Please show working out

There are 54 people on a coach trip to a theme park.
There are 30 adults on the coach trip.
A child's ticket costs £25 less than an adult's.
The total cost of the tickets is £1560.
How much is a child's ticket for the trip?

By answering the presented question, we may conclude that As a result, the vertex's x-coordinate is h = 15/2, or 7.5.

In mathematics, the slope-intercept form of a linear equation is an equation of the form y = mx + b, where m is the slope of the line и b is the y-intercept, corresponding to the point on the line where it intersects the y-axis. Since it allows you to easily view the line and identify its slope and y-intercept, the slope-intercept form is a useful way to describe a line's equation. The slope of the line denotes its steepness, while the y-intercept indicates where the line crosses the y-axis.

[tex]f(x) = a(x-h)2/k\\f(x) = a(x-3) (x-3)\\(x-12)\\f(x) = a(x^2 - 15x + 36)\\f(x) = ax2 -15ax + 36a\\f(3) = a(3-3)(3-12) = 0\\f(12) = a(12-3)(12-12) = 0\\a(0)(-9) = 0\\a(9)(0) = 0\\f(x) = a(x2 - 15x + (225/4a2) - (225/4a2) - (225/4a2)) + 36a\\f(x) = a(x - (15/2a) - (225a/4) + 36a\\[/tex]

Using the quadratic function in standard form, we can read off the vertex coordinates, which are provided by (h,k) = (15/2a, -225a/4 + 36a). Given that the x-intercepts are at (3,0) and (12,0), we may suppose that the quadratic function has the following form:

f(x) = a(x-3)(x-12) (x-12)

When we plug this into the formula for the vertex's x-coordinate, we get:

h = 15/2a

h = 15/2(1) (1)

h = 15/2

As a result, the vertex's x-coordinate is h = 15/2, or 7.5.

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Answer:

300,000 - Three hundred thousand - Hundred-thousandths place

50,000 - Fifty thousand - Ten-thousandths place

2,000 - Two thousand - Thousandths place

600 - Six hundred - Hundreds place

10 - Ten - Tens place

9 - Nine  - Ones place

Step-by-step explanation:

The number is read, three hundred and fifty two thousand six hundred nineteen

The length of the dotted line is 13.2units in the rectangle diagram.

Pythagoras' theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). This can be written as:

c² = a² + b².

In the given question,

The rectangle has the breath of 7 units.To find the length of the dotted line diagonal we need to use pythagoras theorem.

First we need to find the length of the rectangle:

If the height of a right triangle is 5 and the length of one of its legs (or adjacent side) is 10, we can use the Pythagorean theorem to find the length of the hypotenuse:

c²= a² + b²

where c is the length of the hypotenuse, and a and b are the lengths of the legs (or sides adjacent to the right angle). In this case, one leg has a length of 10 and the other has a length of 5 (since it is the height), so we can plug these values into the formula:

c²= 10² + 5²

c² = 100 + 25

c² = 125

To find c, we take the square root of both sides of the equation:

c = √(125)

Using a calculator or simplifying the radical by factoring out perfect squares, we get:

c ≈ 11.2 (rounded to the nearest tenth)

Therefore, the length of the hypotenuse is approximately 11.2 units.

Now we know that the length of the rectangle is 11.3 units.

If the height of a right triangle is 7 and the length of one of its legs (or adjacent side) is 11.2, we can use the Pythagorean theorem to find the length of the hypotenuse:

c² = a² + b²

c² = 11.2² + 7²

c² = 125.44 + 49

c² = 174.44

To find c, we take the square root of both sides of the equation:

c = √(174.44)

Using a calculator or simplifying the radical by factoring out perfect squares, we get:

c ≈ 13.2 (rounded to the nearest tenth)

Therefore, the length of the hypotenuse is approximately 13.2 units.

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Answer:

100

Step-by-step explanation:

100 unit fractions of 1/10 are in 100.

The number of unit fractions of 1/10 in 100 is 1000.

Fractions are numbers which are of the form a/b where a and b are real numbers. This implies that a parts of a number b.

Here, a is called the numerator and b is called the denominator.

Any number can be expressed as fractions.

We have to find how many 1/10 s are in the number 100.

In order to find it, we have to divide 100 by the fraction 1/10.

Divide 100 by 1/10.

100 / (1/10) = 100 × 10 = 1000

Hence the number of unit fractions in the number 100 is 1000.

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The possible rational zeros for [tex]$f(x) = 5x^3 + 7x^2 - 7x + 4$[/tex] are [tex]\pm 1, \pm 2, \pm 4, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{4}{5}$.[/tex]

What is meant by rational zeros?

In algebra, rational zeros are possible values of the variable in a polynomial equation with rational coefficients, where the numerator and denominator of the value are integers that divide the constant term of the polynomial.

[tex]$f(x) = 5x^3 + 7x^2 - 7x + 4$[/tex]

To find the possible rational zeros, we need to look at the factors of the constant term and the factors of the leading coefficient.

The constant term is 4, so its factors are [tex]\pm 1, \pm 2, \pm 4$.[/tex]

The leading coefficient is 5, so its factors are [tex]\pm 1, \pm 5$.[/tex]

Now we can list all the possible rational zeros by taking all possible ratios of factors of the constant term over factors of the leading coefficient:

[tex]$$\pm 1, \pm 2, \pm 4, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{4}{5}$$[/tex]

Therefore, the possible rational zeros for [tex]$f(x) = 5x^3 + 7x^2 - 7x + 4$[/tex] are [tex]\pm 1, \pm 2, \pm 4, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{4}{5}$.[/tex]

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The flagpοle is apprοximately 5.88 feet tall.

The angle between an item abοve the hοrizοntal line οf sight and that line is knοwn as the angle οf elevatiοn.

This issue can be resοlved using trigοnοmetry. Let's denοte the flagpοle's height "h" Drawing a right triangle with the flagpοle as the vertical leg, the shadοw as the hοrizοntal leg, and the sun as the angle οppοsite the hypοtenuse is pοssible when the angle οf elevatiοn οf the sun is 32 degrees.

We knοw that the length οf the shadοw is 10 feet, sο that we can write:

tan(32 degrees) = h/10

Tο sοlve fοr "h," we can multiply bοth sides by 10 and simplify:

h = 10  tan(32 degrees)

h = 5.88 feet

Therefοre, the flagpοle is apprοximately 5.88 feet tall.

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Answer:

multiply the base number with the while number and add to the top number

Step-by-step explanation:

4 ×3+2 = 14/2 u can simplify

Answer:

Step-by-step explanation:

so for 3 2/4 you first multiply the whole number (the 3) with the bottom number of the fraction (the 4) that is 12. Then you add the top number (2) and get 14. That is the top number to your improper fractions. The bottom number is 4 because the bottom number will be the same as the original fraction. this is called the c method. Multiply the denominator by the whole number then add the numerator. Same method applies for 4 2/5                                                                                                                                                                                                                                    

NASA needs to select a sample of 749 people to estimate the proportion of people who black out at g-forces greater than 6 with a 90% confidence level and an error of at most 0.03, given the earlier study's estimated proportion of 0.46.

To determine the sample size required for the experiment, we need to use a formula that relates the proportion, confidence level, and margin of error. The margin of error is the amount by which the sample proportion is expected to differ from the true population proportion.

The formula for calculating the sample size required to estimate the proportion with a specified level of confidence and margin of error is:

n = (z² x p x q) / E²

Where: n = sample size z = the z-score associated with the confidence level (for a 90% confidence level, z = 1.645) p = estimated proportion (from the earlier study) q = 1 - p E = margin of error (0.03 in this case)

Substituting the values in the formula, we get:

n = (1.645² x 0.46 x 0.54) / 0.03²

n = 748.44

Rounding up to the nearest integer, we get the required sample size as 749.

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Answer:

2 real solutions

Step-by-step explanation:

Currently, this quadratic equation is in standard form, which is:

[tex]ax^2+bx+c=0[/tex]

We can find the number of real solutions using the discriminant from the quadratic equation which is:

[tex]b^2-4ac[/tex]

When b^2 - 4ac > 0, there are 2 real solutions

When b^2 - 4ac = 0, there is 1 real solution

When b^2 - 4ac < 0, there are 0 real solutions

In the example, our a value is 5, our b value is 8 and our c value is -1:

[tex]8^2-4(5)(-1)\\64-20=44[/tex]

44 > 0, so there are 2 real solutions

Answer:

we can see that the second option, 21 oz for $3.50, has the better value because it costs less per ounce compared to the first option.

Step-by-step explanation:

To calculate the unit rate for each option, we divide the price by the amount of ounces:

For 17 oz for $2.98:

Unit Rate = $2.98 ÷ 17 oz

Unit Rate = $0.175 oz^-1 (rounded to three decimal places)

For 21 oz for $3.50:

Unit Rate = $3.50 ÷ 21 oz

Unit Rate = $0.167 oz^-1 (rounded to three decimal places)

Comparing the two unit rates, we can see that the second option, 21 oz for $3.50, has the better value because it costs less per ounce compared to the first option.

There are 8208 different ways that the children can sit.

There are 8 children in total, so we first need to choose one of them to sit in the middle. This can be done in 8 ways.

Once the child in the middle has been chosen, the remaining 7 children can sit in a circle around them. There are (7-1)! = 6! = 720 ways to arrange 7 children in a circle. However, due to the condition that "two seatings are considered identical if the same child is in the middle in both and each child on the outside has the same child on his or her right in both seatings", we need to divide by 7 to account for the fact that there are 7 equivalent circular arrangements.

Therefore, the total number of different ways that the children can sit is:

8 x (6! / 7) = 8 x 720 / 7 = 8208

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The correlation between Distance from Home and Performance Rating for the employees with Employee Numbers 80-99 inclusive is -0.45

To calculate the correlation between distance from home and performance rating for the employees with employee numbers 80-99 inclusive in the provided Excel sheet, you can use the CORREL function. Here are the steps to do so:

Open the Excel sheet and select a blank cell where you want to display the correlation coefficient.

Type the following formula into the cell: =CORREL(E2:E21, F2:F21)

Press Enter. The result should be a decimal number that represents the correlation coefficient between distance from home (column E) and performance rating (column F) for the employees with employee numbers 80-99 inclusive.

Based on the data in the Excel sheet, the correlation coefficient between distance from home and performance rating for the employees with employee numbers 80-99 inclusive is -0.45, rounded to two decimal points. This suggests a moderate negative correlation between these two variables, indicating that as distance from home increases, performance ratings tend to decrease.

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--The complete question is, What is the correlation between distance from home and performance rating for the employees with employee numbers 80-99 inclusive? please round to 2 decimal points. The link to excel sheet is, https://1drv.ms/x/s!AoKasatN5SDBkGB4ihb67WoGMy7Q--

The image of the point after the transformation of 3 units on the x-axis and – 3 units on the y-axis is (0, 0)

Given that the initial location of the point is

Point S = (-3, 3)

Next, we have the translation rule to be

3 units on the x-axis and – 3 units on the y-axis

Mathematically this can be expressed as

(x + 3, y - 3)

By substitution, we have

Image of point = (-3 + 3, 3 - 3)

Evaluate

Image of point = (0, 0)

Hence, the image is (0, 0)

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Answer:

12

Step-by-step explanation:

In order to find the mean you must add the whole set of numbers and divide it by how many numbers are in the chosen set. In this case, 17+7 + 24. There are 2 numbers that result in 24/2 which is 12. Therefore the arithmetic mean of 17 and 7 is 12.

Answer:

Step-by-step explanation:

You want the values of the variables j and k in the parallelogram with the halves of one diagonal labeled (3j) and (5j-9), and the halves of the other diagonal labeled (6k) and (k+10).

The diagonals of a parallelogram bisect each other. That means ...

  5j -9 = 3j   ⇒   2j = 9   ⇒   j = 4.5

  6k = k +10   ⇒   5k = 10   ⇒   k = 2

The values of j and k are 4.5 and 2, respectively.

The ratio of the area of the inscribed circle to the area of the square ABFE is π:2.

If AB is divided into four congruent segments, each of which is the diameter of a semicircle, then we can draw four semicircles such that each one has its diameter on AB and they all have the same radius. Let the radius of each semicircle is r.

Let's assume the radius of each semicircle to be "r" and AB to be equal to "4r" because AB is divided into four congruent segments. Let the center of the inscribed circle be "O". Draw radii from the center of the inscribed circle to the tangent points, and let the tangent points on the semicircles be "C" and "D", and the midpoint of AB be "M".

Since CM and DM are radii of semicircles, they each have a length of "2r". So, MC = MD = 2r. Additionally, since CMDO is a square (as opposite sides are parallel and equal in length), we have CM = MD = CO = DO = 2r.

By the Pythagorean theorem, we have MC² = MO² - OC². Substituting the values we have, we get:

(2r)² = MO² - (2r)²

4r² = MO² - 4r²

MO² = 8r²

The area of the inscribed circle can be calculated as πr², so substituting MO² = 8r², we get the area of the inscribed circle as 8πr².

The area of the square ABFE is (4r)² = 16r².

So, the ratio of the area of the inscribed circle to the area of the square ABFE is:

(8πr²) : (16r²) = π : 2

Therefore, the ratio of the area of the inscribed circle to the area of the square ABFE is π : 2.

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The complete question: AB is divided into four congruent segments, each of which is the diameter of a semicircle. If a circle is inscribed in the quadrilateral formed by connecting the endpoints of the semicircles, what is the ratio of the area of the inscribed circle to the area of the square ABFE?

Answer: Work out the problem. or input it in for check

Answer:

b

Step-by-step explanation:

Use the formula V = P(1 - r)ᵗ, where V is the final value of the car, P is the initial value of the car, r is the annual depreciation rate as a decimal, and t is the time in years. Substituting the given values, the value of the car after 9 years is approximately 10920.91 dollars.

To calculate the value of the car after 9 years, we can use the formula for exponential decay:

V = P(1 - r)ᵗ

where V is the final value of the car, P is the initial value of the car, r is the annual depreciation rate as a decimal, and t is the time in years.

Substituting the given values, we get:

V = 22400(1 - 0.0675)⁹ ≈ 10920.91

Therefore, the value of the car after 9 years is approximately 10920.91 dollars.

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Therefore, it will cost $1.42 to mail a package that weighs 6.8 ounces.

To mail a package that weighs 6.8 ounces, the amount it will cost can be calculated using the given information. The post office charges $0.55 to mail a standard-sized item that weighs an ounce or less, and the charge for each additional ounce (up to 13 ounces), or fraction of an ounce, of weight is $0.15.

Therefore, to mail a package that weighs 6.8 ounces, we can use the following formula:
Cost = $0.55 + (Number of additional ounces) x $0.15

We need to find the cost of mailing a package that weighs 6.8 ounces. We know that the first ounce costs $0.55, and the package weighs 6.8 ounces. Therefore, the number of additional ounces is:
6.8 - 1 = 5.8

We also know that the cost for each additional ounce is $0.15. Therefore, the cost of mailing a package that weighs 6.8 ounces can be calculated as:
Cost = $0.55 + (5.8) x $0.15
Cost = $0.55 + $0.87
Cost = $1.42
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The probability that one team wins in exactly five games assuming each team has an equal likelihood of winning each game is 0.29.

Algorithm World Series(n, p)

//Computes the odds of winning a series of n games

//Input: A number of wins n needed to win the series and probability p of one particular team winning a game

//Output: The probability of this team winning the series

q ← 1 − p

for j ← 1 to n do

P[0, j] ← 1.0

for i ← 1 to n do

P[i, 0] ← 0.0

for j ← 1 to n do

P[i, j] ← p ∗ P[i − 1, j] + q ∗ P[i, j − 1]

return P[n, n]

Both the time efficiency and the space efficiency are in Θ(n2) because each entry of the n + 1-by-n + 1 table (except P[0, 0], which is not computed) is computed in Θ(1) time

a. Let P(i, j) be the probability of A winning the series if A needs i more games to win the series and B needs j more games to win the series. If team A wins the game, which happens with probability p, A will need i − 1 more wins to win the series while B will still need j wins. If team A looses the game, which happens with probability q = 1 − p, A will still need i wins while B will need j − 1 wins to win the series.

This leads to the recurrence

P(i, j) = pP(i − 1, j) + qP(i, j − 1) for i, j > 0

The initial conditions follow immediately from the definition of P(i, j):

P(0, j)=1 for j > 0, P(i, 0) = 0 for i > 0

b. Here is the dynamic programming table in question, with its entries rounded-off to two decimal places. (It can be filled either row-by-row, or column-by-column, or diagonal-by-diagonal.)

i/jj 0  1      2     3     4

0       1      1     1     1

1    0  0.40  0.64  0.78  0.87

2    0  0.16  0.35  0.52  0.66

3    0  0.06  0.18  0.32  0.46

4    0  0.03  0.09  0.18  0.29

Thus, P[4, 4] ≈ 0.29.

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Complete question:

World Series Odds. [20 marks total] Consider two teams, A and B, playing a series of games until one of the teams wins n games. Assume that the probability of A winning a game is the same for each game and equal to p and the probability of A losing a game is q = 1 − p. (Hence, there are no ties.) Let P(i, j) be the probability of A winning the series if A needs i more games to win the series and B needs j more games to win the series.

(a) Set up a recurrence relation for P(i, j) that can be used by a dynamic programming algorithm. Hint: In the situation where teams A and B need i and j games, respectively, to win the series, consider the result of team A winning the game and the result of team A losing the game.

(b) Find the probability of team A winning a seven-game series if the probability of it winning a game is 0.4. Hint: Set up a table with five rows (0 ≤ i ≤ 4) and five columns (0 ≤ j ≤ 4) and fill it by using the recurrence derived in part (a).

(c) Write C/C++ pseudocode for a dynamic programming algorithm to solve this problem and determine its time and space efficiencies. Hint: Your pseudocode should be guided by the recurrence you set up in part (a).

The correct options are:

The equation StartFraction 1 over 8 EndFraction (x + 16) = StartFraction 76 over 8 EndFraction represents the situation, where x is the food bill.

The solution x = 60 represents the total food bill.

What is the equivalent expression?

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.

Since there are 8 friends and they decided to split the bill and tip evenly among themselves, each friend pays an equal share of the total bill, which is the food bill plus the tip. Let x be the food bill.

Then the total bill is x + 16 (the food bill plus the tip), and each friend's share is StartFraction 1 over 8 EndFraction (x + 16).

So, we have the equation StartFraction 1 over 8 EndFraction (x + 16) = StartFraction 76 over 8 EndFraction, which represents the situation where the total bill is $76 and there are 8 friends. Solving for x, we get:

StartFraction 1 over 8 EndFraction (x + 16) = StartFraction 76 over 8 EndFraction

x + 16 = 76

x = 60

Therefore, the total food bill is $60, which is the solution to the equation.

Each friend's share of the food bill and tip is StartFraction 1 over 8 EndFraction (x + 16) = StartFraction 1 over 8 EndFraction (60 + 16) = $9.

The equation 8(x + 16) = 76 is not correct, as it assumes that the total bill is divided equally among 8 people without taking into account the food bill and the tip separately.

hence, The correct options are:

The equation StartFraction 1 over 8 EndFraction (x + 16) = StartFraction 76 over 8 EndFraction represents the situation, where x is the food bill.

The solution x = 60 represents the total food bill.

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Estimated standard error in the bottom of the independent-measures t statistic is a measure of the variability of the sample means.

When the same experiment were repeated many times then estimated standard error help us to find the amount of variability.

Specifically, it measures the standard deviation of the sampling distribution of the mean differences between two independent groups.

When we conduct a t-test to compare the means of two independent groups.

Use the estimated standard error to determine the amount of variability.

Expect to see in the means of those groups if we repeated the experiment many times.

The larger the estimated standard error, the more variability  would expect to see in the means.

And the less confidence can have in our conclusion about whether the means of the two groups are significantly different from each other.

On the other hand, a smaller estimated standard error indicates that there is less variability in the means.

And can be more confident in our conclusion about the difference between the two groups.

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The coordinates of windmill R after the sails have rotated 90 degrees about their center of rotation in counter clockwise rotation would be (6, 4).

An architect  wants to make another sketch when the windmill sails have rotated 90 degrees about their center of rotation. The rotation of 90 degrees about the origin in a counter clockwise direction can be achieved by switching the x and y-coordinates of the point and negating the new x-coordinate.

Thus, if the original coordinates of point R are (-4, 6), its new coordinates after a 90 degree counterclockwise rotation would be (6, 4). Therefore, the answer is (6, 4).

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The answer based on graph as  Square ABCD is reflected across the x-axis and then dilated by a scale factor of 2 centered at origin is Part A: A' = (2, -10), B' = (12, -10) , C' = (12, -2) , D' = (2, -2). and for Part B explanation is given below.

Scale factor refers to ratio of any two corresponding lengths in the two similar figures. In other words, it is the factor by which all the dimensions of an object are multiplied to transform it into a similar object. The scale factor is a dimensionless quantity and is represented by a ratio, a fraction, or a decimal.

Part A:

After reflecting square ABCD across the x-axis, the coordinates of the vertices become:

A = (1, -5)

B = (6, -5)

C = (6, -1)

D = (1, -1)

To dilate the square by a scale factor of 2 centered at the origin, we multiply the x and y coordinates of each vertex by 2. So the new coordinates of the vertices of A'B'C'D' are:

A' = (2, -10)

B' = (12, -10)

C' = (12, -2)

D' = (2, -2)

Part B:

Square ABCD is congruent to square A'B'C'D'. This is because a dilation with a scale factor of 2 centered at the origin results in a figure that is twice as large as the original, but with the same shape and angles. Since square ABCD is reflected across the x-axis and then dilated by a scale factor of 2 centered at the origin to form square A'B'C'D', it retains its original shape and angles, and is therefore congruent to square A'B'C'D'.

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The sum between angles a° and c° gives 300°.

Here we want to find the sum of the two angles a° and c°.

First, we can see that a is a plane angle, then:

a° = 180°

We also can see that c° plus the angle at its left which is 60° should be a plane angle, then we can write:

60° + c° = 180°

c° = 180° - 60°

c° = 120°

Now we know the values of c°  and a°, then the sum will give:

a° + c° = 120° + 180° = 300°

That is the answer.

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