can you solve this question?
x=?
the value of this limit=?
y=?

Using the sum-to-product formula we get the value as √2 cos(2α)

What is the sum-to-product formula?

The sum-to-product formula is a trigonometric identity that expresses the sum or difference of two trigonometric functions as a product of two trigonometric functions. There are several versions of this formula, depending on the specific functions involved.

We can use the sum-to-product formula to simplify the expression:

cos(2α - π/16) + cos(9π/16 - 2α)

= cos(2α)cos(π/16) - sin(2α)sin(π/16) + cos(9π/16)cos(2α) + sin(9π/16)sin(2α)

= cos(2α)(cos(π/16) + cos(9π/16)) + sin(2α)(-sin(π/16) + sin(9π/16))

= cos(2α)(2cos(5π/16)cos(3π/16)) - sin(2α)(2cos(7π/16)sin(π/16)) (using the sum-to-product formula again)

= 2cos(2α)cos(5π/16)cos(3π/16) - 2sin(2α)cos(7π/16)sin(π/16)

= 2cos(2α)(cos(5π/16)cos(3π/16) - sin(5π/16)sin(3π/16)) (using the product-to-sum formula)

= 2cos(2α)cos(π/4)

= 2cos(2α)/√2

= √2 cos(2α)

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After solving the question, we can say that diameter of the soap bubble =14.30 mm.

A sphere, like a ball, is a three-dimensional geometrical object that is completely spherical. It is a symmetrical form with all points on its surface being equidistant from the centre. The radius of a sphere is the distance from its centre to its surface. Spheres can be found in nature in the shape of planets, stars, and bubbles, and they are utilised in a variety of applications including sports equipment, ball bearings, and aesthetic things.

volume of a sphere

[tex]V = (4/3) * \pi * r^3\\ r = (3V / 4 \pi)^(1/3)\\ r = (3 * 1696.5 mm^3 / (4 * 3.14))^(1/3)\\ r = 7.15 mm[/tex]

d = 2r

d = 14.30 mm

diameter of the soap bubble =14.30 mm.

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Answer:

To write the equation of the parabola in vertex form, we use the standard form:

y = a(x - h)^2 + k, where (h, k) is the vertex.

Given that the vertex is (-4, -6), we have:

h = -4 and k = -6

Now, we need to find the value of "a" using the point (-14, 13/2) which lies on the parabola.

Substituting the values in the standard form, we get:

13/2 = a(-14 - (-4))^2 - 6

Simplifying, we get:

13/2 = 100a - 6

100a = 13/2 + 6

a = 19/200

Therefore, the equation of the parabola in vertex form is:

y = (19/200)(x + 4)^2 - 6

Step-by-step explanation:

Answer: Ten liter, or one tenth of a hectoliter (2.6418 gallons liquid measure or 1.135 pecks dry measure): abbrev.dal.

The simplified expression is 28 - 4 using distributive property.

To simplify the expression (4 + )(7 – ), we can use the distributive property of multiplication over addition.

First, we have to multiply the 4 by both the terms inside parentheses:

(4 + )(7 – ) = (47) + (4- )

Next, we distribute negative sign to second term inside parentheses:

(47) + (4- ) = 28 - 4

The simplified expression = 28 - 4.

In other words, the phrase is the outcome of adding 4 to an unknown quantity, taking it out of 7 after that, and then taking the outcome of that subtraction out of 28. The operation's outcome is simplified to 24, which can be used in additional calculations or problem-solving.

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Answer: there you go :)

Car 1 consumes 30 gallons of gas.

Car 2 consumes 20 gallons of gas.

Step-by-step explanation:

Let x = gallons consumed by car 1

Let y = gallons consumed by car 2

We set up our equations:

35x+40y = 1850        eq1

x+y = 50                   eq2

Substituting eq 2 into eq 1,

35x+40(50-x) = 1850

35x+2000-40x = 1850

-5x+2000 = 1850

-5x = -150

  x = 30

Substitute value of x into eq 2.

x+y = 50

30+y = 50

     y = 20

Car 1 consumes 30 gallons of gas.

Car 2 consumes 20 gallons of gas.

Answer:

y = 4x^2

Step-by-step explanation:

To find a quadratic function of the form y = ax^2 that passes through the point (-2,16), we need to substitute the coordinates of the point into the equation and solve for the coefficient a.

Substituting x=-2 and y=16 into the equation, we get:

16 = a(-2)^2

16 = 4a

a = 4

Therefore, the quadratic function of the form y = ax^2 that passes through the point (-2,16) is:

y = 4x^2

The piecewise function formed by two functions and a point is not continuous at x = 1.

In this problem we find the case of a piecewise function formed by three expressions in three corresponding intervals: two functions and a point. These expressions are defined in the images attached below. The x-value is on the horizontal axis and the y-value is on the vertical axis.

According to the first figure, the y-value of the function is getting closer to 2 as x approaches 1 from the left side.

According to the second figure, the y-value of the function is getting closer to 3 as x approaches 1 from the left side.

According to the third figure, the y-value of the function is equal to 0.5 when x is equal to 1.

Therefore, the piecewise function is not continuous at x = 1.

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Answer: Kayla sold 1/12 gallons more than Ethan.

Step-by-step explanation:

Let's start by finding out how much lemonade Kayla sold. We know that together, Ethan and Kayla sold 1 1/4 gallons, which is the same as 5/4 gallons. We also know that Ethan sold 2/3 gallons. Therefore, we can find how much Kayla sold by subtracting Ethan's sales from the total sales:

Kayla's sales = Total sales - Ethan's sales

Kayla's sales = 5/4 - 2/3

To subtract these fractions, we need to find a common denominator. The smallest common multiple of 4 and 3 is 12, so we can rewrite the fractions with 12 as the denominator:

Kayla's sales = (5/4)(3/3) - (2/3)(4/4)

Kayla's sales = 15/12 - 8/12

Kayla's sales = 7/12

Therefore, Kayla sold 7/12 gallons of lemonade. Now, we need to compare this to Ethan's sales. We can convert Ethan's sales to twelfths to make it easier to compare:

Ethan's sales = (2/3)(4/4) = 8/12

Now we can see that Kayla sold more lemonade than Ethan. The difference in sales is:

Kayla's sales - Ethan's sales = 7/12 - 8/12

Kayla's sales - Ethan's sales = -1/12

The result is negative, which means that Ethan sold 1/12 gallons less than Kayla. Therefore, Kayla sold 1/12 gallons more than Ethan.

Given the above circles,

1) x = 42°

2) x = 29°

3) x = 101°

4) x = 37°

5) x = 31°

6) x = 120°

1) x = 42° because the Angles at the circumference of a circle

subtended by the same arc are equal. Since 35° and 103° are two of the three angles in a triangle whose third angle subtends the circumference of the circle along with ∠x, and the third angle = 180 - 35 - 103 = 42° therefore,  x = 42°

2) x = 29° because according to the central angle theorem, the angle at the circumference is double the one at the center. Since the angle at the circumference is 61°, the one at the center is 122°. Since the line from the center that couches on both sides of the circumference is the radius, they are equal. Hence, the angles formed at their bases are equal.

Thus, x = (180-122)/2

x = 29°

3) 3) x = 101° because the opposite angles of a cyclic quadrilateral are supplementary.

4) x = 37° based on the angle in the alternate segment theorem.

5) x= 31° is based on the "inscribed angle theorem" or the "theorem of the intercepted arc".

This theorem states that if an angle is inscribed in a semicircle, then the angle is a right angle (i.e., it measures 90 degrees). In other words, if you draw any two chords from the ends of the diameter to any point on the arc of the semicircle, then the angle formed by these two chords at the point on the arc will always be 90 degrees.

6) x = 120° on the basis of  the "Tangent-Secant Angle Theorem" or "Tangent-Chord Angle Theorem".

In the problem given above, the angle x is formed by two tangents intersecting at the vertex outside the circle, and the Tangent-Secant (or Tangent-Chord) Angle Theorem is used to find its measure.

The arc formed at the points where the two tangents meet is twice the angle formed inside the circle which is = 30°
Thus, where the rest of the arc traced from the upper point of tangency  is traced back to the lower point of tangency is called A,

The measure of arc A is 360° - 60° = 300°.

The measure of angle x is half the difference between the measures of the intercepted arcs, which is (300° - 60°)/2 = 120°.

Therefore, the measure of angle x would be 120°.


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The values of the quadratic equation; y = x² - 2·x - 1 can be used to complete the table and draw the graph as follows;

(a) x[tex]{}[/tex]       -2        -1           0          1           2          3           4

    y[tex]{}[/tex]       7         -1           2         -2          -1          2           7

(b) Please find attached the graph of the quadratic equation created with MS Excel

A graph of an equation is a visual representation of the rule of the equation that shows how the output value changes as the input values is varied.

The table of values in the question can be completed by substituting the x-values in the table into the equation; y = x² - 2·x - 1 as follows;

The values that complete the table are;

y when x = -1 is; y = (-1)² - 2 × (-1) - 1 = 2

y when x = 0 is; y = (0)² - 2 × (0) - 1 = -1

y when x = 3 is; y = (3)² - 2 × (3) - 1 = 2

y when x = 4 is; y = (4)² - 2 × (4) - 1 = 7

The completed table is therefore presented as follows;

x[tex]{}[/tex]  -2         -1          0          1           2          3             4

y [tex]{}[/tex]  7          -1         2          -2         -1          2             7

(b) The steps that can be used to draw the graph on the coordinate grid using MS Excel are as follows;

Please find attached the graph of the quadratic equation, y = x² - 2·x - 1, created with MS Excel

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Answer:

To work out the probability that Caroline buys fruit or buys a CD or both, you can use the formula:

P(A or B) = P(A) + P(B) - P(A and B)

where A and B are events.

In this case, A is the event that Caroline buys fruit and B is the event that Caroline buys a CD.

The probability that Caroline buys fruit is 0.4 and the probability that Caroline buys a CD is 0.6.

To find the probability that Caroline buys fruit and a CD, you can multiply the probabilities of the two events:

P(A and B) = P(A) x P(B) = 0.4 x 0.6 = 0.24

Therefore, the probability that Caroline buys fruit or buys a CD or both is:

P(A or B) = P(A) + P(B) - P(A and B) = 0.4 + 0.6 - 0.24 = 0.76

So, the probability that Caroline buys fruit or buys a CD or both is 0.76 .

Answer:

The formula for the surface area of a rectangular prism is:

Surface Area = 2lw + 2lh + 2wh

where l, w, and h are the length, width, and height of the prism, respectively.

Since the length, width, and height of the prism in this case are all x inches, we can substitute x for l, w, and h in the formula to get:

Surface Area = 2(x)(x) + 2(x)(x) + 2(x)(x)

Simplifying the equation, we get:

Surface Area = 2x^2 + 2x^2 + 2x^2

Surface Area = 6x^2

Therefore, the equation for the surface area of a prism with a length, width, and height of x inches is:

Surface Area = 6x^2 square inches

The expression that represents the sentence "A number cubed decreased by 24"​ is given as follows:

x³ - 24.

The sentence for this problem is defined as follows:

"A number cubed decreased by 24".

The number is unknown, hence the variable that represents the number is given as follows:

x.

The cube of the number is given as follows:

x³.

The number is then decreased by 24, hence we subtract the expression x³ by 24 to obtain the algebraic expression as follows:

x³ - 24.

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Answer: Your answer should be

               D

Step-by-step explanation: this is because absolute value makes a number positive, no matter the scenario as long as the number(s) are in the absolute value sign (indicated basically by straight lines) this makes it to where the number can never be negative within the lines.

Log base 19 of 53.875 is approximately 0.97212 when converted to base 10.

To convert log base 19 of 53.875 to base 10, we can use the formula:

log base b of x = log base a of x / log base a of b

where a: current base and b: desired base.

So, in this case, we want to convert log base 19 of 53.875 to base 10. We can use the formula above with a = 19 and b = 10:

[tex]log base 10 of 53.875 = log base 19 of 53.875 / log base 19 of 10[/tex]

Log base 19 of 10 is equal to 1/log base 10 of 19, since the logarithm of 10 to any base is equal to 1. Using a calculator, we can determine that the log base 10 of 19 is roughly 1.278753.

Substituting:

log base 10 of 53.875 = log base 19 of 53.875 / 1.278753

To find the value of log base 19 of 53.875, we can use a calculator to evaluate log base 19 of 53.875, which is approximately 1.244036.

Substitute to equation:

log base 10 of 53.875 = 1.244036 / 1.278753

Simplify to get:

log base 10 of 53.875 = 0.97212

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Answer:

If 60% of the beings surveyed preferred hopt juice and 50 beings were surveyed altogether, then we can calculate the number of beings who preferred hopt juice by multiplying the total number of beings surveyed by the percentage that preferred hopt juice: `50 x 0.60 = 30`. So, **30** beings preferred hopt juice.

Answer:

165inch^3

Step-by-step explanation:

4×7×3=84inch^3

9×3×3=81inch^3

84+81=165inch^3

Answer:

34 ft.²

Step-by-step explanation:

We can find the area of this polygon by splitting it into two rectangles.

We can find the area of each individual rectangle, then add their areas to get the area of the entire polygon.

See the attached image for a diagram.

First, we can find the area of the smaller rectangle.

length × width

5 ft. × 2 ft. = 10 ft.²

Next, we can find the area of the larger rectangle.

4 ft. × 6 ft. = 24 ft.²

Finally, we can add the two rectangles' areas together to get the area of the entire polygon.

10 ft.² + 24 ft.² = 34 ft.²

Answer: 120

Step-by-step explanation:

The angle measures should add up to 360

110+130+x=360

240+x=360

x=120

In the exponential decay, A) r = -0.0839 , B) r = -8.39% , C) Value=$11,800.

What is exponential decay?

The term "exponential decay" in mathematics refers to the process of a constant percentage rate reduction in an amount over time. It can be written as y=a(1-b)x, where x is the amount of time that has passed, an is the initial amount, b is the decay factor, and y is the final amount.

To find the annual rate of change between 1992 and 2006, we can use the formula:

r = [tex](V_2/V_1)^{1/n}-1[/tex]

where V1 is the initial value, V2 is the final value, and n is the number of years between the two values.

=>r = -0.0839

Therefore, the rate of change between 1992 and 2006 is -0.0839.

To express the rate of change in percentage form, we can multiply the result from part A by 100:

=>r = -0.0839 x 100

=> r = -8.39%

Therefore, the rate of change between 1992 and 2006 is a decrease of 8.39%.

To find the value of the car in the year 2009, we can assume that the value continues to drop at the same percentage rate as calculated in part A.

From 2006 to 2009, there are 3 years. So, using the formula for exponential decay, we have:

where V0 is the value in 2006, r is the rate of decrease, and n is the number of years between 2006 and 2009.

=>V = 11792.51

Therefore, the value of the car in the year 2009 would be approximately $11,800 (rounded to the nearest $50).

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Answer : 10.5

Explanation: Divide 168/16

By answering the presented question, we may conclude that As a result, variable P(A) = 0.667.

A variable is something that may be changed in the setting of a math concept or experiment. Variables are often represented by a single symbol. The characters x, y, and z are often used generic symbols for variables. Variables are characteristics that can be examined and have a large range of values. These include things like size, age, money, where you were born, academic status, and your kind of dwelling, to name a few. Variables may be divided into two main categories using both numerical and categorical methods.

To compute E[X], we must integrate the PDF X times over its support:

From x=1 to x=2, E[X] = x fX(x) dx

Because the PDF is only non-zero in the interval (1,2), we have:

[tex]E[X] = x(2x/3) dx between x=1.5 and x=2.E[X] = (2/3) x2 dx between x=1.5 and x=2.E[X] = (2/3) [(2^3)/3 - (1.5)^3]E[X] = (2/3) [8/3 - (27/8)]E[X] = 19/12As a result, E[X] = 1.583.[/tex]

P(A) is calculated by integrating the PDF across the area where X is higher than or equal to 1.5:

[tex]P(A) = fX(x) dx between x=1.5 and x=2.P(A) = (2x/3) dx between x=1.5 and x=2.P(A) = [(2/3)x^2] between x=1.5 and x=2P(A) = (2/3)(2^2 - 1.5^2)P(A) = 2/3As a result, P(A) = 0.667.[/tex]

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Complete question -

Consider a random variable X with PDF fX(x) = (2x/3, if 1 < x ≤ 2, 0, otherwise, and let A be the event {X ≥ 1.5}. Calculate E[X], P(A), and E[X | A].

Answer: 6 songs.

Step-by-step explanation:

Each song is $2 dollars. $12 divided by $2 equals 6.

So he can download 6 songs.

Answer: -3x - 6y = -27 (Equation 1)

4x - 2y = -4 (Equation 2)

We can use the method of elimination, which involves adding or subtracting the equations to eliminate one of the variables.

In this case, we can start by multiplying Equation 2 by 3 to eliminate the y variable:

-3x - 6y = -27 (Equation 1)

12x - 6y = -12 (Equation 2 multiplied by 3)

Now we can add Equation 1 to Equation 2:

-3x - 6y = -27 (Equation 1)

12x - 6y = -12 (Equation 2 multiplied by 3)

9x = -39

Simplifying this expression, we get:

x = -39/9 = -13/3

Now we can substitute this value of x into either Equation 1 or Equation 2 to solve for y. Let's use Equation 1:

-3x - 6y = -27

-3(-13/3) - 6y = -27

13 + 6y = -27

6y = -40

y = -40/6 = -20/3

Therefore, the solution to the system of equations is x = -13/3 and y = -20/3, or (-13/3, -20/3) in coordinate form.

To check the solution, we can substitute the values of x and y into both equations:

-3(-13/3) - 6(-20/3) = -27

4(-13/3) - 2(-20/3) = -4

Simplifying both expressions, we get:

13 + 40 = 27 (which is true)

-52/3 + 40/3 = -4 (which is also true)

Therefore, the solution is correct.

Step-by-step explanation:

Due to its neighbouring perpendicular sides and opposite parallel and equal-length sides, the quadrilateral ABCD is a rectangle.

a) The opposite sides are parallel and equal.

b) The opposite angles are equal.

c) The consecutive or adjacent angles are supplementary.

d) If any one of the angles is a right angle, then all the other angles will be at right angle.

e) The two diagonals bisect each other.

4. To find the area of a parallelogram, you can follow these steps:

Measure the length of the base of the parallelogram. Let's call this measurement "b".

Measure the height of the parallelogram. Let's call this measurement "h".

Multiply the base length "b" by the height "h" to get the area of the parallelogram.

Therefore, the formula to calculate the area of a parallelogram is:

Area = base length (b) [tex]*[/tex] height (h)

In symbols, this can be written as:

[tex]A = b \times h[/tex]

Alternatively, if the length of both adjacent sides is known, you can use the formula:

A = base length (b) [tex]*[/tex] the length of adjacent [tex]side (a) * sin[/tex](θ)

where theta is the angle between the base and adjacent side.

5.  the given vertices on a coordinate plane and see what kind of figure they form.

The figure formed by the vertices A, B, C, and D is a quadrilateral. To determine the precise classification of the quadrilateral, we can use its properties.

To determine if the parallelogram is a rhombus, rectangle, or square, we need to check if the diagonals bisect each other and if the adjacent sides are perpendicular. We can find the midpoint of AC and BD as:

Midpoint of AC:  [tex]((2 + 8)/2, (3 + 7)/2) = (5, 5)[/tex]

Midpoint of BD: [tex]((7 + 3)/2, (2 + 8)/2) = (5, 5)[/tex]

The diagonals AC and BD intersect at the midpoint [tex](5,5)[/tex], so they bisect each other.

We can also find the slopes of adjacent sides AB and BC, and we find that they are:

AB: [tex]-1/5[/tex]

BC: [tex]5[/tex]

The product of their slopes is [tex]-1[/tex], so the adjacent sides are perpendicular.

Therefore, the quadrilateral ABCD is a rectangle, because it has opposite sides that are parallel and equal in length, and adjacent sides that are perpendicular.

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Answer:september

Step-by-step explanation: