Question

The average of three numbers is 16. If one of the numbers is 18, what is the sum of the other two

numbers?

12

14

20

30

The new points A', B', and C' after rotating triangle ABC 90 degrees counterclockwise about the origin and reflecting it over the x-axis are (-3,1), (0,3), and (2,4) respectively.

How rotate and reflect a point?

To rotate a point in a two-dimensional plane counterclockwise about the origin by a certain angle θ, we can use the rotation matrix. This matrix has the values of sine and cosine of the angle θ and can be used to multiply with the original point's coordinate vector to get the new coordinate vector after rotation. To reflect a point over the x-axis, we negate its y-coordinate.

Calculation the coordinates of the new points :

To rotate triangle ABC 90 degrees counterclockwise about the origin, we can use the rotation matrix:

[ cosθ -sinθ ]

[ sinθ cosθ ]

where θ is the angle of rotation, in this case 90 degrees, so we have:

[ 0 -1 ]

[ 1 0 ]

We can apply this matrix to each of the points of the triangle ABC to obtain their new coordinates after the rotation.

Then, to reflect the triangle over the x-axis, we simply negate the y-coordinate of each point.

So, to find the new points A', B', and C', we perform the following operations:

A' = (1, -3) * [ 0 -1 ] = [ (-3) (-1) ] -> reflect over x-axis -> [ (-3) 1 ]

[ 1 0 ]

B' = (3, 0) * [ 0 -1 ] = [ 0 (-3) ] -> reflect over x-axis -> [ 0 3 ]

[ 1 0 ]

C' = (4, -2) * [ 0 -1 ] = [ 2 (-4) ] -> reflect over x-axis -> [ 2 4 ]

[ 1 0 ]

Therefore, the new points are (-3,1), (0,3), (2,4).

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[tex]x=\textit{Liters of solution at 20\%}\\\\ ~~~~~~ 20\%~of~x\implies \cfrac{20}{100}(x)\implies 0.2 (x) \\\\\\ y=\textit{Liters of solution at 50\%}\\\\ ~~~~~~ 50\%~of~y\implies \cfrac{50}{100}(y)\implies 0.5 (y) \\\\\\ \textit{600 Liters of solution at 21\%}\\\\ ~~~~~~ 21\%~of~600\implies \cfrac{21}{100}(600)\implies 126 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{array}{lcccl} &\stackrel{Liters}{quantity}&\stackrel{\textit{\% of Liters that is}}{\textit{acid only}}&\stackrel{\textit{Liters of}}{\textit{acid only}}\\ \cline{2-4}&\\ \textit{1st Sol'n}&x&0.2&0.2x\\ \textit{2nd Sol'n}&y&0.5&0.5y\\ \cline{2-4}&\\ mixture&600&0.21&126 \end{array}~\hfill \begin{cases} x + y = 600\\\\ 0.2x+0.5y=126 \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{using the 1st equation}}{x+y=600}\implies x=600-y \\\\\\ \stackrel{\textit{substituting on the 2nd equation}}{0.2(600-y)+0.5y=126}\implies 120-0.2y+0.5y=126 \\\\\\ 120+0.3y=126\implies 0.3y=6\implies y=\cfrac{6}{0.3}\implies \boxed{y=20} \\\\\\ \stackrel{\textit{since we know that}}{x=600-y}\implies \boxed{x=580}[/tex]

The error interval for the combined mass, m, of the car and the person is: [tex]$$2067.65 kg \leq m \leq 2068.75 kg$$[/tex].

The mass of the car is given as 2006 kg rounded to the nearest kilogram, which means its actual mass could be as low as 2005.5 kg or as high as 2006.5 kg.

The mass of the person is given as 62.2 kg rounded to one decimal place, which means their actual mass could be as low as 62.15 kg or as high as 62.25 kg.

To find the error interval for the combined mass of the car and the person, we need to add the upper and lower bounds of each mass:

Lower bound of combined mass: 2005.5 kg + 62.15 kg = 2067.65 kg

Upper bound of combined mass: 2006.5 kg + 62.25 kg = 2068.75 kg

Therefore, the error interval for the combined mass, m, of the car and the person is: [tex]$$2067.65 kg \leq m \leq 2068.75 kg$$[/tex].

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When the minimum volume of a clown is 60,000 cm³ and the volume of a car is 3 million cm³, 55 is an upper bound on the maximum number of clowns that can fit in a car. The correct answer is Option A.

Bounds are the maximum and minimum limits that are permitted, within which something can or must be performed. Bounds are used to refer to an acceptable range of values that provide safe operation or performance. In mathematics, a set of bounds can define the limits of the amount of things or objects.

The minimum volume of a clown is 60,000 cm³, so to fit 55 clowns in a car we need:

55 clowns × 60,000 cm³/clown = 3,300,000 cm³

This is lower than the volume of the car. Therefore, 55 is an upper bound on the maximum number of clowns that can fit in the car.

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The rate of change of the distance for limousine is less than the rate of change of the convertible.

How much a quantity changes over a specific time period or interval is the subject of the mathematical notion of rate of change. Several real-world occurrences are described using this basic calculus notion.

In mathematics, the ratio of a quantity change to a time change or other independent variable is used to indicate the rate of change. For instance, the rate at which a location changes in relation to time is called velocity, and the rate at which a velocity changes in relation to time is called acceleration.

The equation of the distance travelled by the convertible is given as:

y = 35x

The equation of the limousine can be calculated using the coordinates of the graph (1, 30) and (2, 60).

The slope is given as:

slope = (change in y) / (change in x) = (60 - 30) / (2 - 1) = 30

Using the point slope form:

y - 30 = 30(x - 1)

y = 30x

So the equation of the limousine is y = 30x.

Comparing the rates, that is the slope we observe that, the rate of change of the limousine is lower than the rate of change of the convertible.

Hence, the rate of change of the limousine is less than the rate of change of the convertible.

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The approximate labor cost increase in percentage is approximately 21%, which is closest to option D.

To calculate the approximate labor cost increase in percentage, we can use the following formula:

Labor cost increase percentage = ((new labor cost - old labor cost) / old labor cost) x 100%

Plugging in the given values, we get:

Labor cost increase percentage = ((53,000 - 44,000) / 44,000) x 100%

Labor cost increase percentage = (9,000 / 44,000) x 100%

Labor cost increase percentage = 0.2045 x 100%

Labor cost increase percentage ≈ 20.45%

Therefore, the approximate labor cost increase in percentage is approximately 21%, which is closest to option D.

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We need to add 10 students to the hostel so that the food is enough for only 45 days.

Let's assume that "x" students need to be added to the hostel so that the food is enough for only 45 days.

We know that the total amount of food available is fixed, so the product of the number of students and the number of days that the food will last must also be fixed.

We can use this fact to set up an equation:

50 × 54 = (50 + x) × 45

To solve for x, we can simplify and solve for x:

Expand using FOIL method

50 × 54 = (50 + x)45

50 × 54 = 45 × 50 + 45 × x

2700 = 2250 + 45x

450 = 45x

x = 10

Therefore, number of students to be added is 10.

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Answer:

It would cost $32.00

Step-by-step explanation:

Total square feet:

length * width

4 * 4 = 16 sq ft.

Cost per square foot: $2.00

And so...

16 sq ft. * $2.00 = $32.00

The steps for proof by mathematical induction that P(n) is true for all positive integers n, All options are true.

The steps for a proof by mathematical induction that P(n) is true for all positive integers n are as follows:

a. Verify that P(1) is true.

b. Demonstrate that the conditional statement Plk) implies Plk+1) is true for all positive integers k.

c. Verify that P(1), P(2), P(3), ..., P(k) is all true, where k is a specific large, positive integer.

d. Demonstrate that if P(k) is false, then Plk+1) is false for all positive integers k.

e. Demonstrate that P(k+1) implies Plk) is true for all integers k.

Therefore, option (a), option (b), option (c), option (d), and option (e) are the steps for proof by mathematical induction that P(n) is true for all positive integers n.

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The formula for the total number of factors for a given number is given by; Total Number of Factors for N = (a+1) (b+1) (c+1)

The statement about regression line that is true is "it indicates a linear relationship between dependent variable and independent variable(s)."

In statistics, regression is a method of modeling the connection between a dependent variable (also known as a response variable) and one or more independent variables (also known as explanatory variables or predictors). It's also known as a linear regression model.

A regression model is established in order to identify the linear relationship between the dependent variable and one or more independent variables. The line of best fit is represented by the regression line in a simple linear regression model, which represents the relationship between the dependent variable and independent variable(s).

The line of best fit is determined by minimizing the sum of the squares of the residuals (errors). The objective of the regression model is to obtain a relationship between the dependent variable and the independent variable that explains the variance of the dependent variable based on the independent variable(s).

Hence, the correct option is: it indicates a linear relationship between dependent variable and independent variable(s).

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Answer:

Let's call the cost of a blouse "b" (in dollars). Then, according to the problem, the cost of a dress is 3/4 of the cost of a blouse, or (3/4)b.

We know that Vicky paid $350 for 3 dresses and 4 blouses, so we can set up the equation:

3(3/4)b + 4b = 350

Simplifying this equation, we get:

9/4 b + 4b = 350

Combining like terms, we get:

17/4 b = 350

To solve for b, we can multiply both sides by 4/17:

b = 350 × 4/17 = $82.35 (rounded to two decimal places)

Therefore, the cost of a dress is:

(3/4)b = (3/4)($82.35) = $61.76 (rounded to two decimal places)

So each dress cost $61.76.

A sample of n = 9 basal areas, the two statistics, g1 and g2, such that the probability of the difference between the sample mean and population mean being between them is 90% are g1 = y - u - (1.64 * (s/sqrt(n))) and g2 = y - u + (1.64 * (s/sqrt(n))).

The goal of this exercise is to find two statistics, g1 and g2, such that the probability of the difference between the sample mean, y, and population mean, u, being between them is 90%. To do this, we will use the z-score equation to calculate the standard deviations, which we will then use to calculate the two statistics.

Given that we are using a sample of n = 9 basal areas, we can calculate the sample standard deviation s using the following equation: s = sqrt[ (1/8) * Σ (x - y)^2], where x is the individual sample value and y is the sample mean.

Once we have the sample standard deviation, we can calculate the two statistics using the following equations:

g1 = y - u - (1.64 * (s/sqrt(n)))
g2 = y - u + (1.64 * (s/sqrt(n)))

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Answer:

180

Step-by-step explanation:

Form an equation

27x+17=180

Solve it for x  

x = 6.0recurring

the perimeter is 180

Let me know if it helps

Therefore, 15% of the beads on Tiana's necklace are blue.

Percentage is a way of expressing a fraction or a proportion out of 100. It is denoted by the symbol "%". For example, if we say that 50% of the students in a class are girls, it means that 50 out of every 100 students are girls.

Percentage can be calculated by dividing the given quantity by the total and multiplying by 100. For example, if there are 20 girls out of a total of 40 students in a class, the percentage of girls in the class can be calculated as follows:

Percentage of girls = (number of girls / total number of students) x 100%

= (20 / 40) x 100%

= 50%

by the question.

Tiana's necklace has a total of 3 + 17 = 20 beads.

To find the percentage of blue beads, we need to divide the number of blue beads by the total number of beads and then multiply by 100 to get the percentage:

percentage of blue beads = (number of blue beads / total number of beads) x 100

percentage of blue beads = (3 / 20) x 100

percentage of blue beads = 15

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The number of students that are in the sixth grade is given as follows:

250 students.

The number of students is obtained applying the proportions in the context of the problem.

We know that all students in the sixth grade either purchased their lunch or brought their lunch from home on Monday, and 24% of the students purchased their lunch, hence 76% of the students brought their lunch from home.

190 students brought their lunch from home, which is equivalent to 76% of the number of students, hence the number of students is given as follows:

0.76n = 190

n = 190/0.76

n = 250 students.

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The calculation for the price of 10 books is as follows: 10 × (19.60/88) = 2.21

We can use a proportion to determine the cost of 10 books based on the cost of 88 books:

88 books cost $19.60

1 book costs 19.60/88 dollars

Now we can use this information to find the cost of 10 books:

10 books cost 10 × (19.60/88) dollars

Simplifying the expression on the right side, we get:

10 books cost $2.21

Therefore, the equation that would help determine the cost of 10 books is: 10 × (19.60/88) = 2.21 The cost of books is often calculated based on the number of books purchased and the price per book. In this problem, we are given the cost of 88 books, and we need to find the cost of 10 books. Using a proportion, we can find the cost of 1 book and then multiply it by 10 to find the cost of 10 books. This method is useful for determining the cost of any number of books, given the cost of a different number of books. Book prices can vary widely depending on factors such as the format, edition, and publisher, so it is important to carefully consider book prices when making purchases.

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The solutions for the systems of inequalities are:

a) (0, -100), (0, -150), (0, -1000)

b)  (0, 50), (0, 55) , (0, 1,204).

When we have a system of inequalities, a solution is a point that solves both ienqualities at the same time.

The first one is:

y ≤ x - 8

y < -3x - 9

Here y must be smaller than x, then we can define x like x = 0, and really small values for y, like y = -100, replacing that we will get:

-100  ≤ 0 - 8 = -8

-100 < - 3*0 - 9 = -9

Both of these are true, so (0, -100) is a solution, and trivially, (0, -150) and (0, -1000) are other two solutions.

For the second system:

y > 5x + 1

y > 3

Let's do the same thing, x = 0 and y gets really large values, like y = 50

50 > 5*0 + 1 = 1  this is true.

50 > 3   this is true.

so (0, 50) is a solution, and also are (0, 55) and (0, 1,204).

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Answer:

To divide 8 by 52.08, we can use long division as follows:

      0.15   (quotient)

  __________

52.08 | 8.00

      7.81   (subtract 7.81 from 8.00 to get 0.19)

     ----

      1.19   (bring down the next digit 0, and add a decimal point)

      1.04   (subtract 1.04 from 1.19 to get 0.15)

     ----

      0.15   (no more digits to bring down, and the remainder is 0)

Therefore, 8 divided by 52.08 is approximately 0.15.

Answer:

Step-by-step explanation:

To find the percentage increase in Mr Hall's salary, we can use the following formula:

percentage increase = (new value - old value) / old value x 100%

Substituting the given values, we get:

percentage increase = (28000 - 25000) / 25000 x 100%

Simplifying the expression, we get:

percentage increase = 0.12 x 100%

percentage increase = 12%

Therefore, Mr Hall's salary increased by 12%.

The probability that among 12 randomly selected flights, exactly 10 arrive on time is 0.1667

The probability that, among 12 randomly selected flights, exactly 10 arrive on time is 0.1667. This is calculated by using the binomial probability formula:

[tex]P(x) = nCx (p^x) (1-p)^{(n-x)}[/tex]

Where:

[tex]P(x) = 12C10 (0.815^{10}) (0.185)^2 = 0.1667[/tex]

The probability that among 12 randomly selected flights, exactly 10 arrive on time is 0.1667

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The three positive numbers whose sum is 270 and whose product is a maximum is 67.5, 135, 67.5.

Let x, y, and z be the three positive numbers.

Write an equation relating x, y, and z based on the given constraint: x + y + z = 270.

Use algebra to solve for one of the variables in terms of the others.

Solving for z in terms of x and y: z = 270 - x - y.

Substitute the expression for z into the equation for the product of the three numbers:

P = x * y * z = x * y * (270 - x - y).

Expand the expression for P and simplify as P = 270xy - x²y - xy².

Using calculus to find the critical points of P, which occur when the derivative of P with respect to x is zero:

dP/dx = 270y - 2xy - y²= 0.

Solving for x in terms of y: x = (270y - y²)/(2y).

Simplify the expression for x as x = 135 - (y/2).

Since x and y must be positive, we know that 0 < y < 270/2 = 135.

Therefore, we can choose y = 135.

Using the equation for x in terms of y, we can find x as x = 135/2 = 67.5.

Using the constraint x + y + z = 270, we can find z as z = 67.5.

 Check that x, y, and z are positive and that they add up to 270.

Therefore, the three positive numbers whose sum is 270 and whose product is a maximum are 67.5, 135, and 67.5.

The product of these three numbers is (67.5) * (135) * (67.5) = 614062.5.

So the maximum product is 614062.5.

Thus, the answer is: 67.5, 135, 67.5.

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Based on the information, the pollster's conclusion makes sense.

To know if the pollster is right with his conclusion we must perform the following operation.

In accordance with the above, we can infer that the pollster concludes that 54% of the people surveyed are equivalent to 545 people correctly because 0.86 is a minimum margin of error for this type of statistics. In addition, it can be inferred that the pollster correctly uses the statistics and study parameters of a sample to establish a correct conclusion.

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Answer:

5) 12.2       6) 2

Step-by-step explanation:

Use Pythagoras for both

Rearrange for x in question 6

Answer:

y = 12

Step-by-step explanation:

if a tangent and a secant are drawn from an external point to a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant's external part and the entire secant , that is

y² = 8(8 + 6 + 4) = 8 × 18 = 144 ( take square root of both sides )

y = [tex]\sqrt{144}[/tex] = 12

The expected value of X is -$1.11. This means that on average, Mary can expect to lose $1.11 per game.

The probability that Mary will not win is 0.97; hence, the probability that she will win is 0.03.

Probability is the likelihood of an event happening. It is a way of measuring the chance or the likelihood of an event occurring. Probability is measured on a scale from 0 to 1, where 0 is impossible and 1 is certain.

Probability = (number of favorable outcomes) / (total number of outcomes)

The expected value is the sum of the products of each outcome and its probability. It represents the average value that one can expect to win from a game by placing a bet on that game.

In a game where the probability of winning is 0.03 and the probability of losing is 0.97,

Mary will win $60 if she wins and lose $3 if she loses.

Then, the expected value of X (the amount of money Mary wins or loses) can be calculated as:

E(X) = (0.03)($60) + (0.97)($-3)

E(X) = $1.80 - $2.91E(X) ⇒ $-1.11

The expected value of X is -$1.11 (a negative value).

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Answer:

x = 9

Step-by-step explanation:

[tex]2x + 3 = 3x - 6[/tex]

[tex]3 = x - 6[/tex]

[tex]x = 9[/tex]

Answer:

54 sq in.

Step-by-step explanation:

For a cube, try using the rectangular prism formula. (there's an easier way but I prefer this one)

2lw + 2hw + 2lh

(2 × 3 × 3) + (2 × 3 × 3) + (2 × 3 × 3)

18 + 18 + 18 + 18

SA = 54 sq in

hope i explained it (you can search up the formula for a cube but i like this one)

Answer:

Multiplying the second equation by 5, we get:

15x + 20y = 180

Now, we can add this equation to the first equation:

26x = 208

x = 8

Substituting x = 8 in the second equation:

3(8) + 4y = 36

4y = 12

y = 3

Therefore, the solution to the system is (8, 3).