2. Given: MNOP is a rectangle.
Show: The diagonals bisect each other, and the diagonals are congruent.
M
-5
N

P(A and B) is not equal to P(A) * P(B), we can conclude that events A and B are dependent.

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.

We can determine if events A and B are independent or dependent by comparing the probability of both events occurring together with the product of their individual probabilities.

P(A and B) = 1/7

P(A) = 2/3

P(B) = 3/7

If A and B are independent events, then P(A and B) = P(A) * P(B). However, in this case, P(A and B) is not equal to P(A) * P(B):

P(A) * P(B) = (2/3) * (3/7) = 2/7

Since P(A and B) is not equal to P(A) * P(B), we can conclude that events A and B are dependent.

Events A and B are dependent since the probability of their intersection is not equal to the product of their individual probabilities. Specifically, P(A and B) = 1/7 while P(A)*P(B) = 2/7.

Therefore, P(A and B) is not equal to P(A) * P(B), we can conclude that events A and B are dependent.

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Assume that A and B are independent events. it means that the occurrence of event A does not affect the probability of event B and vice versa.

b. Assume that A and B are independent events. My response is Yes, P(A and B) can be defined as P(B) • P(A|B).

In the event that A and B are independent events, at that point the probability of A happening is one that is influenced by the event of B and vice versa. Hence, P(B) is comparable to P(B|A), showing that the probability of occasion B happening remains unaltered in any case of event A's event. Within the same vein, the likelihood of event A happening remains consistent whether event B has stay put or not, indicated as P(A) = P(A|B).

Therefore in response to question 2, The reason why independent events are not affected  by one another is that the likelihood of one event happening has no effect on the probability of the other event occurring.

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The mean, median, mode and range are 42.2, 30, 87 and 95 respectively.

The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.

Therefore mean = (23+87+19+34+37+37+5+14+100+26)/10

= 422/10 = 43.2

The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.

median = 26+34/2 = 60/2 = 30

Mode is the number with the highest frequency

Therefore, the mode is 87

Range Is the difference between the highest and the lowest

= 100-5 = 95

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The Complement of a 14° angle is 76°, while the supplement of 14° angle is 166°.

Recall that two angles that have a sum of 90 degrees are referred to as complementary angles. This implies that, if x is an angle, its complement is (90 - x).

Also, the supplement of an angle is the angle measure that is added to it to make it 180 degrees. This implies that if x is an angle, its supplement would be (180 - x).

Therefore, we have:

Complement of a 14° angle = 90 - 14 = 76°.

Supplement of 14° angle = 180 - 14 = 166°.

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Answer: 50

Step-by-step explanation
One method to estimate the number of fish in the pond is to use the mark-and-recapture method. This method assumes that the proportion of marked fish in the second sample is the same as the proportion of marked fish in the entire population. Using this assumption, we can estimate the total number of fish in the pond as follows:

Let N be the total number of fish in the pond.

Let M be the number of fish caught in the first sample (50 fish).

Let R be the number of marked fish caught in the second sample (8 fish).

Let r be the unknown total number of marked fish in the pond.

According to the mark-and-recapture method, we have the following proportion:

M/N = R/r

We can rearrange this equation to solve for N:

N = M * r / R

Substituting the values given in the problem, we have:

N = 50 * r / 8

Simplifying, we get:

N = 6.25 * r

This means that the estimated total number of fish in the pond is 6.25 times the number of marked fish in the second sample. Since we know that 8 fish were marked in the second sample, the estimate for the total number of fish in the pond is:

N = 6.25 * 8 = 50

Therefore, the estimated number of fish in the pond is 50

The equation is given by y = 2.5x. It means that the cost of each gallon of gas is $2.5

An exponential equation is an expression that shows how numbers and variables using mathematical operators.

Let y represent the cost in dollars for x gallons of gas she puts in the tank

The graph passes through point (0, 0) and (3, 7.5), hence:

y - 0 = [(7.5 - 0) / (3 - 0)](x - 0)

y = 2.5x

The equation is given by y = 2.5x

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5.) The measure of angle D of the given triangle would be = 59.5°

To calculate the missing angles, sine rule is used which is ;

Sine rule = a/sin A = b/sin B

Where a = 23

A = 56°

b = 24

B = X

That is;

23/sin56° = 24/sinx

Make sin X the subject formula;

Sin X = sin56×24/23

= 0.829037572×24/23

= 19.89690174/23

= 0.865082684

X = Sin-1 (0.865082684)

= 59.89214806

= 59.9° (approximately)

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Answer:

x=0, 5 (-5±√15)/2

Step-by-step explanation:

see images for solution

The evaluated probabilities for the given question are 0.20 for the men who asked for promotion, 0.59 for the men who received the promotion when asked, and  0.118 for men who asked for promotion and received.under the condition that from the men interviewed  20% had asked for a promotion and 59% of the men who had asked for a promotion and  received.
Then,
P(man asked for a promotion)
= 20/100
= 0.20

P(man received promotion, given he asked for one)
= 59/100
= 0.59

P(man asked for promotion and received )
= P(man asked for a promotion) x P(man received promotion, given he asked for one)
= 0.20 x 0.59
= 0.118
Probability is considered as  the percentage of an event taking place in a specific time frame, in a given place. It is said to be a great aid in the horizons of science and mathematics.


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a) the total number of  strings of seven hexadecimal digits that do not have any repeated digits is 450,450

b) The number of strings of seven hexadecimal digits have at least one repeated digit is 267,985,006

c) the probability that a randomly chosen string of seven hexadecimal digits has at least one repeated digit is 99.8%

The total number of strings of seven hexadecimal digits is 16⁷.

For a string to not have any repeated digits, the first digit can be chosen in 16 ways, the second in 15 ways (since one digit has already been used),

the third in 14 ways, and so on, giving a total of 16 * 15 * 14 * 13 * 12 * 11 * 10 = 13,608,960 strings.

So

(16 * 15 * 14 * 13 * 12 * 11 * 10)/ (2 * 2 *2* 2* 2 *2* 2)

=  450,450

b) the number of strings with no repeated strings is 450450, so the nubmer of strings with at least one rpeated digit is

16⁷ - 450450 = 267,985,006

c) the probability that a  a randomly chosen string of seven hexadecimal digits has at least one repeated digit

= 267,985,006/ 16⁷

= 0.99832194298
= 99.8%

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The equation of the circle is (x + 1)² + (y - 1)² = 16.

option D

The radius of the circle is 4, and the center of the circle is (-1, 1).

The equation of a circle with center (a,b) and radius r is given by:

(x - a)² + (y - b)^2 = r²

In this case, the center of the circle is (-1, 1) and the radius is 4. Plugging these values into the equation, we get:

(x - (-1))² + (y - 1)² = 4²

Simplifying the equation, we get:

(x + 1)² + (y - 1)² = 16

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Given that biology is the subject, the Probability of being on time is 32.8%.

In this case:

P(on-time arrival in class | biology) = P(on-time arrival in class and biology) / P(biology) is the result of applying the conditional probability formula.

According to the table, 82.4% of students will be on time for biology class. Since there are four topics listed, each with an equal weight, the likelihood of biology is 25%. Therefore, we have:

P(on-time arrival in class | biology) = 82.4% / 25% = 0.328 = 32.8%

Therefore it can be concluded that the probability of on-time arrival in class given that the subject is biology is 32.8%.

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Answer: The compound interest is $409.49, and the total amount after one year is $3,665.49.

Step-by-step explanation:

We can use the formula for compound interest to solve this problem:

A = P(1 + r/n)^(nt)

where:

P = the principal (in this case, $3,256)

r = the annual interest rate (100%, or 1 as a decimal)

n = the number of times the interest is compounded per year (4, since it's compounded quarterly)

t = the number of years (1)

Plugging in the values we know, we get:

A = $3,256(1 + 1/4)^(4*1)

A = $3,256(1.25)^4

A = $3,665.49 (rounded to the nearest cent)

To find the compound interest, we can subtract the principal from the total amount:

Compound interest = $3,665.49 - $3,256 = $409.49

Therefore, the compound interest is $409.49, and the total amount after one year is $3,665.49.

The volume of the rectangular prism for the given dimensions length , width , and the height is equal to 168 cubic inches.

Dimensions of the rectangular prism is equal to,

length of the rectangular prism = 12 inches

width of the rectangular prism = 8 inches

height of the rectangular prism = 1.75 inches

The formula for the volume of a rectangular prism is equal to,

Volume of the rectangular prism = length x width x height

Plugging in the given values of the dimensions , we have,

⇒ Volume of the rectangular prism  = 12 inches x 8 inches x 1.75 inches

Simplifying this expression to get the volume of rectangular prism, we get,

⇒ Volume of the rectangular prism = 168 cubic inches

Therefore, the volume of the rectangular prism is 168 cubic inches.

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The value of quadratic equation with the help of  hit and trial method is 39

A quadratic equation is a polynomial equation of degree 2, which means that the highest power of the variable (usually x) in the equation is 2. It is in the form:

ax^2 + bx + c = 0

where a, b, and c are constants and a is not equal to zero.

the value of  given quadratic equation is

given

y=4x²+8x+7

using hit and trial method:

The value of the quadratic equation y = 4x² + 8x + 7 depends on the value of x.

To find the value of y for a specific value of x, simply substitute that value of x into the equation and solve for y.

For example, if x = 2, then:

y = 4(2)² + 8(2) + 7

y = 16 + 16 + 7

y = 39

So the value of the quadratic equation y = 4x^2 + 8x + 7 when x = 2 is 39.

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The statement that is true of the pizza is A. About half of the pizza is left.

The answer to the subtraction of the fractions is 7/24.

Miron was 1/3 of an hour faster today than yesterday.

Initially, 2/3 of the pizza was left over. Marcy ate 2/9 of the pizza.

2/3 - 2/9

(2/3) x (3/3) = 6/9

6/9 - 2/9 = 4/9

So, 4/9 of the pizza is left. Since 4/9 is less than half but more than very little.

2/3 - 3/8

Find a common denominator, which in this case is 24:

(2/3) x (8/8) = 16/24

(3/8) x (3/3) = 9/24

Now subtract:

16/24 - 9/24 = 7/24

The answer in simplest form is 7/24

Miron finished his tasks in 3/4 of an hour yesterday and 5/12 of an hour today.

3/4 - 5/12

Find a common denominator, which in this case is 12:

(3/4) x (3/3) = 9/12

9/12 - 5/12 = 4/12

4/12 = 1/3

Miron was 1/3 of an hour faster today than yesterday.

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1) The value of an equation for y in terms of x is, y = 1/5x

And, y = 20 at x = 100

2) y = 184.8 , at x = 14

The value of an equation for y in terms of x is, y = 13.2x

Given that;

1) y is directly proportional to x.

2) y ∝ x and y=132 when x=10

Now, For 1;

1) Since, y is directly proportional to x.

Hence, We get;

y = kx

Plug y = 5 and x = 25

5 = 25k

k = 1/5

Hence, The value of an equation for y in terms of x is,

y = 1/5x

And, Plug x = 100,

y = 1/5 x 100

y = 20

2) Since,  y ∝ x and y=132 when x=10

y = kx

132 = 10k

k = 13.2

Hence, At x = 14;

y = 13.2 x 14

y = 184.8

Thus, The correct equation is,

y = 13.2x

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The ice cream factory could make 720 quarts of ice cream in 36 hours.

We can use a proportion to solve this problem:

quarts / hours = constant rate of production

Let x be the number of quarts that could be made in 36 hours. Then we have:

200 quarts / 10 hours = x quarts / 36 hours

Cross-multiplying, we get:

10x = 200 * 36

Simplifying, we get:

10x = 7200

Dividing both sides by 10, we get:

x = 720

Therefore, the ice cream factory could make 720 quarts of ice cream in 36 hours.

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The angle measure of the arc bounding the sector with area 8π square inches is 45 degrees.

The area of the sector is given by the formula:

[tex]A = (1/2) r^2[/tex]θ

where A is the sector's area,  is the sector's central angle in radians, and r is the circle's radius.

The radius of the circle is 8 inches, and the area of the sector is 8 square inches. When these values are added to the formula, we get:

8π = [tex](1/2) (8^2)[/tex] θ

Simplifying, we get:

8π = 32θ

Dividing both sides by 32, we get:

θ = π/4 radians

To convert this to degrees, we multiply by 180/π:

θ = (π/4) × (180/π) = 45 degrees

As a result, the angle of the arc that divides the 8-square-inch sector is 45 degrees..

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Answer:

Greater than 7.

Step-by-step explanation:

To solve for x in the inequality, we need to isolate x on one side of the equation. First, we need to subtract

$ [tex]\large \boxed{\mathrm{1328}}[/tex]

from each side, giving $30x > $210. Then, we divide both sides by 30 to find that x > 7. Therefore, to make the inequality true, x needs to be greater than 7.

The future value of the account after 25 years is $210,595.95.

We can use the formula for future value of an annuity:

FV = P * (1 + r)^n + PMT * ((1 + r)^n - 1) / r

Where

Substituting the given values, we get:

FV = 19675.13 * (1 + 0.0925)^25 + 3000 * ((1 + 0.0925)^25 - 1) / 0.0925

Simplifying and evaluating, we get:

FV = $210,595.95

Therefore, the future value of the account after 25 years is $210,595.95.

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The  equation that shows the number of calories, c, in terms of the number servings, n  where Han wants to know how many calories he gets from the fruit snacks is  c = 90n.

Based on the provided information from the question, whereby Han wants to know how many calories he gets from the fruit snacks,  and was able to find the multilication of  number of servings, n, with calories per serving.

Therfore, we  can write the equation  as c = 90n since it is about the multiplication of the variables.

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complete question;

Each serving of a certain fruit snack contains 90 calories. Han wants to know how many calories he gets from the fruit snacks. Write an equation that shows the number of calories, c, in terms of the number servings, n. Complete the equation below c=______ Use n as your variable.

The first recommendation among the given options is "Buy more than one ticket at a time".

This recommendation suggests that purchasing multiple tickets in a single transaction can result in cost savings. Many airlines offer discounted fares for group bookings, which means buying multiple tickets at once can save money compared to buying them individually.

Buying tickets in bulk not only saves money, but it also provides a level of flexibility in terms of travel plans. For instance, if a group of travelers plan to visit a particular destination together, they can book their tickets together and have a better chance of securing seats on the same flight. Additionally, in case there are changes in the travel plans, such as the need to reschedule or cancel, group bookings provide more flexibility in terms of making changes or requesting refunds.

Overall, buying more than one ticket at a time can be a wise decision, especially if one is traveling with a group or planning multiple trips in the near future. By doing so, one can save money and have more flexibility in terms of travel plans, making the overall travel experience smoother and more enjoyable.

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Answer:

l = 2w - 3

2(2w - 3) + 2w = 27

4w - 6 + 2w = 27

6w - 6 = 27

6w = 33, so w = 5.5 feet and l = 8 feet

Answer:

Width: 5.5 feet

Length: 8 feet

Step-by-step explanation:

Let's begin by assigning variables to stand in for the rectangle's width and length.

Let x represent the rectangle's width.

The length is three feet shorter than double the width, according to the problem. As a result, the length may be written as:

2x - 3

The lengths of all the sides make up a rectangle's perimeter, which in this instance is:

2w + 2l

By substituting the width and length measurements we discovered:

2x + 2(2x - 3)

Simplifying:

2x + 4x - 6 = 27

6x = 33

x = 5.5

The rectangle is 5.5 feet wide as a result.

We may change x into the expression we discovered earlier to figure out the length:

2(5.5) - 3 = 8

The rectangle is 8 feet long as a result.

As a result, the rectangle's measurements are:

Width: 5.5 feet

Length: 8 feet

Answer:x+2

Step-by-step explanation:

Answer: (a) Let the circle roll along the x-axis with the center at the origin. Then, the parametric equations of the cycloid are given by:

x = t - sin(t)

y = 1 - cos(t)

where t is the angle in radians that the circle has rotated from its initial position.

(b) The arclength of one period of the cycloid is given by the integral:

L = ∫(0,2π) √[dx/dt]^2 + [dy/dt]^2 dt

Substituting the parametric equations from part (a), we get:

dx/dt = 1 - cos(t)

dy/dt = sin(t)

Therefore,

[dx/dt]^2 + [dy/dt]^2 = 2 - 2cos(t)

Substituting back into the arclength integral, we get:

L = ∫(0,2π) √(2 - 2cos(t)) dt

This integral can be evaluated using the half-angle formula:

cos(t) = 1 - 2sin^2(t/2)

Substituting this into the integral and simplifying, we get:

L = 8∫(0,π/2) √(sin^3(t/2)) dt

This integral can be evaluated using a substitution u = cos(t/2), du = -sin(t/2)dt:

L = 16∫(0,1) √[(1-u^2)^3] du

This integral can be further simplified by making the substitution u = sin(θ):

L = 16∫(0,π/2) cos^4(θ) dθ

This integral can be evaluated using integration by parts and trigonometric identities. After some algebraic manipulations, we get:

L = 8π

Step-by-step explanation: