The table describes the quadratic function p(x).


x p(x)
−1 31
0 17
1 7
2 1
3 −1
4 1
5 7

What is the equation of p(x) in vertex form?
p(x) = 2(x − 3)2 − 1
p(x) = 2(x + 3)2 − 1
p(x) = 3(x − 3)2 − 1
p(x) = 3(x + 3)2 − 1

The mean of the data for minutes spent every day = 47mins.

The mean in statistics is calculated by multiplying all the values in a set of data by the total number of values for a given set of observations. To put it another way, all that is necessary to determine the mean of a set of data is to sum up all the values and divide the total by the number of values.

The data as per the question is:

33, 59, 68, 32, 43.

Now mean will be = sum of all data/ number of data

= (33 + 59 + 68 + 32 + 43)/5

= 235/5

= 47mins.

Hence, the mean of the data is 47mins.

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Answer:

k = 150

Step-by-step explanation:

the measure of a secant- secant angle is half the difference of the intercepted arcs , that is

45 = [tex]\frac{1}{2}[/tex] (k - 60) ← multiply both sides by 2 to clear the fraction

90 = k - 60 ( add 60 to both sides )

150 = k

Answer:

x = 4

Step-by-step explanation:

Use Pythagoras theorem

Using the graph, the answers to the questions are

Using the graph f(1) is the value of y when x is zero.

This is solved by tracing point x = 1 to the graph then tracing the corresponding value of y

Doing this give f(1) to be equal to 2 and this is not negative

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The equation Hats 2030=230.09+23x represents a linear relationship between the number of years since 2010 (x) and the projected number of hats sold in the year 2030.

The slope of the equation, 23, indicates that for each additional year since 2010, the projected number of hats sold in 2030 increases by 23. The y-intercept of the equation, 230.09, represents the projected number of hats sold in 2030 if x is equal to zero (i.e., in the year 2010). This equation can be used to forecast future hat sales based on past trends and can help businesses and organizations plan for inventory and production needs. However, it is important to note that unforeseen events, changes in consumer behavior, and other factors can impact actual hat sales and should be considered when making business decisions.

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The given cylindrical wind chime of dimensions 16 inches long will not fit inside the box in the shape of rectangular-prism having dimensions 10 inches x 8 inches x 6 inches.

1)shorter than

2) shorter than

3)will not fit.

What is a rectangular-prism?

Rectangular prism also called as cuboid, is a closed three dimensional solid having six faces. This prism has eight vertices or corners and 12 edges. The perimeter of this shape is the total length of 12 edges. The area of this shape is the sum of area of for or six surfaces depending on the given situation.

Given dimensions of cuboid/rectangular prism:

Length=10 inches, height= 6 inches & Width=8 inches

Dimensions of cylindrical wind chime: Length of cylinder = 16 inches

Case I: comparision of diagonal of base of the box & length of wind chime.

Diagonal of base can be found using pythagorean theorem:

Diagonal=[tex]\sqrt{L^{2} +W^{2} }[/tex]

              =[tex]\sqrt{(10)^{2} +(8)^{2} }[/tex]

              =[tex]\sqrt{100+64\\}[/tex]

               =[tex]\sqrt{164}[/tex]

               =12.8 inches ≠ 16 inches

Diagonal of rectangular prism base shorter than length of wind chime.

Case II:comparision of interior diagonal of the box & length of wind chime.

Interior diagonal of box=[tex]\sqrt{L^{2} +W^{2} +H^{2} }[/tex]

                                       =[tex]\sqrt{(10)^{2} + (8)^{2} +(6)^{2} }[/tex]

                                       =[tex]\sqrt{100+64+36}[/tex]

                                        =[tex]\sqrt{200}[/tex]

                                        =14.14 inches ≠ 16 inches

Interior Diagonal of rectangular prism is shorter than length of wind chime.

As the wind chime is greater than both the diagonals of cuboid, the chime will not fit into the prism.

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Refer to the complete question attached below.

Answer:We want to find the average rate of change of the graph on the interval 1 ≤ x ≤ 3.

The correct option is B: -4

We know that for a given function f(x) the average rate of change on the interval a ≤ x ≤ b is given by:

Here we have the interval  1 ≤ x ≤ 3, and by looking at the graph we can see that:

f(3) = 1

f(1) = 9

Then the average rate of change is:

So the correct option is B.

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Step-by-step explanation:

Answer: are you mental

Step-by-step explanation:

The exponential function that relates the mass of the substance (y) to the number of years since 2004 (t) is: y = 450 * (0.923) ^ t

In this function, 0.923 is the decimal equivalent of 1 - 0.077, which represents the decay factor of 7.7%. The exponent t represents the number of years since 2004.

So, for example, if we want to know the mass of the substance in 2010 (6 years after 2004), we can plug t = 6 into the function:

y = 450 * (0.923) ^ 6 = 298.8 milligrams

Therefore, the mass of the substance in 2010 would be approximately 298.8 milligrams.

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Note that the exponential function that shows the relationship between the mass y of the substance and the number of years t since 2004 is:

y = 450 * 0.923^t

The general formula for exponential decay is:

y = a * (1 - r)^t

where:

y = final amount

a = initial amount

r = decay rate

t = time elapsed

In this problem, the initial amount is 450 milligrams, and the decay rate is 7.7% per year, or 0.077 as a decimal. We can use the formula above to write an exponential function for the relationship between y and t:

y = 450 * (1 - 0.077)^t

Simplifying this expression, we get:

y = 450 * 0.923^t

Therefore, the exponential function that shows the relationship between the mass y of the substance and the number of years t since 2004 is:

y = 450 * 0.923^t

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Full Question:

Although part of your question is missing, you might be referring to this full question:

In 2004, a sample of a radioactive substance had a mass of 450 milligrams. Since then, the sample has decayed by 7.7% each year. Let t be the number of years since 2004. Let y be the mass of the substance in milligrams. Write an exponential function showing the relationship between y and t.

Answer: 24 grapes

Step-by-step explanation:

By metric system it can be determined that Jasοn lives οn 0.075 hectare.

The metric system is a prοcess οf measuring things. The measurements are used wοrldwide in calculatiοns and research. Sοme examples οf measurements οf things using the metric system are given belοw:

Meter is the unit οf measuring distances and lengths.

Kilοgram οr Gram is used tο measure weight.

Jasοn lives οn 750m plοt .

The given prοblem can be sοlved by metric system.

By metric system

1 hectare =10000m

10000m = 1 hectare

1m= 1/10000 hectare

750m= 750/10000 hectare

Hence,

Jasοn lives οn 0.075 hectare.

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Answer:

Step-by-step explanation:

You want the distances from the ground to the top of the ladder, and from the wall to the base of the ladder when the 47-inch ladder makes an angle of 51° with the ground.

The trig functions Sine and Cosine relate the sides of a right triangle to the angle and the hypotenuse:

  Sin = Opposite/Hypotenuse

  Cos = Adjacent/Hypotenuse

Multiplying by the hypotenuse gives ...

  opposite (ground to top) = (47 in) × sin(51°) = 36.53 in

  adjacent (wall to base) = (47 in) × cos(51°) = 29.58 in

__

Additional comment

These trig relations are summarized in the mnemonic SOH CAH TOA.

Answer:

x = 3,

y = 3

Step-by-step explanation:

{5x + 6y = 33,

{-3x + 4y = 3;

Make x or y the subject from either the first or the second equation (it is advisable to choose one where the numbers can be written as a finite decimal fraction after division, I chose to make x the subject from the 1st equation):

5x = 33 - 6y / : 5

x = 6,6 - 1,2y

Replace x in the 2nd equation with its value from the 1st one (the underlined expression):

-3 × (6,6 - 1,2y) + 4y = 3

Multiply every term inside the bracket by the term on the outside:

-19,8 + 3,6y + 4y = 3

Collect like terms and add them together:

7,6y = 22,8 / : 7,6

y = 3

x = 6,6 - 1,2 × 3 = 3

Every element x in S has a left inverse y that is also a right inverse, and therefore S is a group.

In abstract algebra, a semigroup is a mathematical structure consisting of a set equipped with an associative binary operation. That is, given any three elements a, b, and c in the set, the operation is such that (a * b) * c = a * (b * c), where * denotes the binary operation. This is known as the associative law of the operation.

Semigroups are a fundamental concept in algebraic structures, and they arise in many areas of mathematics and computer science, including group theory, ring theory, semigroup theory, and automata theory. Examples of semigroups include the set of non-negative integers under addition, the set of matrices of a fixed size under matrix multiplication, and the set of all finite strings over a given alphabet under concatenation.

One important property of semigroups is that they need not have an identity element, which is an element that acts as a neutral element under the binary operation.

Let S be a semigroup with a left identity e and a left inverse for every element x in S. We need to show that S is a group.

Let x be an arbitrary element of S. We want to show that x has a two-sided inverse.

Since e is a left identity, we have ex = x for all x in S.

Now, let y be the left inverse of x. This means that yx = e.

We can then use the associativity property of the semigroup to rearrange these equations:

(yx)x⁻¹ = ex⁻¹

y(xx⁻¹) = x⁻¹

ye = x⁻¹

y = x⁻¹

Thus, every element x in S has a left inverse y that is also a right inverse, and therefore S is a group.

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The equation in slope-intercept form y = x - 1

In general, to find the equation of a trend line in a scatter plot, you can use linear regression. Linear regression is a statistical method that finds the line of best fit through a set of data points. The equation of the trend line can then be written in the slope-intercept form, y = mx + b, where "m" is the slope of the line and "b" is the y-intercept.

The yellοw pοints are lοcated at the pοints (1, 0) and (8, 7)

Fοr a linear equatiοn that pases trοugh the pοints (x1, y1) and (x2, y2), the slοpe is:

s = (y₂ - y₁)/(x₂ - x₁)

Then, fοr οur equatiοn, the slοpe is

s = (7-0)/(8-1) = 1

Then οur linear equatiοn is

y = 1 × x + b

where b is the y-axis intercept.

Tο find b, we can use that when x = 1, y = 0

0 = 1 × 1 + b

b = -1

Then οur equatiοn is y = x - 1

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Complete question:

What is the equation of the trend line in the scatter plot?

Use the two yellow points to write the equation in slope-intercept form. Write any coefficients as integers, proper fractions, or improper fractions in simplest form.

The angles that have the given tangents, correct to 2 significant figures, are: 45°, 30° and 15°

To find the angles that have the given tangents, we can use the inverse tangent function (tan⁻¹) or the arctan function on a calculator. Remember that the arctan function gives an angle in radians, so we need to convert the result to degrees.

[tex]\tan \theta = 1.000\ \theta = \tan^{-1}(1.000)\\\\ \theta \simeq 45^{\circ}[/tex]

tan θ = 0.577

θ = tan⁻¹(0.577)

θ ≈ 30°

tan θ = 0.268

θ = tan⁻¹(0.268)

θ ≈ 15°

tan θ = 4

θ = tan⁻¹(4)

θ ≈ 75.96°

θ ≈ 76° (rounded to 3 significant figures)

tan θ = 2.747\\θ = tan⁻¹(2.747)

θ ≈ 70.53°

θ ≈ 71° (rounded to 3 significant figures)

tan θ = 3.732

θ = tan⁻¹(3.732)

θ ≈ 75.89°

θ ≈ 76° (rounded to 3 significant figures)

Therefore, the angles that have the given tangents, correct to 2 significant figures, are:

45°

30°

15°

And the angles that have the given tangents, correct to 3 significant figures, are:

76°

71°

76°

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The type of the function in the question is a linear function

The range of -∞ to +∞ means that the equation is valid for all real numbers.

This range alone provides enough information to determine whether the equation is linear, exponential, or quadratic.

Because only a linear function can have this range

A linear equation is of the form y = mx + b, where m and b are constants and x is the independent variable.

And it always have a range of -∞ to +∞ , otherwise stated

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The truth value is OFT T which is option D.

Truth value is a term used in logic to describe the truth or falsehood of a statement or proposition. Every statement or proposition is either true or false, which are the two possible truth values. In symbolic logic, the truth values are usually represented by the letters "T" for true and "F" for false

The truth value of p - q depends on the truth values of p and q, and the meaning of the "-" operator.

Assuming that "-" represents the logical operator of implication, then p - q means "if p is true, then q is true", or "p implies q".

In this case, both p and q are false, which means that p implies q is true. So the truth value of p - q is T (True).

Therefore, the correct answer is OFT T.

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Answer: In dilation, the parent function is scaled or stretched by a given value, and can be presented by the equation y = a*x^2 where "a" is the dilation factor and a ≠ 0.

When a > 1, the dilation stretches the graph vertically, making the parabola narrower and taller. When a < 1, the dilation compresses the graph vertically, making the parabola wider and shorter. When a is negative, the dilation reflects the graph across the x-axis.

Step-by-step explanation:

Answer:

To solve for Z in the equation X = Y + Z, we need to isolate Z on one side of the equation. We can do this by subtracting Y from both sides:

X - Y = Y + Z - Y

Simplifying the right-hand side, we get:

X - Y = Z

Therefore, the equation X = Y + Z solved for Z is:

Z = X - Y

The monthly mortgage payment on a $500,000 6% 30-year mortgage is approximately $2,997.75.

The angle measures in the figure are 45°, 55° and 145° respectively.

Angles are measured in degrees (°) using a protractor, which is a measuring device used to calculate or draw angles in terms of degrees

From the figure,

7x + 5 + 35 = 180      (angle on a straight line)

7x = 180-35-5

7x = 180-40

7x = 140

Making x the subject of the relation by dividing both sides by 7

7x/7 = 140/7

x = 20

Now, 7(20) +5 = 140+5 = 145°

Also, 90 -35 = 55

Therefore the angles that are measures of angles are 45°, 55°,145°

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Based on the information, the ordinate to the origin is -3.

In mathematics, an "ordinate" refers to the vertical coordinate of a point on a graph, while the "origin" typically refers to the point (0, 0) where the x- and y-axes intersect. Therefore, the phrase "ordinate to the origin" means the vertical distance of a point from the origin, which is simply the value of its y-coordinate.

For example, if you have a point with coordinates (3, 5), the ordinate to the origin would be 5, because that is the distance from the point to the x-axis (which passes through the origin).

The first question is incomplete. Regarding the second question, the equation y = -6x - 3 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept (or the ordinate to the origin). Therefore, the slope is -6 and the y-intercept is -3. So the ordinate to the origin is -3.

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The SUMIF function in excel combines a condition and a sum of the values which meet the stated condition.  The SUMIF statement required on Cell A17 would be =SUMIF(C1:C50, '>=40')

How to write functions in Excel?

In Microsoft Excel, formulas are usually used to generate results or values for a cell.

Now, there are different ways that a formula can be written but then the formula must start with an equal sign. The formula to be used for the cells given has the function "SUMIF" and this is used to provide a sum of numeric values based on a condition.

The SUMIF syntax goes thus ; SUMIF(row_range, condition)

Assume the total membership count is in cells A1 to A50.

Cells, where the values are greater than or equal to 40, should be summed.

Hence, the required formula would be =SUMIF(C1:C50, '>=40')

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A sample size of at least 54,444 guests to ensure a 95% confidence level and a margin of error of 3% and a sample size of at least 217777 guests to ensure a 95% confidence level and a margin of error of ±1.5%.

A. To calculate the sample size needed for a 95% confidence level and a margin of error of 3%, we can use the following formula:

n = (Z^2 * p * q) / E^2

Where:

n = sample size

Z = Z-score (which corresponds to the level of confidence) = 1.96 (for a 95% confidence level)

p = proportion of the population that has the characteristic we are interested in (we don't know this value, so we use 0.5 for a conservative estimate)

q = 1 - p

E = margin of error

Plugging in the values, we get:

n = (1.96^2 * 0.5 * 0.5) / 0.03^2 = 54,444.44

Therefore, we need a sample size of at least 54,444 guests to ensure a 95% confidence level and a margin of error of 3%.

B. If Ouranos Resorts wants to reduce the margin of error to ±1.5%, the required sample size will increase. Using the same formula as above, we get:

n = (1.96^2 * 0.5 * 0.5) / 0.015^2 = 217777

Therefore, we need a sample size of at least 217777 guests to ensure a 95% confidence level and a margin of error of ±1.5%. As we can see, the required sample size has increased by a factor of 4, which means that the survey will be more expensive and time-consuming to conduct.

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Answer:

The key features of the parabola formed by the equation F(x) = -x² + 4x + 12 are: x-intercepts : x = 6 and x = -2, vertex: (2, -8) and opens downward.

What is parabola?

A parabola is a symmetrical plane curve that is shaped like an arch. It is a quadratic function and is defined by the equation y = ax² + bx + c, where a, b, and c are constants.

To identify the key features of the parabola formed by the equation F(x) = -x² + 4x + 12, we can start by finding the x-intercepts, which are the points where the graph of the parabola intersects the x-axis.

To find the x-intercepts, we set F(x) equal to zero and solve for x:

F(x) = -x² + 4x + 12 = 0

We can factor this quadratic equation:

F(x) = -(x² - 4x - 12) = 0

F(x) = -(x - 6)(x + 2) = 0

So the x-intercepts are x = 6 and x = -2.

Next, we can use the vertex form of the quadratic equation, which is:

y = a(x - h)² + k

where (h,k) is the vertex of the parabola and a is a constant that determines whether the parabola opens up or down.

To put F(x) into vertex form, we can complete the square:

F(x) = -x² + 4x + 12

F(x) = -(x² - 4x - 12)

F(x) = -(x² - 4x + 4 - 4 - 12)

F(x) = -(x - 2)² + (-8)

So the vertex of the parabola is (2, -8), and because the coefficient of x² is negative, the parabola opens downward.

Therefore, the key features of the parabola formed by the equation F(x) = -x² + 4x + 12 are:

x-intercepts: x = 6 and x = -2

vertex: (2, -8)

opens downward.

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The equation is 9x + 8y = 81.

The answer of the given question based on circle is we have shown that XY = WZ, since both line segments are equal to XM.

The Pythagorean theorem is fundamental concept in mathematics that relates to sides of right triangle. It states that square of the length of hypotenuse (the side opposite the right angle) is equal to sum of the squares of lengths of the other two sides (the legs).

To prove that XY = WZ, we will use the fact that the line segment connecting the center of a circle to the midpoint of a chord is perpendicular to the chord.

Let M be the midpoint of XY, and N be the midpoint of WZ. Since X = Z, we know that XM is parallel to ZN, and therefore MZNX is a parallelogram.

Let O be center of circle k(O). Then, by the perpendicularity property mentioned above, we know that OM is perpendicular to XY, and ON is perpendicular to WZ.

Since M and N are midpoints of XY and WZ respectively, we have:

XM = MY, and ZN = NW

Therefore, by the Pythagorean theorem, we have:

OX² = OM² + MX², and OZ² = ON² + NZ²

Adding these two equations gives:

OX² + OZ² = OM² + MX² + ON² + NZ²

Since MZNX is a parallelogram, we know that OM = ON, and MX = NZ. Substituting these values gives:

OX² + OZ² = 2OM² + 2MX²

But since OM is perpendicular to XY and ON is perpendicular to WZ, we have:

OM² + MX² = XM²/4, and ON² + NZ² = ZN²/4

Substituting these values gives:

OX² + OZ² = XM²/2 + ZN²/2

But since MZNX is a parallelogram, we know that XM = ZN. Therefore:

OX² + OZ² = XM²

Thus, we have shown that XY = WZ, since both line segments are equal to XM.

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