triangles are congruent. A triangle's size is determined by its three sides, and its shape by its three angles. Pairs of corresponding sides and corresponding angles in two triangles are said to be equal if they are congruent. They share a similar size and shape. In triangles, there are numerous congruence conditions.

The triangles are congruent by SAS congruency because

AC = CD (Given)

∠ACB = ∠DCE ( Vertically opposite angles)

BC = CE  (Given)

Thus, Triangle ABC ≅ Triangle DEC

Since, the triangles are congruent, therefore

⇒AC = CD

⇒4y-6 = 2x+6   ...(1)

Also, BC = CE

⇒3y+1 = 4x

⇒(3y+1)/4 = x    ...(2)

Now, substituting the value of x in (1), we get,

⇒4y-6 = 2(3y+1)/4+(6)

⇒4y-6 = (6y+2 +24)/4

⇒16y-24 = 6y+2 +24

⇒10y = 50

⇒y = 5

Now putting the value of y in (2), we get

⇒(3(2)+1)/4 = x

⇒7/4 = x

⇒ x = 1.75

Hence, the value of x is 1.75 and y is 5.

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Answer:

The fraction 14/16 is greater than 7/8.

Step-by-step explanation:

Answer:

no cause y value repeat

number 3

Step-by-step explanation:

Answer:

3.1kg

Step-by-step explanation:

4.7+5.6+6.4+2.8+3.5+4.4+4.5 = 31.9

4.375 x 8 = 35

35 - 31.9 = 3.1

The equation is P = 2 ( L + W )

The width of the field is 60 feet.

The length of the field is 600 feet.

Given,

A rectangular field is ten times as long as it is wide.

The perimeter of the field is 1320 feet.

Area of rectangle = Length x wide.

Perimeter = 2 ( Length + width )

Let the length of the rectangle be L

Width = W

L = 10W

Perimeter + 2 ( L + W )

1320 = 2 ( 10W + W )

Divide it by 2 into both sides.

1320/2 = 2/2 x 11W

660 = 11W

Divide both sides by 11.

660/11 = 11/11 W

60 = W

W = 60 feet

L = 10W = 10 X 60 = 600 feet

L = 600 feet

Thus,

The equation is P = 2 ( L + W )

The width of the field is 60 feet.

The length of the field is 600 feet.

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Answer:

Step-by-step explanation:

\(1-\frac{4}{2\cdot \:7+4}\\\\\frac{4}{2\cdot \:7+4} = \frac{2}{9} \\\\=1-\frac{2}{9}\\\\\mathrm{Convert\:element\:to\:fraction}:\quad \:1=\frac{1\cdot\:9}{9}\\\\=\frac{1\cdot \:9}{9}-\frac{2}{9}\\\\Since\:the\:denominators\:are\:equal,\\\:combine\:the\:fractions}:\\\\\quad \frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}\\\\=\frac{1\cdot \:9-2}{9}\\\\1\cdot \:9-2 =7\\\\=\frac{7}{9}\)

Answer:

its 430

Step-by-step explanation:

Answer:

57.3

Step-by-step explanation:

\(\huge\text{Hey there!}\)

\(\mathsf{10 \dfrac{2}{3}\times 5\dfrac{3}{8}}\)

\(\mathsf{= \dfrac{10\times3 + 2}{3}\times\dfrac{5\times8 + 3}{8}}\)

\(\mathsf{= \dfrac{30 + 2}{3} \times \dfrac{40 + 3}{8}}\)

\(\mathsf{= \dfrac{32}{3}\times \dfrac{43}{8}}\)

\(\mathsf{=\dfrac{32\times43}{3\times 8}}\)

\(\mathsf{= \dfrac{172}{3}}\)

\(\mathsf{= 57\dfrac{1}{3}}\)

\(\huge\text{Therefore, your answer should be:}\)

\(\huge\boxed{\mathsf{57\dfrac{1}{3}}}\huge\checkmark\)

\(\huge\text{Good luck on your assignment \& enjoy your day!}\)

The shaded region of the rectangle is 242.9 cm² and the shaded region of the sector is 7.1 square units.

Question 17) is a figure of a rectangle and two inscribed circles.

The area of a rectangle is expressed as: A = length × width

The area of a circle is expressed as: A = πr²

Where r is the radius.

To determine the area of the shaded region, we simply subtract the areas of the two circles from the area of the rectangle.

Area = ( Length × width ) - 2( πr² )

Area = ( 40 × 10 ) - 2( π × 5² )

Area = ( 400 ) - 2( 25π )

Area = 400- 50π

Area = 242.9 cm²

Area of the shaded region is 242.9 squared centimeters.

Question 18) is the a figure a sector of a circle and a right triangle.

The area of a sector is expressed as: A = (θ/360º) × πr²

The area of a triangle  is expressed as: A = 1/2 × base × height

To determine the area of the shaded region, we simply subtract the areas of the triangle from the area of the sector.

Hence:

Area = ( (θ/360º) × πr² ) - ( 1/2 × base × height )

Plug in the values:

Area = ( (90/360º) × π × 5² ) - ( 1/2 × 5 × 5 )

Area = 25π/4 - 12.5

Area = 7.1

Therefore, the area of the shaded region is 7.1 square units.

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Answer:

  x = 36

Step-by-step explanation:

Given a tangent segment of length 24, and a secant segment from the same point with an external length of 12 and a total length of (12+x), you want to find the value of x.

The product of lengths from the common point to the two intersections with the circle are the same for both segments. In the case of the tangent, the two intersections with the circle are the same point, so the square of the length is used.

  24² = 12(12 +x)

  2·24 = 12 +x . . . . . . . divide by 12

  36 = x . . . . . . . . . subtract 12

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Answer:

y=2/5x+3

Step-by-step explanation:

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The equivalent expression is 5 - 3x/2. Option D

Algebraic expressions are described as expressions that are made up of variables, their coefficients, factors and constants.

these algebraic expressions are also composed of mathematical operations. These operations includes;

From the information given, we have that;

1/ 4(8 - 6x + 12)

expand the bracket, we have;

Add the like terms

1/4(20 - 6x)

now, multiply the values, we have;

20 - 6x/4

Divide by the denominator

5 - 3x/2

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The complete question:

Simply the expression:

1/ 4(8 - 6x + 12)

The exact expression for g(x) is 2.25cos((2π/4.5)×(x-3.5)) - 4.

A function can be represented using a formula or an equation, and it can be graphed on a coordinate plane. The input values are typically represented on the x-axis and the output values on the y-axis.

From the given information, we know that the function g(x) has a maximum point at (3.5, -4) and a minimum point at (-1, -5).

The general form of a cosine function is f(x) = A×cos(Bx + C) + D, where A is the amplitude, B is the period, C is the phase shift, and D is the vertical shift.

Since the function has a maximum point at (3.5, -4), we know that the graph has been shifted to the left by 3.5 units. Therefore, we can write the function as g(x) = A×cos(B(x - 3.5)) + D.

Similarly, since the function has a minimum point at (-1, -5), we know that the graph has been shifted upwards by 1 unit. Therefore, we can write the function as g(x) = A×cos(B(x - 3.5)) - 4.

To determine A and B, we can use the fact that the period of the function is 4.5 units (the distance between the maximum and minimum points). Therefore, we have B = 2*pi/4.5.

To determine A, we can use the fact that the amplitude is half the distance between the maximum and minimum points, which is 0.5*(5-(-4)) = 4.5. Therefore, we have A = 4.5/2 = 2.25.

Substituting these values into the equation for g(x), we have:

g(x) = 2.25cos((2π/4.5)×(x-3.5)) - 4

Therefore, the exact expression for g(x) is:

g(x) = 2.25cos((2π/4.5)×(x-3.5)) - 4.

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Answer:

Graph A

So it has two distinct real roots.

Graph B

It has one repeated real root

Graph C

So it has two complex roots.

Graph D

One real root and one complex root

Step-by-step explanation:

For graph A

The value of the roots is x= 1 and x= 3

And the minimum value = -3

It's a positive graph

So it has two distinct real roots.

For graph B

The value of the roots is x = 2 and x= 2

That is x= 2 twice

Has a maximum value of 0

It's an inverse graph

It has one repeated real root

For graph C

It's a positive graph but on the negative of x

Has a minimum value of 1

It didn't touch x at y = 0

And it's root will be negative

So it has two complex roots.

For Graph D

Value of the roots is x= 2 and x= -2

It's a positive graph

Minimum value of -4

One real root and one complex root

The equation of the line is y = 7x + 6.

What is slope intercept form?

The slope intercept form of an equation is represented as follows:

y = mx + c ,where m = slope c = y-intercept

The  of a line can be written in slope-intercept form:

y = mx + b

where m is the slope of the line, b is the y-intercept (where the line crosses the y-axis), and (x,y) are the coordinates of any point on the line.

We are given that the line crosses the y-axis at the point (0,6), which means that the y-intercept is 6. We are also given that the slope of the line is 7.

Substituting these values into the slope-intercept form of the equation, we get:

y = 7x + 6

Therefore, the equation of the line is y = 7x + 6.

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The best predicted weight of a person that is 156 cm is 285.4kg

y= 106+1.15

When the adult male is 156 cm tall the weight of this person would be calculated as:

y= 106+1.15*156

y = 106 + 179.4

y = 285.4

Hence we can arrive at the conclusion that the best predicted weight of a person that is 156 cm is 285.4kg

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Answer:

5.55 million cells per microliter

4.68 million cells per microliter

Step-by-step explanation:

First, use a chart or calculator to find the z-score for reach percentage.

P(Z > z) = 0.24

P(Z < z) = 1 − 0.24 = 0.76

z = 0.706

Next, use definition of z-score to find the value.

z = (x − μ) / σ

0.706 = (x − 5.2) / 0.5

x = 5.55

Repeat for the other percentage.

P(Z < z) = 0.15

z = -1.036

z = (x − μ) / σ

-1.036 = (x − 5.2) / 0.5

x = 4.68

The slope of the cubic polynomial and it's y-intercept are respectively; f(1) = -4 and f(0) = -6

We are given the equation;

f(x) = 2x³ - 6

Now, because the highest power of x is 3, it means that this is a cubic polynomial and not linear because it has a degree of 3

Thus, let us differentiate to get the slope;

f'(x) = 6x² - 6

At f'(x) = 0; x = 1

Thus, slope is;

f(1) = 2(1)³ - 6

f(1) = -4

y-intercept is f(x) at x = 0. Thus;

f(0) = 2(0³) - 6

f(0) = -6

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Answer:

To investigate the different sizes of ponds and their effects on the garden, I followed these steps:

1. I drew several sizes of ponds on a grid paper, using the given dimensions of 18 ft2, 24 ft2, and 30 ft2. I made sure that the length and width of each pond were whole numbers. I also drew borders around the ponds to represent the tiles needed for each pond. Here is an example of how I drew a 24 ft2 pond:

| | | | | | | | | | |

| | | | | | | | | | |

|X|X|X|X|X|X|X|X|X|X|

|X|O|O|O|O|O|O|O|O|X|

|X|O|O|O|O|O|O|O|O|X|

|X|O|O|O|O|O|O|O|O|X|

|X|X|X|X|X|X|X|X|X|X|

In this drawing, X represents a tile and O represents water.

2. I recorded the number of tiles needed for each pond in a table, as well as the length and width of each pond. Here is the table I made:

Pond size (ft2) Length (ft) Width (ft) Number of tiles

18 6 3 22

24 6 4 28

30 6 5 34

3. I looked for patterns in the table and noticed that the number of tiles increased by 6 for every increase of 1 ft in width. I also noticed that the number of tiles was always equal to twice the length plus twice the width minus four.

4. I came up with a numerical expression that relates the length and width to the number of tiles needed. The expression is:

Number of tiles = 2 x Length + 2 x Width - 4

This expression works for any pond size with whole number dimensions.

By following these steps, I was able to compare different pond sizes and their tile requirements. I hope this helps you decide on the best size for your pond and garden.

________________________________________________________

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Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length  \(\sqrt{89}\) units and diagonals that differ by 5 units.

Diagonals of Rhombus:

A diamond's Rhombus is a line segment that connects two non-adjacent vertices of the diamond. A rhombus has two diagonals that bisect it at right angles. That is, they form four congruent right triangles. The rhombic diagonal formula is based on the area of ​​the diagonal, expressed as p = (2 × area)/q. where 'p' and 'q' are the two diagonals of the rhombus.

Now,

Label each half of the short diagonal as X.  

Label each half of the long diagonal as X+ 3.

The sides of the rhombus as \(\sqrt{89}\).

Each of the internal triangles will be figured the same.

We have a right triangle with one side X, one side X+3, and the hypotenuse \(\sqrt{89}\) .

According to the Pythagorean theorem, in a right triangle (90 degrees), the square of the hypotenuse equals the sum of the squares of the other two sides. The triangle ABC where BC2 = AB2 + AC2. where AB is the base, AC is the height (height), and BC is the hypotenuse. Note that the hypotenuse is the longest side of a right triangle.

Therefore,

Pythagoras' Theorem  :

(X)² + (X+3)² =   \((\sqrt{89} )^2\)

X² + X² + 6X + 9 = 89

2X² + 6X – 80 = 0  

This will factor to          

(2X + 16)(X – 5) = 0

Disregard the negative solution.  

Half of the short diagonal is 5 units.  

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