Determine the number of elements of a set and represent the number of elements using a symbol

As we have proved that if one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

To prove this, we can use the concept of alternate angles. Alternate angles are angles that are on opposite sides of a transversal line and are congruent (equal). We can draw a line segment AC between points A and C, which divides the quadrilateral into two triangles, namely triangle ABC and triangle ACD.

Since AB and CD are parallel, line segment AC acts as a transversal line, and the angle formed by AB and AC is equal to the angle formed by CD and AC. These angles are alternate angles, and therefore they are equal. Similarly, the angle formed by BC and AC is equal to the angle formed by AD and AC, since they are also alternate angles.

Now, we know that triangle ABC and triangle ACD share a common side AC, and two pairs of angles are equal in both triangles. Therefore, by the angle-angle-side (AAS) theorem, the two triangles are congruent to each other. Congruent triangles have equal corresponding sides, and since AB and CD are equal, we can conclude that BC and AD are also equal in length.

Therefore, we have shown that if one pair of opposite sides of a quadrilateral are equal and parallel, then the other pair of opposite sides are also equal and parallel.

This means that the quadrilateral is a parallelogram. In this case, we can conclude that quadrilateral ABCD is a parallelogram.

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Answer:

  see attached

Step-by-step explanation:

You want to graph the function f(x) = 3/5x -5.

For graphing purposes, it is convenient to choose values of x that result in integer values of y. In this case, the multiplier of x (the slope) has a denominator of 5, so it is convenient to choose x-values that are multiples of 5.

For x = 0, y = 3/5·0 -5 = -5

For x = 5, y = 3/5·5 -5 = 3 -5 = -2

Suitable points for your plot are (0, -5) and (5, -2). These are shown in the attachment.

The CH3-CIOI-CNI molecule contains three carbon atoms with different electron geometries and bond angles. The CH3 and CIOI carbon atoms have tetrahedral geometry with a bond angle of approximately 109.5 degrees, while the CNI carbon atom has a trigonal planar geometry with a bond angle of approximately 120 degrees.

Using this Lewis structure, we can determine the electron geometry and bond angle for each carbon atom in the molecule as follows.

The carbon atom in the CH3 group has four electron domains (three bonding pairs and one non-bonding pair). The electron geometry around this carbon atom is tetrahedral, and the bond angle is approximately 109.5 degrees.

The carbon atom in the CIOI group has four electron domains (two bonding pairs and two non-bonding pairs). The electron geometry around this carbon atom is also tetrahedral, and the bond angle is approximately 109.5 degrees.

The carbon atom in the CNI group has three electron domains (one bonding pair and two non-bonding pairs). The electron geometry around this carbon atom is trigonal planar, and the bond angle is approximately 120 degrees.

Therefore, the electron geometry and bond angle for each carbon atom in the structure CH3-CIOI-CNI are:

CH3 carbon atom tetrahedral geometry, bond angle of approximately 109.5 degrees

CIOI carbon atom tetrahedral geometry, bond angle of approximately 109.5 degrees

CNI carbon atom trigonal planar geometry, bond angle of approximately 120 degrees

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_____The given question is incomplete, the complete question is given below:

Determine the electron geometry and bond angle for each carbon atom in the structure CH3-CIOI-CNI

The probability that a randomly chosen card that is less than 7 is the 5 of diamonds is 5%.

There are four suits in a standard deck of 52 cards: diamonds, clubs, hearts, and spades. Each suit has 13 cards, with ranks ranging from 2 (low) to 10, jack, queen, king, and ace (high).

If a randomly chosen card from the deck is less than 7, there are only two possibilities: it is either a 2, 3, 4, 5, or 6 of any suit, or it is the 5 of diamonds.

There are 20 cards that are less than 7 in the deck (4 cards of each of the 5 ranks). Out of these 20 cards, only one is the 5 of diamonds.

Therefore, the probability that a randomly chosen card that is less than 7 is the 5 of diamonds is:

1/20 = 0.05 = 5%

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The equation of the plane passing through the point (-1,0,1) and containing the lines x = 5t, y=1+t, z= -t is y + z = 1 .

We substitute , t=0 in the given line equations ,

we get; x=0 , y = 1 and z = 0 ;

So, the plane contain the line , Thus plane will also pass through (0,1,0);

Now, we have that plane passes through (-1,0,1) and (0,1,0), direction ratios of line joining these 2 points are ;

⇒ DR₁ = (-1-0 , 0-1 , 1-0) = (-1,-1,1);

So , the line can be written as x/5 = (y-1)/1 = z/-1 = t;

So, the direction ratio of this line will be :

⇒ DR₂ = (5 , 1 , -1);

The Direction Ratio of normal to the plane is = DR₁ × DR₂;

= (-1,-1,1) × (5,1,-1);

= [tex]\left|\begin{array}{ccc}i&j&k\\-1&-1&1\\5&1&-1\end{array}\right|[/tex]

On solving ,

We get;
= 4j + 4k = (0,4,4) ;

We know that for a normal with direction ratios (a,b,c), equation of plane is written as ax + by + cz = d;

We got direction ratio for plane normal = (0,4,4);

So, equation of plane is 0x+ 4y + 4z = d;

the plane passes through the point (-1,0,1) ;

⇒ 4(0) + 4(1) = d ⇒ d = 4;

we get the equation of plane is 4y + 4z = 4;

⇒ y + z = 1.

Therefore, the equation of the required plane is  y + z = 1.

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The inverse of the coefficient matrix is A^-1 = [-2 2]. The solution to the system of equations is x = -1 and y = 1/5.

To solve the system of equations:

2x + 2y = 1

2x - 3y = 0

We can write this system in matrix form as:

[2 2] [x] [1]

[2 -3] [y] = [0]

The coefficient matrix is:

[2 2]

[2 -3]

To find the inverse of the coefficient matrix, we can use the following formula:

A^-1 = (1/|A|) adj(A)

where |A| is the determinant of A and adj(A) is the adjugate of A.

The determinant of the coefficient matrix is:

|A| = (2)(-3) - (2)(2) = -10

The adjugate of the coefficient matrix is:

adj(A) = [-3 2]

[-2 2]

Therefore, the inverse of the coefficient matrix is:

A^-1 = (1/-10) [-3 2]

[-2 2]

Multiplying both sides of the matrix equation by A^-1, we get:

[x] 1 [-3 2] [1]

[y] = -10 [-2 2] [0]

Simplifying the right-hand side, we get:

[x] [-1]

[y] = [1/5]

Therefore, the solution to the system of equations is:

x = -1

y = 1/5

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_____The given question is incomplete, the complete question is given below:

solve the systems of equations by finding the inverse of the coefficient matrix. a. 2x+2y=1 2x-3y-0

Answer:

19/20 , 1 1/4 , 1 4/5

Step-by-step explanation:

1.8 = 1 4/5 (fraction)

1.8 converts to 18/10. This can be simplified twice, firstly by making it 9/5 since both 18 and 10 are divisible by two, but can be simplified further to 1 4/5

1.25 = 1 1/4 (fraction)

1.25 converts to 125/100. This can be simplified to 5/4 or 1 1/4

0.95 = 19/20 (fraction)

0.95 converts to 95/100. This can be simplified to 19/20

Ascending Order (smallest to largest)

smallest - 19/20

middle - 1 1/4

largest - 1 4/5

I believe this is the right answer, but haven't done fractions in a while so may want to double check to make sure

So Abbie was asked to randomly select at least 368 of her high school unitary method students to estimate the percentage of students planning to go to college. With a 90% confidence level and a 5% error rate. 

The unit method is an approach to problem solving that first determines the value of a single unit and then multiplies that value to determine the required value. Simply put, the unit method is used to extract a single unit value from a given multiple. For example, 40 pens cost 400 rupees or pen price. This process can be standardized. single country. Something that has an identity element. (Mathematics, Algebra) (Linear Algebra, Mathematical Analysis, Matrix or Operator Mathematics) Adjoints and reciprocals are equivalent. 

To determine the required sample size, the following formula should be used:

[tex]n = (Z^2 * p * (1-p)) / E^2[/tex]

where:

N:sample size required

Z:The Z-score corresponds to the desired confidence level and is 1.645 at the 90% confidence level.

Pa:Estimated Percentage of Students Planning to Go to College

1-p:Percentage of students not planning to go to college

E:Desired error margin of 0.05

Since we don't know the actual percentage of students who want to go on to college, we must use estimates based on past studies and surveys. Let's assume the estimated proportion is 0.6 (her 60% of students).

After plugging in the values ​​it looks like this:

[tex]n = (1.645^2 * 0.6 * 0.4) / 0.05^2\\n = 368.03[/tex]

So Abbie was asked to randomly select at least 368 of her high school students to estimate the percentage of students planning to go to college. With a 90% confidence level and a 5% error rate. 

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The distance between the points P and Q is 10 units using the Pythagorean theorem.

The right-angled triangle's three sides are related in accordance with the Pythagoras theorem, commonly known as the Pythagorean theorem. The Pythagorean theorem states that the hypotenuse square of a triangle is equal to the sum of the squares of the other two sides. According to the Pythagoras theorem, if a triangle has a right angle, the hypotenuse's square is equal to the sum of the squares of the other two sides.

The coordinates of the point P and Q are (3, 2) and (9, 10).

Using the Pythagoras theorem:

c² = (x2 - x1)² + (y2 - y1)²

Substitute the values:

c² = (9 - 3)² + (10 - 2)²

c² = 36 + 64

c² = 100

c = 10 units.

Hence, the distance between the points P and Q is 10 units using the Pythagorean theorem.

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Answer:

D

Step-by-step explanation:

If their reflections are congruent to each other then looking at the diagram we can see a reflection just like a mirror where its flipped on the other side of the dotted line. When flipping it and aligning one triangle to the other we find that BC is congruent to DF

Jadi, lebar jalan adalah 14 meter.

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The solution to the polynomial equation 8x⁴ - 32x² = 0 by factoring are x = 0 or x = -2 or x = 2

To solve the equation 8x⁴ - 32x² = 0 by factoring, we can factor out the greatest common factor of the terms:

8x²(x² - 4) = 0

Now we can use the difference of squares identity to factor the quadratic expression:

8x²(x + 2)(x - 2) = 0

Using the zero product property, we know that the product of two factors is zero if and only if at least one of the factors is zero.

Therefore, we can set each factor equal to zero and solve for x:

8x² = 0 or x + 2 = 0 or x - 2 = 0

Solving for x, we get:

x = 0 or x = -2 or x = 2

So the solution set for the equation is {0, -2, 2}.

To check the solution, we can substitute each value into the original equation and verify that it equals zero.

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The expected amount of time until the ant returns to its starting vertex is 2.5 minutes.

Vertex in math is the point at which two lines, curves, or edges meet. In the case of a triangle, for example, the vertex is the point where all three lines meet. In higher dimensional shapes, such as a parallelogram, it is the point at which all the lines meet. Vertex can also be used to refer to the highest point of a graph, or the point of maximum or minimum value.

The expected amount of time until an ant returns to its starting vertex after traversing the edges of a tetrahedron can be calculated by applying the principle of expected value. The expected value of a random variable is the sum of the probability of each outcome multiplied by its associated value.
In this case, the ant has four possible outcomes, with each outcome being the length of time it takes to traverse the edge to an adjacent vertex. Since the ant has an equal probability of going to each vertex, each outcome has a probability of 0.25. Thus, the expected value can be calculated as:
Expected Value = (1 minute x 0.25) + (2 minutes x 0.25) + (3 minutes x 0.25) + (4 minutes x 0.25)
Expected Value = 2.5 minutes
Therefore, the expected amount of time until the ant returns to its starting vertex is 2.5 minutes.

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Complete questions as follows-
an ant starts at one vertex of a tetrahedron. each minute it walks along a random edge to an adjacent vertex. what is the expected amount of time until it returns to its starting vertex?

We cannot determine the actual value of the average rate of change without knowing the function f(x) or having a graph of the function.

The visual representation of mathematical functions or data points on a Cartesian coordinate system is an x-y axis graphic. The vertical or dependent variable is represented by the y-axis, while the horizontal or independent variable is represented by the x-axis. The difference between the change in output values and the change in input values is known as the average rate of change of a function over a period.

Let's assume that the function is denoted by f(x). Then, the average rate of change on the interval (a, b) can be calculated as

average rate of change = (f(b) - f(a)) / (b - a)

Using this formula, we can calculate the average rate of change on the given intervals as follows:

For the interval (-3, -2):

average rate of change = [tex]\frac{[f(-2) - f(-3)]}{[-2 - (-3)]}[/tex]

For the interval (1, 3):

average rate of change = [tex]\frac{(f(3) - f(1))}{(3 - 1)}[/tex]

For the interval (-1, 1):

average rate of change = [tex]\frac{(f(1) - f(-1))}{ (1 - (-1))}[/tex]

Note that we cannot determine the actual value of the average rate of change without knowing the function f(x) or having a graph of the function. If you provide the function or the graph, I can help you find the actual values of the average rate of change on these intervals.

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Yes, there is an analogous ratio between 2:1 and 20:10.

We just cancel by a common factor. So 4:2=2:1 . The simplest representation of the ratio 4 to 2 is the ratio 2 to 1. Also, since each pair of numbers has the same relationship to one another, the ratios are equivalent.

By dividing the terms of each ratio by their greatest common factor, we may simplify both ratios to explain why.

As the greatest common factor for the ratio 2:1 is 1, additional simplification is not necessary.

The greatest common factor for the ratio 20:10 is 10. When we multiply both terms by 10, we get:

20 ÷ 10 : 10 ÷ 10

= 2 : 1

As a result, both ratios have the same reduced form, 2:1, making them equal.

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The probability that the sample mean of ln(LC50) will differ from the true population mean by no more than 0.5 is 0.0019.

The concept of independent random variables is very similar to independent events. Recall that two events A and B are independent if we have P(A,B) = P(A)P(B) and remember that comma means sum, i.e.

         P(A, B)=P( A and B)=P (A∩B).

Similarly, we have the following definition for independent discrete random variables.

Assuming 'X' and 'Y' are independent random samples

X : mean value =  , variance = 0.4 /10 = 0.04

Y : mean value =  , variance = 0.8/10 = 0.08

since the values are independent

V [ X - Y ] = V [ X ] + V [Y ] = 0.04 + 0.08 = 0.12

Now,

The probability that the sample mean for species A exceeds the sample mean for species B by at least 1 unit

  [tex]P(X-Y\geq 1) = P{ \frac{(X-Y)-(U_1-U_2)}{\sqrt{V(X-Y} } \geq \frac{1-0}{\sqrt{0.12} }[/tex]

⇒ P{Z ≥ 2.8858}

⇒ 1 -P{Z≤ 2.8858} = 1 - 0.9981

⇒ P{X-Y≥ 1} = 0.0019.

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If a₁ = 5 and an 5an-1 then The value οf a₄ is 625.

An arithmetic sequence is a sequence οf numbers in which each term after the first is fοund by adding a fixed cοnstant number, called the cοmmοn difference, tο the preceding term. Fοr example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with a cοmmοn difference οf 3, since each term after the first is fοund by adding 3 tο the preceding term.

The nth term οf an arithmetic sequence can be fοund using the fοrmula:

an = a1 + (n-1)d

where an is the nth term, a1 is the first term, and d is the cοmmοn difference. The sum οf the first n terms οf an arithmetic sequence can be fοund using the fοrmula:

Sn = n/2 (a1 + an)

We are given that a₁ = 5, and that the nth term is 5 times the (n-1)th term. We can use this infοrmatiοn tο find the value οf a₄ as fοllοws:

a₂ = 5a₁ = 5(5) = 25

a₃ = 5a₂ = 5(25) = 125

a₄ = 5a₃ = 5(125) = 625

Therefore, the value of a₄ is 625.

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In response to the stated question, we may state that The other numbers range - 371 jelly beans, 517 jelly beans, and 522 jelly beans

The variable's range is calculated by taking its greatest observed value and subtracting its minimum observed value (minimum). Possible range or variational bounds: a variety of steel costs; a variety of styles; The scope or size of a method or action: perception. The maximum or anticipated range of a weapon's projectile. A list or set's range is the number between the lowest and maximum. Align all of the numbers before determining the region. Next, subtract (reduce) the lowest number from the greatest number. The solution includes the range of the list. The solution specifies the range of the list.

The interquartile range (IQR) indicates how far the consumers' predictions differed. In this instance, the IQR is 381 jelly beans. The IQR is a measure of the spread of predictions around the median since it shows the range of the middle 50% of the guesses.

The other numbers - 371 jelly beans, 517 jelly beans, and 522 jelly beans - are not measurements of how much the consumers' predictions changed, but rather individual data points (the median guess, the actual number of jelly beans, and the winning guess, respectively).

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Step-by-step explanation:

'Yoself'  drank 1/4 of 16 oz so he got 1/4 of 250%  

   1/4 * 250% = 62.5 %

Answer: y = 12 - 1.25x

Step-by-step explanation:

Let y be the number of yards of fabric remaining after Tara sews x tote bags.

Initially, Tara has 12 yards of fabric. For every tote bag she sews, she uses 1.25 yards of fabric. Therefore, the total amount of fabric used after sewing x tote bags is 1.25x yards.

The number of yards of fabric remaining is the difference between the initial amount of fabric and the total amount of fabric used:

y = 12 - 1.25x

This equation shows how the number of yards of fabric remaining depends on the number of tote bags Tara sews, x. As she sews more tote bags, the amount of fabric remaining decreases.

Answer:

5.75 yds left

1.25 yards each bag multiplied by 5 bags is 6.25 yards utilized

Tara's yardage was 12-6.25 = 5.75.

She has 5.75 yards to go.

Step-by-step explanation:

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To calculate the molarity of Pb(NO3)2(aq), you need to know the molar mass of the compound. The molar mass of Pb(NO3)2 is 331.21 g/mol.

Using the equation, concentration = moles/liters, we can calculate the molarity of the Pb(NO3)2(aq) solution.

First, we need to calculate the moles of Pb(NO3)2. We can do this by converting the mass of the precipitate (19.40 g) to moles. Moles = mass (g) / molar mass (g/mol).

Therefore, moles of PbCl2 = 19.40 g / 331.21 g/mol = 0.05833 moles.

Next, we can calculate the molarity of Pb(NO3)2. Molarity = moles/liters.

Therefore, the molarity of Pb(NO3)2 = 0.05833 moles/ 0.2 liters = 0.29165 M.

Triangle pair (A, B) can be mapped to each other using a translation and rotation about point A, which is calculated using the formula T(x, y) = (x + a, y + b) followed by a rotation of θ about the point A(a, b).

The triangle pair (A, B) can be mapped to each other using a translation and rotation about point A. The formula for the transformation is T(x, y) = (x + a, y + b) followed by a rotation of θ about the point A(a, b). The calculation for this transformation would be x' = xcosθ - ysinθ + a and y' = xsinθ + ycosθ + b. For example, if the coordinates of point A are (4, 5), point B is (6, 2) and the rotation angle is 30°, the coordinates of point B' would be (7.9, 5.4) after the transformation.

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Answer:

●B. The numbers -1,0,1 are zeros of multiplicity 1.

Step-by-step explanation:

So first, understand that when you are asked for roots, zeros, solutions, or x-intercepts...all of these, they are essentially asking for the same thing. Roots ARE solutions ARE zeros ARE x-intercepts. Maybe its oversimplifying a little bit; there are tiny nuanced differences to a mathematician but if you are just learning this, go ahead and over simplify. They are all the same. So you set it equal to 0 and solve.

Yes, literally, change y to a 0 and solve. See image.

You can factor out a 2x and then you have a "difference of squares" so factor that too.

see image.

"multiplicity" is a cool word. It just means how many times a number is the answer. It sort of doesn't even apply here. 0, -1, and 1 are the answer just one time each...so multiplicity 1. Also, on the graph, the curve will cross the x-axis like a line, so there's that. (See multiplicity 2 is cooler, because the curve will "bounce" at the x-intercept instead, but that's not happening here)

Anyway, set the problem equal to 0 and solve. Ta-da! You're done! Hope this helps! See image.

a. In this exercise, we are given that Mystery Pizza has an average οf 4.2 teIephοne orders per hour between 11 P.M. and midnight on Thursday night. Nοw using these given vaIues, we wiII caIcuIate the probabiIity that at Ieast 30 minutes wiII eIapse before having the next teIephone οrder.

Accοrding to probabiIity theory and statistics, the exponentiaI distributiοn is the probabiIity distribution of time between occurrences in the Poissοn distribution. It is a distribution of probabiIities that frequentIy correIates to the amount of time before a specific event takes pIace. It is a prοcess in which events take pIace continuousIy, independentIy, and at an average pace that remains constant throughout the process.

In caIcuIating the area under the curve of its graph (CDF), we wiII have to use the fοIIowing formuIa for the mean and the standard deviation,

Mean and Standard Deviatiοn:

[tex]$\begin{array}{r}{\mu={\frac{1}{\lambda}};}\\ {\sigma={\frac{1}{\lambda}}\,.}\end{array}$[/tex]

where,

First, let us calculate the mean and standard deviation using the given Pοisson mean [tex]$\lambda=4.2.$[/tex] Using the fοrmula, we have,

[tex]$\begin{aligned}\rm{\mu=\sigma={\frac{1}{\lambda}}}\\ {={\frac{1}{4.2}\\{=0.2381}\end{aligned}$[/tex]

Sο we have the mean and the standard deviation of 0.2381 hours.

Nοw, we wiII caIcuIate the probabiIity that at Ieast 30 minutes or 0.50 hοurs wiII eIapse before the next teIephone order. Keep in mind that we are caIcuIating the probabiIity for "more than" the x so we wiII use the right-taiIed formuIa for this which is given by,

Right-tailed area(Mοre than x) :

[tex]$P(X\gt x)=e^{-\lambda x};$[/tex]

where,

Using the fοrmula, we have:

[tex]$\begin{array}{r l}{P(X\gt 0.50)=e^{-\lambda x}}\\ {=e^{-42(0.50)}}\\ {=0.1225\,.}\end{array}$[/tex]

Therefοre, we can concIude that there is a 12.25% chance that at Ieast 30 minutes or 0.5 hours wiII eIapse before another teIephone order.

b. Next, we wiII caIcuIate the probabiIity that Iess than 15 minutes wiII eIapse befοre the next teIephone order. Remember that we are caIcuIating the probabiIity of Iess than x. This means that we wiII be using the fοrmuIa for the Ieft-taiIed area which is given by,

Left-tailed area(Less than οr equal to x):

[tex]$P(X\leq x)=1-e^{-\lambda x}$[/tex]

where,

Using the fοrmula, we have:

[tex]$\begin{array}{r l}{P(X\leq0.25)=1-e^{-\lambda x}}\\ {=1-e^{-42(0.25)}}\\ {=1-0.3499}\\ {=0.6501\,.}\end{array}$[/tex]

Therefοre, there is a 65.01% that Iess than 15 minutes wiII eIapse before the next teIephοne caII.

c. In this part, we wiII caIcuIate the probabiity that between 15− 30 minutes wiII eIapse befοre the next teIephone order. MathematicaIIy, we have,

[tex]$P(0.25\lt X\lt 0.5)=P(X\lt 0.50)-P(X\lt 0.25)$[/tex]

Frοm part a, we have the value for P(X>0.50) which is 0.1225. Now using the cοmplement rule, we can get P(X<50}

[tex]$\begin{array}{c}{{P(X\lt 50)=1-P(X\gt 50)}}\\ {{=1-0.1225=0.8775\,.}}\end{array}$[/tex]

We have nοw the value of P(X<0.50) which is 0.8775.

We can nοw get the P(0.25<X<0.50)  by subtracting P(X<0.50) by P(X<0.25) frοm part b.. So we have,

[tex]$\begin{array}{r}{P(0.25\lt X\lt 0.50)=P(X\lt 0.50)-P(X\lt 0.25)}\\ {=0.8775-0.6501}\\ {=0.2274\,.}\end{array}$[/tex]

Sο we have P(0.25<X<0.50)=0.2274. Therefore, we can concIude that there is a 22.74% chance that between 15 and 30 minutes wiII eIapse before having a teIephone order.

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[tex]T(11) = -1[/tex] represents the initial negative term inside the series.

A phrase that acknowledges a characteristic or trait in anything is said to be positive. Negative terms are those that downplay some aspect or characteristic of something. The positive or negativizes of a phrase is termed its quality.

Positive simply denotes excellent or the polar opposite of negative. For instance, you're less likely to receive favorable feedback on your scorecard if you've a good mindset toward your assignments. It might be difficult to keep track of all the different definitions of the word positive.

T(n) is negative when [tex]32 - 3n < 0[/tex], which can be rearranged as:

[tex]3n > 32[/tex]

[tex]n > 32/3[/tex]

Since [tex]n[/tex] is an integer, the smallest value of n that satisfies this inequality is [tex]n = 11[/tex].

Therefore, the [tex]11th[/tex] term of the sequence is:

[tex]T(11) = 32 - 3(11) = -1[/tex]

So the first negative term in the sequence is [tex]T(11) = -1.[/tex]

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Answer:

Step-by-step explanation:

To prove that z^2 + 1/2z is a real number, we need to show that the imaginary part of z^2 + 1/2z is equal to zero.

We know that z = (a+bi)/(a-bi)

Multiplying the numerator and denominator by the complex conjugate of the denominator, we get

z = (a+bi)(a+bi)/(a-bi)(a+bi)

z = (a^2 + 2abi - b^2)/(a^2 + b^2)

Expanding z^2, we get:

z^2 = [(a^2 + 2abi - b^2)/(a^2 + b^2)]^2

z^2 = (a^4 + 2a^2b^2 + b^4 - 2a^2b^2 + 4a^2bi - 4b^2i)/(a^4 + 2a^2b^2 + b^4)

Simplifying, we get:

z^2 = (a^4 - b^4 + 2a^2bi)/(a^4 + 2a^2b^2 + b^4)

Now, let's compute z^2 + 1/2z:

z^2 + 1/2z = (a^4 - b^4 + 2a^2bi)/(a^4 + 2a^2b^2 + b^4) + 1/2[(a+bi)/(a-bi)]

To simplify this expression, we need to find a common denominator:

z^2 + 1/2z = (2a^5 - 2a^3b^2 + 3a^4b - 3ab^4 - 2b^5 + 3a^3bi + 3ab^3i)/(2(a^4 + 2a^2b^2 + b^4))

We can see that the imaginary part of z^2 + 1/2z is (3a^3b - 3ab^3)/(2(a^4 + 2a^2b^2 + b^4))

However, we know that a and b are real numbers, so the imaginary part of z^2 + 1/2z is zero.

Therefore, z^2 + 1/2z is a real number.