W(–1, 4), X(4, 0), Y(–5, 1), and Z(0, –3) are vertices of quadrilateral WXZY. What is true of wx and yz?

The weight of the package is 56 using arithmetic operations.

Mathematicians use multiplication to calculate the product of two or more integers. It is a fundamental operation in mathematics that is frequently used in everyday living. When we need to combine sets of similar sizes, we use multiplication. The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the factors that are compounded are referred to as the factors. Repeated adding of the same number is made easier by multiplying the numbers.

let weight of package be= x

x*5/7=40

x=(40*7)/5

x=56

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Step-by-step explanation:

4x + y^2 + 2y =  - 5

4x = - y^2 -2y - 5          

4x = - (y^2 + 2y)   - 5      'complete the square' for y

4x = - ( y +1)^2   + 1    - 5

 x = - 1/4 ( y+1)^2  - 1             vertex is at -1, -1

-14x+2y^2-8y -20 = 0

14x = 2y^2 -8y-20

14x = 2 ( y^2 - 4y) - 20     complete the square for y

14x = 2(y-2)^2  -8 - 20

x = 1/7 ( y-2)^2 - 2                Vertex is at   -2 , 2

m∠R is therefore 76 degrees as a triangle's total sides equal 180 degrees.

The degree of rotation between two lines or two planes around a central point is measured by an angle. Typically, it is expressed in radians or degrees. Angles are used in many mathematical and scientific uses, including trigonometry, physics, and engineering, where they are crucial in determining the shape and characteristics of geometric figures. Angles come in four different varieties: acute (less than 90 degrees), right (exactly 90 degrees), oblique (more than 90 degrees), and straight (exactly 180 degrees).

given

Because a triangle's total sides equal 180 degrees, we have:

R, S, and T add up to 180.

Inputting the numbers provided yields:

m∠R + 72 + 32 = 180

Simplifying the equation:

m∠R = 76

m∠R is therefore 76 degrees as a triangle's total sides equal 180 degrees.

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Step-by-step explanation:

A. To find f(x+h), we substitute (x+h) for x in the equation f(x) = 4x + 7:

f(x+h) = 4(x+h) + 7

Expanding the brackets:

f(x+h) = 4x + 4h + 7

Simplifying, we get:

f(x+h) = 4x + 7 + 4h

Therefore, f(x+h) = 4x + 7 + 4h.

B. To find f(x+h)-f(x)/h, we use the formula for the difference quotient:

[f(x+h) - f(x)] / h

Substituting the expressions we derived earlier:

[f(x+h) - f(x)] / h = [(4x + 7 + 4h) - (4x + 7)] / h

Simplifying, we get:

[f(x+h) - f(x)] / h = (4x + 4h - 4x) / h

Canceling out the 4x terms, we get:

[f(x+h) - f(x)] / h = 4h / h

Simplifying further, we get:

[f(x+h) - f(x)] / h = 4

Therefore, f(x+h)-f(x)/h = 4.

Roy purchased whole pizzas for P 190.00 each. To determine the number of whole pizzas he bought, we can divide the total cost of the pizzas by the cost of each pizza. Therefore, the calculation P 950.00 / P 190.00 results in 5, indicating that Roy bought five whole pizzas.

Roy spent a total of P 1070 for pizza with 3 sets of additional toppings. Since each set of additional toppings costs P 40.00, then the total cost of the toppings is 3 x P 40.00 = P 120.00. Subtracting this from the total amount spent gives us P 950.00, which is the cost of the pizzas alone.

Since each whole pizza costs P 190.00, we can divide the cost of the pizzas by the cost of each pizza to find the number of whole pizzas Roy bought. Therefore, P 950.00 / P 190.00 = 5.

Thus, Roy bought 5 whole pizzas.

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Dan’s mistake is that he said the probability of Deshawn not doing basketball or not talking to friends after school is 35% which is not true.

Given:

Let’s consider the probability of Deshawn not doing basketball after school:Probability of Deshawn not doing basketball after school = 100% - Probability of Deshawn doing basketball after school= 100% - 20% = 80%

Similarly, let’s consider the probability of Deshawn not talking to friends after school: Probability of Deshawn not talking to friends after school = 100% - Probability of Deshawn talking to friends after school= 100% - 45% = 55% Probability of Deshawn doing neither basketball nor talking to friends after school:Probability of Deshawn not doing basketball after school * Probability of Deshawn not talking to friends after school= 80% * 55% = 44%

The probability of Deshawn doing either basketball or talking to friends after school is 65%, and the probability of Deshawn doing neither basketball nor talking to friends after school is 44%, which is greater than 35% which is Dans mistake. Hence, Dan’s mistake is that he said the probability of Deshawn not doing basketball or not talking to friends after school is 35% which is not true.

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c) The locus of z is a straight line parallel to the y-axis for x = 4.

The locus of a line is the set of all points that satisfy a given geometric condition related to that line. The term "locus" refers to the path or trajectory followed by a point or set of points that satisfy the given condition.

To determine the locus of z in the given equation |z-2| = |z-6|, we can use the definition of the absolute value of a complex number which is

[tex]|x + iy| = \sqrt{(x^2 + y^2)}[/tex]

So, we can square both sides of the given equation to get:

[tex]|z-2|^2 = |z-6|^2[/tex]

put z = (x + iy)

[tex]|x+iy-2|^2 = |x+iy-6|^2\\|(x-2)+iy|^2 = |(x-6)+iy|^2\\[/tex]

[tex][\sqrt{((x-2)^2 + y^2)} ]^{2} = [\sqrt{((x-6)^2 + y^2)} ]^{2}[/tex]

x² + 4 - 4x = x² + 36 - 12x

after simplification, x = 4

Therefore, the locus of z is a straight line parallel to the y-axis passing through the point x = 4.

Hence, the correct option is (c) a straight line parallel to the y-axis.

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The solution to the system of equations is x = 16/5 and y = 11/5.

Graphing, substitution, elimination, and matrices are the four methods for solving systems of equations.

Using the substitution method, we can find x and y.

To begin, let us solve the first equation for x:

2x + 3y = 13

2x = 13 - 3y

x = (13 - 3y)/2

This expression for x can now be substituted into the second equation:

x - y = 1

[(13 - 3y)/2] - y = 1

To remove the denominator, multiply both sides by 2:

13 - 3y - 2y = 2

Simplifying:

13 - 5y = 2

Taking 13 off both sides:

-5y = -11

-5 divided by both sides:

y = 11/5

Now that we've discovered y, we can plug it back into either of the original equations to find x. Let's look at the first equation:

2x + 3y = 13

2x + 3(11/5) = 13

To eliminate the fraction, multiply both sides by five:

10x + 33 = 65

Taking 33 away from both sides:

10x = 32

Divide both sides by ten:

x = 16/5

As a result, the system of equations solution is x = 16/5 and y = 11/5.

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Answer:

Step-by-step explanation:

X+125=180    (angles on a straight line add to 180 (supplementary) ).

so x=55°

The number of different selections possible if at least 3 balls must be blue is 35


1. Calculate the total number of possible selections (6 red + 7 blue = 13 total): 13C4 = 715
2. Calculate the number of possible selections that have fewer than 3 blue balls: 6C4 = 15
3. Subtract the number of possible selections with fewer than 3 blue balls from the total number of possible selections: 715 - 15 = 700
4. Divide the answer by the number of selections with exactly 3 blue balls: 700/7 = 100
5. Multiply the result by the number of selections with exactly 4 blue balls: 100 * 7 = 700
6. Finally, subtract the number of selections with 4 blue balls from the total number of possible selections: 715 - 700 = 35

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The Height of tree is around 8.33 feet tall.

In science, we work out the Height of an item utilizing distance and points. Distance is the even distance between the items, and point is the point over the level of the article's top, which gives the item's level.

We can utilize the rule of comparable triangles.

The hypotenuse of this triangle would be the line associating Jenny's eyes to the highest point of the tree (we should refer to this distance as "h").

We can set up the accompanying extent between the two triangles:

h / 20 = (h + 5) / 8

To solve for "h", we can cross-multiply and simplify:

8h = 20(h + 5)

8h = 20h + 100

12h = 100

h = 8.33 feet

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Answer:

5/7

Step-by-step explanation:

Make all the denominators 42.

26/42, 27/42, 28/42, 29/42

The pattern is the numerator increases by one each time.

30/42 = 5/7

Hope this helps!

Answer:

Step-by-step explanation:

sorry I don't know

The induced emf in a region of the brain when a conducting loop is held near a person's head and the current in the loop is changed rapidly, is equal to -525 V.

The induced emf can be calculated using Faraday's law of electromagnetic induction, which states that the emf induced in a loop of wire is equal to the rate of change of magnetic flux through the loop.

The magnetic flux (Φ) is equal to the product of the magnetic field (B) and the area (A) through which it passes. Therefore, the induced emf (ε) is given by:

ε = -dΦ/dt ⇒ -B dA/dt.

Where the negative sign indicates that the emf is induced in a direction that opposes the change in magnetic flux.

In this problem, the magnetic field changes at a rate of 3.00 × 10^4 T/s over an area of 1.75 × 10^-2 m^2. Therefore, the induced emf is:
Plugging in our values, we get:
E = (-3.00 10^4 T/s)(1.75 10^(-2) m^2)/(1 s)
E = -525 V

Therefore, the induced emf, in this case, is -525 V. Here, the negative sign shows that the emf is induced in a direction that opposes the change in magnetic flux

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Answer: 96

Step-by-step explanation:

M = 180 - 84 = 96

m<k = 96

The total number of pairwise comparisons for k is (k*(k-1))/2.

There are (k*(k-1))/2 pairwise comparisons in a completely randomized design with k treatments. For example, if there are 4 treatments, there will be 6 pairwise comparisons (4C2 = 6). Here's the explanation:In a completely randomized design, the treatments are randomly assigned to the experimental units. The main objective of such a design is to determine whether the treatment means are different from each other or not.To compare the treatment means, we use the mean square between treatments (MST) and mean square error (MSE). The test statistic used to compare the means is F = MST/MSE.The ANOVA table for a completely randomized design has the following format:Source of variationSum of SquaresDegrees of freedomMean SquareF-testtreatmentSS(k-1)k-1MST=MSTrMSEResidualSSTn-kMSEReference: https://www.stat.yale.edu/Courses/1997-98/101/anovar.htmNow, we need to compare each pair of treatments using a multiple comparisons procedure. A pairwise comparison involves comparing the means of two treatments only.The total number of pairwise comparisons is given by the combination formula:$$ \frac{k!}{2!(k-2)!} = \frac{k(k-1)}{2} $$Therefore, the total number of pairwise comparisons for k is (k*(k-1))/2.

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When one of the coin is flipped it lands on heads with probability 0.6 and when the other is flipped, it lands on heads with probability 0.3, then the probability that the coin lands on heads is 0.45 and the coin lands on heads but the probability that the chosen coin was the one that lands on heads with probability 0.6 is 0.67.

a) The probability of getting heads, we can use the law of total probability.

There are two coins, and each has a probability of landing on heads. So we can calculate the probability of getting heads by weighting each coin's probability by its probability of being chosen.

Therefore,

P(heads) = P(heads from coin 1) * P(choose coin 1) + P(heads from coin 2) * P(choose coin 2)

Plugging in the values, we have:

P(heads) = 0.6 * 0.5 + 0.3 * 0.5 = 0.45

Therefore, the probability of getting heads is 0.45.

b) The probability that the chosen coin was the one that lands on heads with probability 0.6, given that the coin lands on heads, we need to use Bayes' theorem. Specifically, we have:

P(choose coin 1 | heads) = P(heads from coin 1 | choose coin 1) * P(choose coin 1) / P(heads)

Plugging in the values, we have:

P(choose coin 1 | heads) = 0.6 * 0.5 / 0.45 = 0.67

Therefore, the probability that the chosen coin was the one that lands on heads with probability 0.6, given that the coin lands on heads, is 0.67.

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Answer: One possible solution is to work 12.3 hours landscaping and 3.7 hours clearing tables, earning approximately $167.1.

Step-by-step explanation:

Let x be the number of hours landscaping and y be the number of hours clearing tables.

The system of inequalities can be written as:

x ≤ 13 (maximum of 13 hours landscaping)

x + y ≤ 16 (maximum of 16 hours total)

13x + 8y ≥ 160 (minimum earnings of $160)

To graph this system of inequalities, we can start by graphing the boundary lines of each inequality as follows:

x = 13 (vertical line at x = 13)

x + y = 16 (line with intercepts at (0, 16) and (16, 0))

13x + 8y = 160 (line with intercepts at (0, 20) and (12.3, 0))

Note that we only need to graph the portion of the lines that are in the feasible region (i.e. where x and y are non-negative).

The feasible region is the triangle formed by the intersection of the three boundary lines, as shown below:

    |

20 --|--------------------------

    |              /|

    |           /   |

    |        /      |

    |     /         |

16 --|------------------

    |  /              |

    |/                 |

    -------------------------

           13         12.3

Let x be the number of hours landscaping and y be the number of hours clearing tables.

The system of inequalities can be written as:

x ≤ 13 (maximum of 13 hours landscaping)

x + y ≤ 16 (maximum of 16 hours total)

13x + 8y ≥ 160 (minimum earnings of $160)

To graph this system of inequalities, we can start by graphing the boundary lines of each inequality as follows:

x = 13 (vertical line at x = 13)

x + y = 16 (line with intercepts at (0, 16) and (16, 0))

13x + 8y = 160 (line with intercepts at (0, 20) and (12.3, 0))

Note that we only need to graph the portion of the lines that are in the feasible region (i.e. where x and y are non-negative).

The feasible region is the triangle formed by the intersection of the three boundary lines, as shown below:

lua

Copy code

    |

20 --|--------------------------

    |              /|

    |           /   |

    |        /      |

    |     /         |

16 --|------------------

    |  /              |

    |/                 |

    -------------------------

           13         12.3

The vertices of the feasible region are (0, 16), (12.3, 3.7), and (13, 0).

To determine one possible solution, we can evaluate the objective function (total earnings) at each vertex:

(0, 16): 13(0) + 8(16) = $128

(12.3, 3.7): 13(12.3) + 8(3.7) ≈ $167.1

(13, 0): 13(13) + 8(0) = $169

Therefore, one possible solution is to work 12.3 hours landscaping and 3.7 hours clearing tables, earning approximately $167.1.

Answer:

When a point is reflected over the x-axis, the x-coordinate remains the same, but the y-coordinate is multiplied by -1.

So, the coordinates of X' are (9, -5) since the x-coordinate remains the same and the y-coordinate is multiplied by -1.

Similarly, the coordinates of Y' are (-3, -6) and the coordinates of Z' are (-8, -4).

Therefore, X′ is (9,−5), Y′ is (−3,−6), and Z′ is (−8,−4).

Jake is presently [tex]0[/tex] years old and the year is [tex]2036[/tex], which is not a relevant response or there may be some missing details in the question.

A span of time that is equivalent to one year on the Calender but starts at a different period. A cycle of 365 and 366 day split into twelve month starting in January and ending in December.

Every monthly on the calender has four complete weeks since every month has at least 28 days. A few month have a few more days, but these extra days don't add up to a full week, therefore they aren't counted.

Let's first find the current year, given that the year in which their ages will sum up to [tex]16x[/tex] is[tex]2036[/tex].

[tex]2036 - (y - 2036) = 2*2036 - y[/tex]

Simplifying this expression, we get:

[tex]2*2036 - y = 4072 - y[/tex]

[tex]2y = 4072[/tex]

[tex]y = 2036[/tex]

Now, let's find Jake's current age by subtracting his birth year from the current year [tex]y - (2036 - 7x)[/tex]

Since their ages sum up to 16x in 2036, we have:

[tex]x + 7x = 16x[/tex]

[tex]8x = 16x[/tex]

[tex]x = 0[/tex]

This means that Jake is currently [tex]0[/tex] years old, which is not a meaningful answer.

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Answer:

Step-by-step explanation:

1. add 14 to 24

2. ur done!

Answer :

1. We are given with a parallelogram whose opposite sides are :-

We know that opposite sides and angles of the parallelogram are equal.

PS = RQ

[tex] \implies \sf \: (-1 + 4x) = (3x + 3) \\ [/tex]

[tex] \implies \sf \: 4x - 3x = 3 + 1\\ [/tex]

[tex] \implies \sf \: x = 4 \\ [/tex]

Length of RQ is (3x + 3)

Substituting value of x = 4 in RQ.

[tex] \implies \sf \: (3x +3) \\ [/tex]

[tex] \implies \sf \: 3(4) + 3 \\ [/tex]

[tex] \implies \sf \: 12 + 3 \\ [/tex]

[tex] \implies \sf \: 15 \\ [/tex]

Therefore, Length of RQ will be 15

2. Find [tex] \sf \angle G [/tex]

We are given with :-

Since, Opposite angles of parallelogram are also equal :

[tex]\angle G = \angle E \\ [/tex]

[tex]\implies \sf \: 5x -9 = 3x + 11 \\ [/tex]

[tex]\implies \sf \: 5x - 3x = 11 +9 \\ [/tex]

[tex] \implies \sf \: 2x = 20 \\ [/tex]

[tex] \implies \sf \: x = \dfrac{20}{10} \\ [/tex]

[tex] \implies \sf \: x = 10 \\ [/tex]

Since [tex] \sf \angle G [/tex] is 5x - 9.

Substituting value of (x = 10) in [tex] \sf \angle G [/tex]

[tex] \implies \sf \: 5(10) - 9 \\ [/tex]

[tex] \implies \sf \: 50 -9 \\ [/tex]

[tex] \implies \sf \: 41 \\ [/tex]

Therefore, [tex] \sf \angle G [/tex] is 41.

Answer:

19. QR = 15 units.

20. m ∡G=41°

Step-by-step explanation:

The properties of a parallelogram are:

Question No. 19.

We know that the opposite side of parallelogram is equal, So

PS=QR

-1+4x=3x+3

First of all we will find the value of x.

-1 + 4x = 3x + 3

Subtracting 3x from both sides, we get:

x - 1 = 3

Adding 1 to both sides, we get:

x = 4

SO, QR=3*4+3=15

Side QR is 15 units.

Question No. 20.

We know that the opposite angle of parallelogram is equal, So

EF=GD

3x+11=5x-9
Subtracting 3x from both sides:

3x + 11 - 3x = 5x - 9 - 3x

11 = 2x - 9

Next, we can add 9 to both sides to isolate the variable term on one side:

11 + 9 = 2x - 9 + 9

20 = 2x

Finally, we can solve for x by dividing both sides by 2:

20/2 = 2x/2

x=10

Now

m ∡G=5x-9=5*10-9=41°

therefore m ∡G=41°

Probability of the number being part of A∩B= 2/9

The Venn diagram, a style of diagram that illustrates the logical connection between sets, was created by John Venn in the 1880s. The examples are used to illustrate fundamental set connections and introduce basic set theory in the fields of probability, logic, statistics, linguistics, and computer science.

A Venn diagram is a type of visual representation that uses circles to show the relationships between various items or constrained groups of objects. Circles that coincide and those that intersect have certain properties in common. Venn diagrams are helpful for visually illustrating the relationship and differences between two ideas.

The probability of an occurrence is a figure used to indicate the likelihood that the event will occur. It is expressed as a figure between 0 and 1, or between 0% and 100%, when expressed as a percentage. The likelihood that the occurrence will occur increases with the probability.

In this question,

A∩B= 4,9

Probability of the number being part of A∩B= 2/9

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The set [1 0 −3 , −3 1 6 , 1 −1 0] is not a basis for set of real numbers R cubed(ℝ3).

The reason why it is not a basis is because it is not linearly independent. However, the set does span set of real numbers R cubed(ℝ3).To determine if the given set is a basis for set of real numbers R cubed(ℝ3), we need to test for linear independence and for span.Linear independence

The given set is said to be linearly independent if and only if the only solution to the equation a[1 0 −3] + b[−3 1 6] + c[1 −1 0] = [0 0 0] is the trivial solution where a, b and c are constants.If the given set is linearly independent then it is a basis for R3; if it is not linearly independent, it is not a basis for R3.

SpanThe given set is said to span R3 if every vector in R3 can be written as a linear combination of vectors in the given set.

If the given set spans R3, then it can be considered a basis for R3.For us to test if the given set is linearly independent, we can form a matrix by placing the three given vectors into the columns of a 3 x 3 matrix as follows:[1 0 1] [−3 1 −1] [−3 6 0]

By expanding the determinant of the matrix above, we get: det(A) = 0 - 0 - (-3) = 3

Since the determinant is non-zero, we can say that the given set is linearly independent. Since the given set is linearly independent, we can then use it to span R3. Hence the given set does not form a basis for R3 but it is linearly independent and spans R3.

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Answer: 0000

Step-by-step explanation:

Answer: 4x(x - 4)

Step-by-step explanation:

4x² - 16x = 4x(x - 4)

Now we can see that the expression inside the parentheses can also be factored:

x - 4 = (x - 4)

So the fully factorized expression is:

4x² - 16x = 4x(x - 4) = 4x(x - 4)

Answer:

ㅤ

- 4x( x + 4 )

ㅤ

Step-by-step explanation:

ㅤ

[tex]\large{\pmb{\sf{ - 4x^{2} - 16x}}}[/tex]

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[tex]\large{\underline{\underline{\sf{Taking \: Out \: {\green{4}} \: As \: Common:-}}}}[/tex]

ㅤ

[tex]\large{\pmb{\sf{\leadsto{- 4(x^{2} + 4x)}}}}[/tex]

ㅤ

[tex]\large{\underline{\underline{\sf{Taking \: Out \: {\green{x}} \: As \: Common:-}}}}[/tex]

ㅤ

[tex]\large{\purple{\boxed{\pmb{\sf{\leadsto{- 4x(x + 4)}}}}}}[/tex]

━━━━━━━━━━━━━━━━━━━━━━

[tex]\star \: {\large{\underline{\underline{\pink{\mathfrak{More:-}}}}}} \: \star[/tex]

ㅤ

[tex]\large{\dashrightarrow}[/tex] Two positive always makes positive sign when multiplied.

ㅤ

[tex]\large{\dashrightarrow}[/tex] Two negatives always makes positive sign when multiplied.

ㅤ

[tex]\large{\dashrightarrow}[/tex] A positive and a negative always makes negative sign when multiplied.

ㅤ

[tex]\large{\dashrightarrow}[/tex] The sum of two positives is always positive with a positive sign.

ㅤ

[tex]\large{\dashrightarrow}[/tex] The sum of two negatives is always positive with a negative sign.

ㅤ

[tex]\large{\dashrightarrow}[/tex] The sum of a positive and a negative is always negative with the sign of whose number is greater.

The x-intercepts are (0, 0) and (4, 0), and the y-intercept is (0, 0).

By graphing linear equatiοns, yοu can explain the relatiοnship between twο variables visually. We can easily see what happens tο οne variable as the οther grοws by using a graph. The value οf the x variable rises as we mοve tο the right οn a graph.

[tex]y = (3/2)x^2 - 6x[/tex] is the given equatiοn.

We can use the fοrmula tο find the x-cοοrdinate(s) οf the vertex οf this parabοla:

x = -b/2a

where a and b are the cοefficients οf the equatiοn's x² and x terms, respectively.

In this case, a = 3/2 and b = -6, resulting in:

x = -(-6)/(2*3/2) = 4

As a result, the vertex's x-cοοrdinate is 4.

Tο find the y-cοοrdinate οf the vertex, enter this value οf x intο the fοllοwing equatiοn:

[tex]y = (3/2)(4)^2 - 6(4) = -12[/tex]

As a result, the parabοla's vertex is at the pοint (4, -12).

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Since the calculated t-value (3.14) is greater than the critical value (1.98), we reject the null hypothesis.

In statistical hypothesis testing, the null hypothesis is a statement about a population parameter that is assumed to be true until there is sufficient evidence to suggest otherwise. The null hypothesis is typically denoted by H0 and represents the status quo or default assumption.

The null hypothesis often takes the form of an equality or a statement of "no difference" or "no effect" between two or more groups, variables, or populations. For example, the null hypothesis could be that the mean score of a group of students on a test is equal to a certain value, or that there is no difference in the average height of males and females in a population.

We want to test the hypothesis that the mean amount of time spent in extracurricular activities per week is the same in the suburban and city school districts. Set up the null and alternative hypotheses is as given by:

Null hypothesis: u1 - u2 = 0

Alternative hypothesis: u1 - u2 ≠ 0

To test this hypothesis, we can use a two-sample t-test. We first calculate the test statistic:

t = ((x1 - x2) - (u1 - u2)) / √(s1²/n1 + s2²/n2)

where x1, s1, and n1 are the sample mean, standard deviation, and sample size for the suburban school district, and x2, s2, and n2 are the sample mean, standard deviation, and sample size for the city school district.

Plugging in the values, we get:

t = ((6 - 4) - 0) / √((3²/60) + (2²/40)) ≈ 3.14

This test's degrees of freedom are given by:

df = (s1²/n1 + s2²/n2)² / ( (s1²/n1)² / (n1 - 1) + (s2²/n2)² / (n2 - 1) )

Plugging in the values, we get:

df = ((3²/60) + (2²/40))² / ( (3²/60)² / 59 + (2²/40)² / 39 ) ≈ 93.24

Using a t-distribution table with 93 degrees of freedom and a level of significance of 0.05, we find the critical values to be approximately -1.98 and 1.98.

Since the calculated t-value (3.14) is greater than the critical value (1.98), we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean amount of time spent in extracurricular activities per week is different between the suburban and city school districts.

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Using proportions, the real height of the telephone mast is estimated to be 9 meters.

A proportion is a fraction of a total amount, and equations are constructed using these fractions and estimates to find the desired measures in the problem using basic arithmetic operations like multiplication and division. Because the telephone box and the mast are similar figures in this problem, their side lengths are proportional.

The following proportional relationship is established as a result:

x / 1.5 cm = 10.8 cm / 1.8 cm.

The relationship's left side can be simplified as follows:

6 = x / 1.5 cm.

The estimate is then calculated using cross multiplication, as shown below:

6 x 1.5 cm = 9.5 cm².

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Given that a 0.625 kg basketball is dropped from a height of 3 meters straight down. The coefficient of restitution between the ball and the ground is e = 0.87. At 0.15 seconds after the basketball is dropped, a 58.5-gram tennis ball is dropped from the same 3-meter height straight onto the basketball. If the coefficient of restitution between the tennis ball and the basketball is e = 0.9,

    determine how high the tennis ball bounces above the ground after the collision. Neglect the size of the balls.

The balls meet at a height of 0.5073 m above the ground. Hence the height that the tennis ball bounces above the ground is to be calculated.

Given that the coefficient of restitution between the ball and the ground is e = 0.87The coefficient of restitution between the tennis ball and the basketball is e = 0.9

The coefficient of restitution(e) is defined as the ratio of the relative velocity of separation and relative velocity of approach between two objects.

When a ball falls from height, it gains potential energy.

Potential energy (PE) = mghWhere, m = mass of the object, g = acceleration due to gravity, h = height

PE of Basketball = mgh= 0.625 kg * 9.81 m/s² * 3 m= 18.4 Joules

Initial kinetic energy (KE) = PE of Basketball

KE of Basketball = 18.4 J

Let the velocity of the basketball before collision be u1 and the velocity of the tennis ball before collision be u2. After the collision, let the velocity of the basketball be v1 and the velocity of the tennis ball be v2.

Using the coefficient of restitution (e) we can find the velocity of the balls after collision

   v1 - v2 = -e(u1 - u2)

Initial momentum (P) = Final momentum (P)

before the collision  P = m1u1 + m2u2

after collision  P = m1v1 + m2v2P = (0.625 kg * u1) + (0.0585 kg * u2)P

                           = (0.625 kg * v1) + (0.0585 kg * v2)

        Using the above two equations and the given coefficients of restitution, we can find the velocity of the balls after collisionv1 = (m1u1 + m2u2 + e * m2 * (u2 - u1)) / (m1 + m2)v2 = (m1u1 + m2u2 + e * m1 * (u1 - u2)) / (m1 + m2)

     Here, m1 = mass of basketball

                     = 0.625 kg,

                m2 = mass of tennis ball

                      = 0.0585 kg,

  u1 = 0, u2 = 0P = m1v1 + m2v2 => v1 + v2 = P / (m1 + m2)

Also given that the time taken by the balls to meet is 0.15 seconds

Let h be the height to which the tennis ball bounces after the collision.

When the tennis ball bounces to height h, it gains potential energy.

KE + PE = Total energy

             = Constant Using the principle of conservation of energy we can find the height to which the tennis ball bounces after collision(1/2) * m2 * v2² + (1/2) * m2 * g * h

              = (1/2) * m2 * u2² + (1/2) * m2 * g * 0(1/2) * m2 * v2²

              = (1/2) * m2 * u2² - (1/2) * m2 * g * h(1/2) * v2²

              = (1/2) * u2² - g * h

Substituting the values of u1, u2, m1, m2, e, t and g, we get:

v1 = (0 + 0.0585 kg * 9.81 m/s² + 0.9 * 0.0585 kg * (0 - 0)) / (0.625 kg + 0.0585 kg)v1

   = 1.96 m/sv2

    = (0.625 kg * 0 + 0 + 0.9 * 0.625 kg * (0 - 0)) / (0.625 kg + 0.0585 kg)v2

    = 5.5 m/s

Here, u1 = u2 = 0.

Using the above equation, we can find the height to which the tennis ball bounces after collision (1/2) * (0.0585 kg) * (5.5 m/s)² = (1/2) * (0.0585 kg) * (0 m/s)² - (0.0585 kg) * 9.81 m/s² * h12.83 J

                = -0.286 J - 0.572 h0.572 h

                = -12.83 J / 2h

                = -12.83 J / (2 * 0.572)

                = 11.2 m

Hence the height to which the tennis ball bounces above the ground after the collision is 11.2 m.

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