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6Aonly valid solution is 5π/4 on the interval [0, 2π]

63,cos x = sin x - 1 is x = 30°

To solve the following trigonometric equations, follow the steps given below:sec x + 1 = tan xcos x = sin x - 12π is the solution to the trigonometric equation sec x + 1 = tan x on the interval [0, 2π]. Let's try to solve the given equation:sec x + 1 = tan xLet's use the identity for secant and tangent in terms of cosine and sine:sec x = 1/cos xtan x = sin x / cos xSubstitute these into the given equation:1/cos x + 1 = sin x / cos xThe denominator of the right-hand side can be cleared by multiplying both sides by cos x. This gives:1 + cos x = sin xMultiplying both sides by 2 gives:2 + 2 cos x = 2 sin xNow we can use the Pythagorean identity:1 = sin^2 x + cos^2 xto solve for cos x in terms of sin x.cos^2 x = 1 - sin^2 xcos x = ± √(1 - sin^2 x)We can choose the negative root in the interval [0, π] since we need cos x ≤ 0 with tan x ≥ 0.Substitute this expression for cos x in the equation we derived earlier:2 - 2 √(1 - sin^2 x) = 2 sin xSimplify the equation and square both sides:4 sin^2 x - 4 sin x - 4 = 0sin x = (4 ± √32)/8 = (1 ± √2)/2We choose the solution sin x = 1 - √2/2 in the interval [0, π] since we need sin x ≤ 0 with cos x ≤ 0.Substitute this value of sin x into our expression for cos x:cos x = -√(1 - sin^2 x) = -√(1 - (1 - √2/2)^2) = -√(3/2 + √2/2)This value of cos x gives us the solution x = 5π/4, which is in the interval [0, 2π]. Therefore, the only valid solution is 5π/4 on the interval [0, 2π].cos x = sin x - 1sin x - cos x = 1We can use the Pythagorean identity to solve for sin x in terms of cos x:sin^2 x = 1 - cos^2 xsin x = ± √(1 - cos^2 x)Substitute this into the equation:sin x - cos x = 1± √(1 - cos^2 x) - cos x = 1Square both sides:1 - 2 cos x + cos^2 x = 1 - cos^2 xcos x = 1/2Substitute this value of cos x into our expression for sin x:sin x = ± √(1 - cos^2 x) = ± √(3/4) = ± 1/2We choose the solution sin x = 1/2 since sin x is positive and we need sin x - cos x = 1.Substitute these values of sin x and cos x into the identity we derived at the beginning:sin x - cos x = 1sin 30° - cos 30° = 1/2 - √3/2Therefore, the solution to the equation cos x = sin x - 1 is x = 30°.

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Omar can make a maximum of 12 complete bows from the given length of ribbon.

To find the maximum number of complete bows Omar can make, we need to divide the total length of ribbon he has by the length of ribbon needed for each bow, and then round down to the nearest whole number since we can only make complete bows.

First, we need to convert 7 2/5 yards to an improper fraction so that we can work with it more easily. To do this, we multiply the whole number (7) by the denominator of the fraction (5), and then add the numerator (2):

7 2/5 = (7 x 5) + 2/5 = 35/5 + 2/5 = 37/5 yards

Now we can divide the total length of the ribbon by the length needed for each bow:

37/5 ÷ 3/5 = 37/5 x 5/3 = 37/3

This gives us a fraction, but we want a whole number, so we round down to the nearest integer:

37/3 ≈ 12.33

Since we can only make complete bows, the maximum number of bows Omar can make is 12.

Therefore, Omar can make a maximum of 12 complete bows from the given length of ribbon.

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Answer:

12

Step-by-step explanation:

The maximum number of complete bows omar can make is 12

Answer:

here it is

Step-by-step explanation:

This trapezoid has been divided into two right triangles and a rectangle.

How can the area of the trapezoid be determined using the area of each shape?

Enter your answers in the boxes.

The area of the rectangle is in², the area of the triangle on the left is in², and the area of the triangle on the right is in².

The area of the trapezoid is the sum of these areas, which is in².

Trapezoid ABCD with parallel sides DC and AB. Points F and E are between D and C. FEBA form a rectangle with 4 right angles. D F is 2 inches, F E is 14 inches, E C is 2 inches, A B is 14 inches., and E B is 12 inches.

The Coordinates of A′′, B′′, and C′′ are (1, -1),(-2, 3) and (3, 4).

What is Coordinate system ?

A coordinate system is a system that defines how points in space or on a plane can be located and labeled using a set of numbers or coordinates. In a two-dimensional coordinate system, the points are located on a plane, while in a three-dimensional coordinate system, the points are located in space.

First, we apply the translation of moving all points down by 2 units.

A(1, 3) is transformed to A'(1, 1) by subtracting 2 from the y-coordinate.

B(−2, −1) is transformed to B'(-2, −3) by subtracting 2 from the y-coordinate.

C(3,−2) is transformed to C'(3, −4) by subtracting 2 from the y-coordinate.

Next, we apply the reflection across the x-axis.

A'(1, 1) is transformed to A''(1, -1) by negating the y-coordinate.

B'(-2, −3) is transformed to B''(-2, 3) by negating the y-coordinate.

C'(3, −4) is transformed to C''(3, 4) by negating the y-coordinate.

Therefore, the coordinates for A'', B'', and C'' are:

A''(1, -1)

B''(-2, 3)

C''(3, 4)

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The x-intercept of g(x) is (3,0) and the vertex of g(x) is (3,0).

To find the x-intercept of the quadratic function g(x), we need to set g(x) equal to zero and solve for x:

0 = -5(x - 3)²

Dividing both sides by -5, we get:

0 = (x - 3)²

Taking the square root of both sides, we get:

x - 3 = 0

x = 3

So the x-intercept of the function g(x) is (3,0).

To find the vertex of the function g(x), we can use the formula:

vertex = (h, k)

where;

For a quadratic function in the form:

f(x) = a(x - h)² + k

the vertex is located at the point (h, k).

In the given function g(x), we can see that a = -5, h = 3, and k = 0.

So the vertex of the function g(x) is:

vertex = (h, k) = (3, 0)

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The cost of two apples is 2A = 2(9.95) = $19.90.

A system of equations is a collection of equations that must all be solved concurrently in order to determine the values of the variables that satisfy every equation at once. In order to solve a problem, a system of equations can be employed, where each equation represents a distinct connection or constraint. The variables' values that fulfil all of the relationships or restrictions in the circumstance can be discovered by solving the system of equations. Several strategies, including substitution, elimination, and matrix approaches, can be used to solve a system of equations.

Let us suppose the cost of pineapples = P.

The cost of apples = A.

Thus, from the given statement we have the equation as:

5P + 8A = 42 (equation 1)

7P + 4A = 48 (equation 2)

Rearranging the equation 2 we have:

7P = 48 - 4A

P = (48 - 4A) / 7

Substitute this expression for P into equation 1:

5[(48 - 4A) / 7] + 8A = 42

240/7 - 20A/7 + 8A = 42

44A/7 = 438/7

A = 9.95

Therefore, the cost of two apples is 2A = 2(9.95) = $19.90.

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Answer:

n

Step-by-step explanation:

less

The associative law of addition states that (A + B) + C = A + (B + C). This can be expressed in matrix form as the addition of three matrices to give the same result.Let A, B, and C be 3x3 matrices.

The associative law of addition states that (A + B) + C = A + (B + C).

This can be expressed in matrix form as:

[tex]\begin{bmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{bmatrix}[/tex]

+

[tex]\begin{bmatrix}b_{11} & b_{12} & b_{13} \\b_{21} & b_{22} & b_{23} \\b_{31} & b_{32} & b_{33}\end{bmatrix}[/tex]

+

[tex]\begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23} \\c_{31} & c_{32} & c_{33}\end{bmatrix}[/tex]

[tex]\begin{bmatrix}a_{11} + b_{11} + c_{11} & a_{12} + b_{12} + c_{12} & a_{13} + b_{13} + c_{13} \\a_{21} + b_{21} + c_{21} & a_{22} + b_{22} + c_{22} & a_{23} + b_{23} + c_{23} \\a_{31} + b_{31} + c_{31} & a_{32} + b_{32} + c_{32} & a_{33} + b_{33} + c_{33}\end{bmatrix}[/tex]

=

[tex]\begin{bmatrix}a_{11} + (b_{11} + c_{11}) & a_{12} + (b_{12} + c_{12}) & a_{13} + (b_{13} + c_{13}) \\a_{21} + (b_{21} + c_{21}) & a_{22} + (b_{22} + c_{22}) & a_{23} + (b_{23} + c_{23}) \\a_{31} + (b_{31} + c_{31}) & a_{32} + (b_{32} + c_{32}) & a_{33} + (b_{33} + c_{33})\end{bmatrix}[/tex]

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The volume of air in liters is in the room is 29901.7 litre.

Dimension :

s 11. 0 ft x 12. 0 ft x  8 ft

Volume = l * b ' h

V = 11.0 * 12.0* 8.0

V = 1056 cu. ft.

Covert of unit

1 cu. ft = 28.316 litre

1056 cu. ft. = 1056 * 28.316 litre

1056 cu. ft. = 29901.7 litre

Thus, the volume of air in liters is in the room is 29901.7 litre.

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Answer:

baby formulaa for sure (wut ur school I got same question)

Step-by-step explanation:

add sum purple to yo nails

46. k=4 we have -4, k= 8 we have -64, k=12 we have -64, k=16 we have 256

48. (sqrt(2)/2) + (sqrt(2)/2)i is a valid solution of the given equation

46. Evaluate (1 + i)k - (1 - i)k for k = 4, 8, and 12. Predict the value for k = 16.

The given expression can be expanded using the binomial theorem as follows:

(1 + i)k - (1 - i)k = [(1 + i)2]k/2 - [(1 - i)2]k/2 = (2i)k/2 = 2k/2 × ik/2 = 2k/2 × eiπk/4

For k = 4, we have:

(1 + i)4 - (1 - i)4 = 2^2 × e^iπ = -4

For k = 8, we have:

(1 + i)8 - (1 - i)8 = 2^4 × e^2iπ = 16

For k = 12, we have:

(1 + i)12 - (1 - i)12 = 2^6 × e^3iπ = -64

For k = 16, we can predict the value using the same formula:

(1 + i)16 - (1 - i)16 = 2^8 × e^4iπ = 256 × 1 = 256

Thus, the predicted value is 256.

48. Show that a solution of x8 - 1 = 0 is (sqrt(2)/2) + (sqrt(2)/2)i.

Given: x^8 - 1 = 0

Let z = x^4

z^2 - 1 = 0

z^2 = 1

z = ±1

Thus, x^4 = ±1

Let a = x^2

a^2 - 1 = 0

a^2 = 1

a = ±1

Thus, x^2 = ±1

Let b = x

b^2 - 1 = 0

b^2 = 1

b = ±1

Thus, x = ±1 or x^2 = -1

Let x^2 = -1

x^2 + 1 = 0

(x + i)(x - i) = 0

x = ±i

Thus, the solutions of x^8 - 1 = 0 are x = 1, -1, i, -i, (sqrt(2)/2) + (sqrt(2)/2)i, (sqrt(2)/2) - (sqrt(2)/2)i, (-sqrt(2)/2) + (sqrt(2)/2)i, and (-sqrt(2)/2) - (sqrt(2)/2)i.

Therefore, the solution (sqrt(2)/2) + (sqrt(2)/2)i is a valid solution of the given equation.

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To subtract 2/3 from 11/12, we need to find a common denominator for the two fractions. The least common multiple (LCM) of 3 and 12 is 12. We can rewrite each fraction with a denominator of 12:

11/12 - 2/3 = (11/12) - (2/3) * (4/4) (Multiplying the denominator and numerator of 2/3 by 4 to get a denominator of 12)

11/12 - 8/12 = 3/12

Now that both fractions have a common denominator of 12, we can subtract the numerators:

11/12 - 2/3 = (11 - 8)/12 = 3/12

Simplifying this fraction by dividing both the numerator and denominator by 3, we get:

3/12 = 1/4

Therefore, 11/12 - 2/3 = 1/4.

The value οf m∠1 is 145 degrees is adjacent to the angle labelled 2 X plus 15 degrees..

Pairs οf angles that adds uptο 180 degrees . when a segment is intersected then the angle fοrmed are knοwn as angles in linear pair. the angles add uptο 180 degrees

We are given twο angle in linear pair

And we knοw that there additiοn is 180 degrees.

we have,

x- 30 + 2x+15= 180

3x-15= 180

3x =195

x=65 degrees

2x +15 and m∠1 are vertically οppοsite angle.

Hence they are οf equal measure

Hence

m∠1 = 2x+15

m∠1 = 2(65)+15

m∠1 = 145 degrees

Hence, The value οf m∠1 is 145 degrees.

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Answer:

Proofs attached to answer

Step-by-step explanation:

proofs attached to answer

The veterinarian can accurately measure the dog's weight up to two decimal places (i.e., 35.16 pounds).

What is Precision?

Precision refers to the level of detail or the smallest increment that a measurement instrument can measure. It is the degree to which repeated measurements under the same conditions show the same results.

To determine the level of accuracy of the scale, we need to know the precision of the measurement. Precision refers to the smallest increment that the scale can measure.

If the scale measures in whole pounds, then the precision is 1 pound. If the scale measures in half-pound increments, then the precision is 0.5 pounds. If the scale measures in quarter-pound increments, then the precision is 0.25 pounds.

Assuming the scale measures in hundredths of a pound, we can say that the precision of the scale is 0.01 pounds. Therefore, the level of accuracy of the scale is 0.01 pounds.

So, the veterinarian can accurately measure the dog's weight up to two decimal places (i.e., 35.16 pounds).

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Answer:

understand

Step-by-step explanation:

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

The equivalent expressiοn in the given οptiοns is -13/14t - 17/12.

In mathematics, an expressiοn that incοrpοrates variables, cοnstants, and algebraic οperatiοns is knοwn as an algebraic expressiοn (additiοn, subtractiοn, etc.). Terms cοmprise expressiοns.

The cοncept οf algebraic expressiοns is the use οf letters οr alphabets tο represent numbers withοut prοviding their precise values. We learned hοw tο express an unknοwn value using letters like x, y, and z in the fundamentals οf algebra. Here, we refer tο these letters as variables. Variables and cοnstants can bοth be used in an algebraic expressiοn. A cοefficient is any value that is added befοre a variable and then multiplied by it.

frοm the questiοn:

We can simplify the given expressiοn as fοllοws:

-8/7t + 4/12 - (-3/14t + 7/4)

= -8/7t + 1/3 - (-3/14t + 7/4) (4/12 = 1/3)

= -8/7t + 1/3 + 3/14t - 7/4 (dοuble negative becοmes pοsitive)

= (-16/14)t + (2/6) + (3/14)t - (49/14)

= (-24/14)t - (47/14) (cοmbining like terms)

= (-12/7)t - (47/14) (simplifying)

The equivalent expressiοn in the given οptiοns is -13/14t - 17/12.

Thus, the apprοpriate chοice is:

negative 13 οver 14 times t minus 17 οver 12

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In the equatiοn [tex]f(x) = ab^{(x-h)}+ k:[/tex]

The letter h cοntrοls the hοrizοntal shift οf the graph.

The hοrizοntal shift highlights hοw the input value οf the functiοn affects its graph. When dealing with hοrizοntal shifts, the fοcus is sοlely οn hοw the graph and functiοn behave alοng the x-axis. Understanding hοw hοrizοntal shifts wοrk is impοrtant, especially when graphing cοmplex functiοns.

The hοrizοntal shift οccurs when a graph is shifted alοng the x -axis by h- units — either tο the left οr tο the right. Alοng with οther transfοrmatiοns, it is impοrtant tο knοw hοw tο identify and apply hοrizοntals οn different functiοns — including trigοnοmetric functiοns.

A mathematical equatiοn is a fοrmula that uses the equals sign (=) tο express the equality οf twο expressiοns. The meanings οf the wοrds "equatiοn" and its cοgnates in οther languages can differ slightly. Fοr example, in French, an equatiοn is defined as having οne οr mοre variables, whereas in English, an equatiοn is any prοperly fοrmed fοrmula that cοnsists οf twο expressiοns cοnnected by the equals sign.

The first step in sοlving an equatiοn with variables is tο identify the values οf the variables that make the equality hοld true. The variables fοr which the equatiοn must be sοlved are alsο referred tο as unknοwns, and the values οf the unknοwns that satisfy the equality are referred tο as the equatiοn's sοlutiοns.

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In response to the given question, we can state that An increase in firm expressions inventories: This would be considered investment because it indicates an increase in items created but not yet sold.

In mathematics, you can multiply, divide, add, or subtract. An expression is formed as follows: Expression, number, and math operator Numbers, parameters, and functions make up a mathematical expression. It is possible to contrast phrases and expressions. Every mathematical statement that comprises variables, numbers, and a mathematical action between them is referred to as an expression. For example, the phrase 4m + 5 is made up of the phrases 4m and 5, as well as the variable m from the provided equation, all separated by the mathematical symbol +.  

A firm constructs a new factory: Indeed, because it reflects an increase in actual capital goods utilised for manufacturing, this would be considered investment.

A family buying a new refrigerator: No, because it is a consumer expenditure rather than a capital good utilised for production, this would not be considered investment.

Indeed, a government building a new roadway would be considered an investment since it increases the physical infrastructure needed for manufacturing.

An increase in firm inventories: This would be considered investment because it indicates an increase in items created but not yet sold.

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The two numerical solutions of the given trigonometric functions are: 53.13° or 306.87°

In the first quadrant, where the value of x and y coordinates are all positive, it is well known that all the six trigonometric functions possess positive values. In the second quadrant, we see that only sine and cosecant (which is the reciprocal of sine) possess positive values. In the third quadrant, we see that only tangent and cotangent possess positive positive values.

The cosine function is positive in the first and fourth quadrants.

Now, we are told that;

cos(x) = 0.6

Thus;

x = cos⁻¹(0.6)

x = 53.13° or 306.87°

These are the numerical solutions required.

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Answer:

Akani

2+3+5=10

2/10*30000/1=60000/10=6000

Lwazi

5/10*30000/1=150000/10=15000

Akani will receive R6000 while Lwazi will receive R150000

The required expression has been proved to [tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]

Starting with the given equation:

[tex]$$\sqrt{x-3a}+\sqrt{x-3b}=\sqrt{x-3c}$$[/tex]

Squaring both sides, we get:

[tex]$$(x-3a)+(x-3b)+2\sqrt{(x-3a)(x-3b)}=x-3c$$[/tex]

Simplifying and rearranging terms:

[tex]$$2x-3a-3b-3c+2\sqrt{(x-3a)(x-3b)}=0$$[/tex]

Dividing both sides by 2 and rearranging:

[tex]$$\sqrt{(x-3a)(x-3b)}= \frac{3a+3b+3c-2x}{2}$$[/tex]

Squaring both sides again:

[tex]$$(x-3a)(x-3b)=\left(\frac{3a+3b+3c-2x}{2}\right)^2$$[/tex]

Expanding the right side:

[tex]$(x-3a)(x-3b)=\frac{9a^2+9b^2+9c^2+2(3ab+3ac+3bc)-4(3a+3b+3c)x+4x^2}{4}$$[/tex]

Simplifying:

[tex]$$4(x-3a)(x-3b)=9a^2+9b^2+9c^2+2(3ab+3ac+3bc)-12(a+b+c)x+16x^2$$[/tex]

Expanding:

[tex]$$4x^2-12(a+b+c)x+9(a^2+b^2+c^2+2ab+2bc+2ac)-36ax-36bx+36ab-36cx+36bc=0$$[/tex]

Rearranging:

[tex]$$4x^2-8(a+b+c)x+9(a+b+c)^2-12(ab+bc+ac)=0$$[/tex]

Using the quadratic formula:

[tex]$x = \frac{8(a+b+c) \pm 2\sqrt{16(a+b+c)^2-4\cdot4\cdot(9(a^2+b^2+c^2-ab-bc-ac))}}{8}$$[/tex]

Simplifying:

[tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]

Therefore, we have proven that:

[tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]

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The probability of pulling a red marble two times in a row when each marble has equally chance of being pulled would be = 5/7 multiplied by 4/6. That is option B.

Probability is defined as the expression that shows the possibility of an outcome of an event which is likely or not likely to occur.

The total number of marbles in the bag = 7 marbles

The total number of red marbles = 5

The total number of blue marbles = 2

After the first pull the probability = 5/7

After the second pull = 4/6 ( there is subtraction of a red marble from the total after the first pull making it 4 red marbles and 6 total).

Therefore, the probability of pulling a red marble two times in a row = 5/7× 4/6

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Answer:

he could work 11 hours

Step-by-step explanation:

if he worked 11 hours he would have earned 66 dollars and had to to take out 4 dollars for an equipment fee therefore leaving him with 62 dollars.

The cylinder has a volume of about 30.55 cubic feet. Consequently, option D, which is 30.55 cubic feet, is the right response.

As a result, the product of the base area and height can be used to determine the cylinder's volume. The volume of any cylinder with base radius "r" and height "h" equals base times height. A cylinder's volume is therefore equal to r²h cubic units.

We use the following formula to determine a cylinder's volume:

V = πr²h

The height of the cylinder is 3.8 feet, according to the issue description.

The distance in feet between two points on the circular base that pass through the centre is designated as the 3.2 feet segment.

When these values are added to the formula, we obtain:

V = π(1.6)²(3.8)

V ≈ 30.55 cubic feet

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Answer:

option D, which is 30.55 cubic feet, is the right response.

Step-by-step explanation:

y = (x - 1)² - 1

y = (x + 13)² - 9.

The given equation is y = x² - 2x. To put the equation in the form y = (x - h)² + k, we will complete the square as follows:Step 1: First, we need to find the value of h by dividing the coefficient of x by 2 and squaring the result. This value will be added inside the parenthesis to complete the square. So, h = (-2/2)² = 1.Step 2: Now, we will add and subtract the value of h inside the parenthesis. y = x² - 2x + 1 - 1 = (x - 1)² - 1Step 3: Finally, we need to find the value of k by subtracting the constant term outside the parenthesis. So, k = -1. Therefore, the equation y = x² - 2x can be written in the form y = (x - 1)² - 1. Answer: y = (x - 1)² - 1.The given equation is y = x² + 26x + 160. To put the equation in the form y = (x - h)² + k, we will complete the square as follows:Step 1: First, we need to find the value of h by dividing the coefficient of x by 2 and squaring the result. This value will be added inside the parenthesis to complete the square. So, h = (26/2)² = 169.Step 2: Now, we will add and subtract the value of h inside the parenthesis. y = x² + 26x + 169 - 169 + 160 = (x + 13)² - 9Step 3: Finally, we need to find the value of k by subtracting the constant term outside the parenthesis. So, k = -9. Therefore, the equation y = x² + 26x + 160 can be written in the form y = (x + 13)² - 9. Answer: y = (x + 13)² - 9.

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The points are vertices of an isosceles triangle by using distance formula.

What is distance formula?

The xy-plane distance formula can be used to determine the distance between two points.

We know that the distance formula is

d = [tex]\sqrt{(x_{2} -x_{1} )^{2} +(y_{2} -y_{1} )^{2} }[/tex]

Using this, we get

⇒AB = [tex]\sqrt{(7 -3)^{2} +(8 -4 )^{2} }[/tex]

⇒AB = [tex]\sqrt{(4)^{2} +(4 )^{2} }[/tex]

⇒AB = √16 + 16

⇒AB = √32

Similarly,

⇒AC = [tex]\sqrt{(11 -3)^{2} +(4 -4 )^{2} }[/tex]

⇒AC = [tex]\sqrt{(8)^{2} +(0 )^{2} }[/tex]

⇒AC = √64

⇒AC = 8

Similarly,

⇒BC = [tex]\sqrt{(11 -7)^{2} +(4 -8 )^{2} }[/tex]

⇒BC = [tex]\sqrt{(4)^{2} +(4 )^{2} }[/tex]

⇒BC = √16 + 16

⇒BC = √32

Since. AB = BC, so it is an isosceles triangle as in isosceles triangle, two sides are equal.

Hence, the given points are the vertices of an isosceles triangle.

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Number of chocolate bars that Ron eat is 30

To solve the problem, we can work backwards from the information provided and use variables to represent unknown quantities.

Let's start with Betty, who ate 10 chocolate bars. We know that Billy had half as many as Betty, so we can represent Billy's chocolate bars as:

Billy = 1/2 × Betty

Billy = 1/2 × 10

Billy = 5

Next, we know that Bob ate 5 more chocolate bars than Billy, so we can represent Bob's chocolate bars as:

Bob = Billy + 5

Bob = 5 + 5

Bob = 10

Finally, we know that Ron eats 3 times as many chocolate bars as Bob, so we can represent Ron's chocolate bars as:

Ron = 3 × Bob

Ron = 3 × 10

Ron = 30

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The polynomial function is therefore a quadratic function with the formula (3x - 10)²/25 + 10.

Unlike the graph that depicts the stock market, the functions for roller coasters need to be smooth curves. The polynomial function[tex]f(x)= -0.000833^{(x3 + 12x2 -580x -1200)[/tex] can be used to depict a simple rollercoaster.

According to the information provided, the original function was as follows: vertical stretch by a factor of 2 horizontal compression by a factor of 1/0.3 = 10/3 horizontal shift right by one unit.

Starting with a common function like f(x) = x², we can perform the following transformations:

Replace x with 3x/10 for the horizontal compression factor, and with 3x/10 for the horizontal shift to the right. Replace x with 3x/10 - 1 for the vertical stretch factor. Vertically move up 10 units by replacing y with 2y: y = 2y + 10.

When you combine everything, you get:

2 f(0.3x - 1) + 10

= 2 (2(3x/10 - 1)² + 10)

= (3x - 10)²/25 + 10

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Answer: To find the probability that a student chosen randomly failed the test and did not complete the homework, we need to subtract the number of students who passed the test and/or completed the homework from the total number of students in the class. Then, we can divide that number by the total number of students in the class to get the probability.

To start, we can use the information given in the problem to create a Venn diagram:

```

Test

|------|------|

| | |

Fail | | |

| | |

--------|------|------|

| | |

| | |

Pass | | |

Assignment|------|------|

| | |

| | |

```

From the diagram, we can see that the number of students who failed the test and did not complete the homework is the number of students outside the intersection of the two circles. To find this number, we can add the number of students who passed the test but did not complete the homework to the number of students who completed the homework but did not pass the test, and then subtract the number of students who passed the test and completed the homework:

Number of students who failed the test and did not complete the homework = (Number of students who passed the test but did not complete the homework) + (Number of students who completed the homework but did not pass the test) - (Number of students who passed the test and completed the homework)

Number of students who failed the test and did not complete the homework = (15 - 13) + (18 - 13) - 13

Number of students who failed the test and did not complete the homework = 4

Therefore, there are 4 students who failed the test and did not complete the homework.

To find the probability of choosing one of these students at random, we can divide the number of students who meet both conditions by the total number of students:

Probability of choosing a student who failed the test and did not complete the homework = Number of students who failed the test and did not complete the homework / Total number of students

Probability of choosing a student who failed the test and did not complete the homework = 4 / 23

Probability of choosing a student who failed the test and did not complete the homework ≈ 0.17

Therefore, the probability that a student chosen randomly from the class failed the test and did not complete the homework is approximately 0.17, or 17% (rounded to two decimal places).