One-third of a number decreased by six equals eight. What is the number?
6

Answer:

statement 2, conduction is the transfer of heat through direct contact

The extreme points (x, y) along the curve are (-1, -1) and (0, 0).

The given function f(I, y) = e² x² + xy + y² = 1 represents a quadratic equation in two variables, x and y. To find the extreme points, we need to determine the values of x and y that satisfy the equation and minimize or maximize the function.

a) The function defined by f(x, y) = e² x² + xy + \(y^2\) - 1 takes on a minimum and a maximum value along the curve.

To find the extreme points, we need to find the critical points of the function where the gradient is zero.

Step 1: Calculate the partial derivatives of f with respect to x and y:

∂f/∂x = 2\(e^2^x\) + y

∂f/∂y = x + 2y

Step 2: Set the partial derivatives equal to zero and solve for x and y:

2\(e^2^x\) + y = 0

x + 2y = 0

Step 3: Solve the system of equations to find the values of x and y:

Using the second equation, we can solve for x: x = -2y

Substitute x = -2y into the first equation: 2(-2y) + y = 0

Simplify the equation: -4e² y + y = 0

Factor out y: y(-4e^2 + 1) = 0

From this, we have two possibilities:

1) y = 0

2) -4e²  + 1 = 0

Case 1: If y = 0, substitute y = 0 into x + 2y = 0:

x + 2(0) = 0

x = 0

Therefore, one extreme point is (x, y) = (0, 0).

Case 2: If -4e^2 + 1 = 0, solve for e:

-4e²  = -1

e²  = 1/4

e = ±1/2

Substitute e = 1/2 into x + 2y = 0:

x + 2y = 0

x + 2(-1/2)x = 0

x - x = 0

0 = 0

Substitute e = -1/2 into x + 2y = 0:

x + 2y = 0

x + 2(-1/2)x = 0

x - x = 0

0 = 0

Therefore, the second extreme point is (x, y) = (0, 0) when e = ±1/2.

b) The equation (1+x)e = (1+y)e* is satisfied along the line y = x.

To find a critical point on this line where the equation is neither locally uniquely solvable for x nor y, we need to find a point where the equation has multiple solutions.

Substitute y = x into the equation:

(1+x)e = (1+x)e*

Here, we see that for any value of x, the equation is satisfied as long as e = e*.

Therefore, the equation is not locally uniquely solvable for x or y along the line y = x.

c) Taylor expansion up to degree 2:

To understand the solution set of the equation in the vicinity of the critical point, we can use Taylor expansion up to degree 2.

2. Expand the function f(x, y) = e²x²  + xy + \(y^2\) - 1 using Taylor expansion up to degree 2:

f(x, y) = f(a, b) + ∂f/∂x(a, b)(x-a) + ∂f/∂y(a, b)(y-b) + 1/2(∂²f/∂x²(a, b)(x-a)^2 + 2∂²f/∂x∂y(a, b)(x-a)(y-b) + ∂²f/∂y²(a, b)(y-b)^2)

The critical point we found earlier was (a, b) = (0, 0).

Substitute the values into the Taylor expansion equation and simplify the terms:

f(x, y) = 0 + (2e²x + y)(x-0) + (x + 2y)(y-0) + 1/2(2e²x² + 2(x-0)(y-0) + 2(\(y^2\))

Simplify the equation:

f(x, y) = (2e² x² + xy) + ( x² + 2xy + 2\(y^2\)) + e² x² + xy + \(y^2\)

Combine like terms:

f(x, y) = (3e² + 1)x² + (3x + 4y + 1)xy + (3 x² + 4xy + 3 \(y^2\))

In the vicinity of the critical point (0, 0), the solution set of the equation, given by f(x, y) = 0, looks like a second-degree polynomial with terms involving  x² , xy, and  \(y^2\).

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Answer:

\(x=-1,5\)

Step-by-step explanation:

\(x^2-4x-5=0\)

In order to solve this quadratic, we have many methods. We can factor, complete the square, or use the quadratic formula. I'm going to factor since it's the easiest method.

To factor, find two numbers that when multiplied equal a(c) and when added equal b.

a=1, b=-4, and c=-5.

So we want two numbers that when multiplied equals 1(-5)=-5 and when added equals -4.

-5 and 1 are the possible numbers. Therefore:

\(x^2-4x-5=0\\x^2+x-5x-5=0\\x(x+1)-5(x+1)=0\\(x-5)(x+1)=0\\x=5, -1\)

Given, if 1/6 of the students are fourth graders, and 1/4 of the remaining number of students are second grade and  if there are 96 students in the auditorium, then there are 20 second graders in the auditorium.

To determine the number of second graders in the auditorium, we need to calculate the number of remaining students after accounting for the fourth graders.

Given that 1/6 of the students are fourth graders, we can calculate the number of fourth graders as follows:

Number of fourth graders = (1/6) * 96 = 16

To find the number of remaining students, we subtract the number of fourth graders from the total number of students:

Remaining students = Total students - Number of fourth graders

Remaining students = 96 - 16 = 80

Now, we can calculate the number of second graders by taking 1/4 of the remaining students:

Number of second graders = (1/4) * 80 = 20

Therefore, there are 20 second graders in the auditorium.

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The distance between the centers of the two wheels is approximately 11.86 feet.

The length of the conveyor belt is equal to the circumference of the circle formed by each wheel plus the distance between the centers of the two wheels.

Let's call the distance between the centers of the two wheels "d". The circumference of each wheel can be calculated using the formula

C = πd

Since each wheel has a diameter of 1 foot, its radius is 0.5 feet. Therefore, the circumference of each wheel is:

C = πd = π(0.5) = 1.57 feet (approx.)

The length of the conveyor belt is given as 15 feet. So we can write

2C + d = 15

Substituting the value of C, we get

2(1.57) + d = 15

3.14 + d = 15

d = 11.86 feet

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The median number of siblings is 3.

We have,

To find the median number of siblings, we need to arrange the data in ascending order and determine the middle value.

Arranging the data in ascending order:

0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4

The total number of data points is the sum of the frequencies:

4 + 2 + 5 + 6 + 8 = 25.

Since there are an odd number of data points (25), the median is the middle value.

The middle value is the 13th value in the ordered data set, which is 3.

Therefore,

The median number of siblings is 3.

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Answer:

c(2) = -10

Step-by-step explanation:

The first equation says that the first term of the sequence is -20.

The second equation is saying that to find any term of the sequence, add 10 to the previous term.

c(2) = c(2-1) + 10

c(2) = c(1) + 10

c(2) = -20  + 10 = -10

When the second derivative of a function equals zero, it indicates a possible point of inflection or a critical point where the concavity of the function changes. It is a significant point in the analysis of the function's behavior.

The second derivative of a function measures the rate at which the slope of the function is changing. When the second derivative equals zero at a particular point, it suggests that the function's curvature may change at that point. This means that the function may transition from being concave upward to concave downward, or vice versa.

Mathematically, if the second derivative is zero at a specific point, it is an indication that the function has a possible point of inflection or a critical point. At this point, the function may exhibit a change in concavity or the slope of the tangent line.

Studying the second derivative helps in understanding the overall shape and behavior of a function. It provides insights into the concavity, inflection points, and critical points, which are crucial in calculus and optimization problems.

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When the second derivative of a function equals zero, it indicates a critical point in the function, which can be a maximum, minimum, or an inflection point.

The second derivative of a function measures the rate at which the slope of the function is changing. When the second derivative equals zero, it indicates a critical point in the function. A critical point is a point where the function may have a maximum, minimum, or an inflection point.

To determine the nature of the critical point, further analysis is required. One method is to use the first derivative test. The first derivative test involves examining the sign of the first derivative on either side of the critical point. If the first derivative changes from positive to negative, the critical point is a local maximum. If the first derivative changes from negative to positive, the critical point is a local minimum.

Another method is to use the second derivative test. The second derivative test involves evaluating the sign of the second derivative at the critical point. If the second derivative is positive, the critical point is a local minimum. If the second derivative is negative, the critical point is a local maximum. If the second derivative is zero or undefined, the test is inconclusive.

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Answer:

Rhombus

Step-by-step explanation:

This isnt a rectangle because it doesnt have right angles. and the precise name would be a rhombus

Answer:

Here you go

Step-by-step explanation:

The increase in amount is $351.00 ($5400 x 0.065 = $351). The amount in the account at the end of the year was $5751.00 ($5400 + $351 = $5751).

Answer:

Let x represent the number of packs of paper Mr. Valentino can purchase.

The inequality can be written as:

3.75x ≤ 20

To solve the inequality, divide both sides of the equation by 3.75.

x ≤ 5.33

The graph of the solution will show a line with a y-intercept of 0 and x-intercept of 5.33.

Answer:

4 + 9 = 13

+ +

12 - 8 = 4

= =

16 17

Took a while to solve but hope this helps!!

Answer:

top left=4, top right =9, bottom left=12, bottom right =8 -- set it up as a system of small equations.  

Step-by-step explanation:

x+y=13

x+w=16

y+z=17

w-z=4

x=16-w

y=17-z

then substitute.

(16-w)+y=13

-w+y= -3

-w+(17-z)= -3

-w-z = -20 add that to w-z=4

-2z= -16

z=8

then plug back in until you have all 4 values

w-(8)=4 --> w=12

-(12)+y= -3 --> -12+y= -3 --> y=9

x+(9)=13 --> x=4

The fraction of chocolates that Malcom ate was 12/31.

Fractions are used to represent the portion/part of the whole. It shows the equal parts of a whole. A fraction has two parts, namely numerator and denominator. The number on the top is called the numerator, and the number on the bottom is called the denominator. The numerator defines the number of equal parts taken, whereas the denominator defines the total number of equal parts in a whole.

1 – 5/7 = 2/7 (milk chocolates eaten)

1 – 2/5 = 3/5 (dark chocolates eaten)

2/7 (milk chocolates eaten) = 3/5 (dark chocolates eaten)

2 x 3 = 6

2/7 (milk chocolates eaten) ——- 6u

1/7 (milk chocolates eaten) ——- 6u/2 = 3u

7/7 (milk chocolates eaten) ——- 3u x 7 = 21u

3/5 (dark chocolates eaten) ——- 6u

1/5 (dark chocolates eaten) ——- 6u/3 = 2u

5/5 (dark chocolates eaten) ——- 2u x 5 = 10u

(6u + 6u)/(21u + 10u) = 12/31

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Answer:

False

Step-by-step explanation:

it's because congruent triangles are 180 degrees and that added up is more than 180 degrees.

The given statement is false.

We have given that,

The triangles shown below must be congruent.

60°

60°

30°

150

We have to determine whether the given statement is true or false.

The sum of all angles is 180 degrees

\(\angle A+\angle B+\angle C=180\)

it's because congruent triangles are 180 degrees and that added up is more than 180 degrees.

Therefore the given statement is false.

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The mean is 18.6.

Given the following information; z = 2.25, x = 22.2, and σ = 1.6, to find the mean, we have to apply the formula for z-score. z = (x - μ)/σWhere; z-score is represented by z, the value of X is represented by x, the mean is represented by μ and the standard deviation is represented by σSubstituting the values into the equation above;2.25 = (22.2 - μ)/1.6Multiplying both sides of the equation by 1.6, we have;1.6(2.25) = (22.2 - μ)3.6 = 22.2 - μ Subtracting 22.2 from both sides of the equation;3.6 - 22.2 = - μ-18.6 = - μ Multiplying both sides of the equation by -1, we have;μ = 18.6

Simply said, a z-score, also known as a standard score, informs you of how far a data point is from the mean. Technically speaking, however, it's a measurement of how many standard deviations a raw score is from or above the population mean.

You can plot a z-score on a normal distribution curve. Z-scores range from -3 standard deviations, which would fall to the extreme left of the normal distribution curve, to +3 standard deviations, which would fall to the far right. You must be aware of the mean and population standard deviation in order to use a z-score.

The z-score can show you how that person's weight compares to the mean weight of the general population.

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Answer:

a) A reduction

b) The scale factor is 1/2

Step-by-step explanation:

The vertices of the original figure, ABCD are;

A(-2, 4), B(4, 4), C(2, -2), and D(-4, -2)

The vertices of the image, A'B'C'D' are;

A'(-1, 2), B'(2, 2), C'(1, -1), and D'(-2, -1)

a) The length of the side AB = 4 - (-2) = 6 (Distance between points having the same 'x' or 'y' coordinates

The length of the side A'B' = 2 - (-1) = 3

The length of AB = 6 is larger than the length of A'B' = 3, therefore the image A'B'C'D' is a reduction of (is smaller) the figure ABCD

b) The scale factor = The ratio of the lengths corresponding sides of the image to the original figure;

∴ The scale factor =A'B'/AB = A'D'/AD = B'C'/BC = D'C'/DC

AD = √((-4 - (-2))² + (-2 - 4)²) = 2·√10

A'D' = √((-2 - (-1))² + (-1 - 2)²)  = √10

A'D'/AD= √10/(2·√10) = 1/2

Therefore, given that the coordinates of the preimage are 2 × the coordinates of the image, we have;

The scale factor = A'B'/AB = B'C'/BC = D'C'/DC = A'D'/AD =1/2

The scale factor is 1/2, therefore, we multiply the dimensions of the original figure by 1/2 to get the dimensions of the image.

The scale factor = 1/2.

The correct answer is (d) The price must be replaced with a relative price, and the marginal values must be replaced with the corresponding Marginal Rates of Substitution.

When the value of a public good cannot be expressed in monetary terms, the Samuelson condition still holds, but some modifications are required. In this case, the per-unit price (p) used in the Samuelson condition needs to be replaced with a relative price, which represents the trade-off between the public good and other goods or services. Additionally, the marginal values (MV) of the public good need to be replaced with the Marginal Rates of Substitution (MRS), which measure the rate at which one person is willing to substitute the public good for another good.

Therefore, to determine the efficient level of the public good, the modified Samuelson condition uses a relative price and the corresponding Marginal Rates of Substitution.

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a) The probability that the murder was committed with a knife: 0.55

b) The probability that the cook did it, given that the murder was committed with a knife: 0.275

To determine the probability that the murder was committed with a knife, we first need to find the total number of weapons.

The butler has four poison tipped pens, two crowbars and four knives and the cook has three rolling pins and seven knives.

So, the total knives: 4 + 7 = 11

crowbars: 2

poison tipped pens: 4

and rolling pins: 3

So, the total number of weapons: 11 + 2 + 4+ 3 = 20

The probability that the murder was committed with a knife:

p = 11/20

p = 0.55

Now we need to find the probability that the cook did it, given that the murder was committed with a knife.

P = 0.55 × 0.5

P = 0.275

The required probability is 0.275

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Answer:

The answer is 1.5

Step-by-step explanation:

Hope it helps :)

The length of y and x in the given figure comes out to be \(4\frac{4}{9}\) units and \(3\frac{5}{9}\) units respectively.

According to the angle bisector theorem, an angle bisector divides the opposite side in equal proportions to the other two sides.

Given:

BC = 15 units

AC = 8 units

AB = 12 units

AC = x + y

8 = x + y ---- (1)

According to the angle bisector theorem,

x : y = 12 : 15

15x = 12y

5x = 4y

x = 0.8y

Put this in equation (1)

8 = 0.8y + y

1.8y = 8

y = 8/1.8

= 40/9 = \(4\frac{4}{9}\) units

x = 32/9 = \(3\frac{5}{9}\) units

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Answer:

1)\(x^2+x-12=0\)

2)\(x^2+8x-20=0\)

3)\(x^2+0.9x-0.36=0\)

4)\(2x^2-6x-36=0\)

5)\(x^2-5x-84=0\)

Step-by-step explanation:

1) The length of a wooden frame is 1 foot  longer than its width and its area is equal  to 12 sq. ft.

Let the width be x

We are given that length of a wooden frame is 1 foot  longer than its width

So, Length = x+1

Area of rectangular frame =\(Length \times Breadth = x(x+1)\)

ATQ

x(x+1)=12

\(x^2+x-12=0\)

2)The length of the floor is 8 m longer than  its width and there is 20 square meters.

Let the width be x

We are given that length of the floor is 8 m longer than  its width

So, Length = x+8

Area of floor= \(Length \times Breadth = x(x+8)\)

ATQ

x(x+8)=20

\(x^2+8x-20=0\)

3)The length of a plywood is 0.9 m more  than its width and its area is 0.36 m2.

Let the width be x

We are given that length of a plywood is 0.9 m more  than its width

So, Length = x+0.9

Area of plywood=\(Length \times Breadth = x(x+0.9)\)

ATQ

x(x+0.9)=0.36

\(x^2+0.9x-0.36=0\)

4)The area of rectangle whose length is six  less than twice its width is thirty-six

Let the width be x

We are given that length is six  less than twice its width

So, Length = 2x-6

Area of rectangle = Length \times Breadth = x(2x-6)

ATQ

x(2x-6)=36

\(2x^2-6x-36=0\)

5)The width of a rectangular plot is 5 m  less than its length and its area is 84 m2.

Let the length be x

Width = x-5

Area =\(Length \times Breadth = x(x-5)\)

ATQ

x(x-5)=84

\(x^2-5x-84=0\)

Hope this was helpful

\(\textbf {The answer is h = -2/9.}\)

\(\textrm {We are given that 9/2h = -1. Hence, divide by 9/2 on both sides.}\)

\(\frac{9/2h}{9/2} = \frac{-1}{9/2}\)

\(\boxed {h = -\frac{2}{9}}\)

Answer:

1*10^2

Step-by-step explanation:

Answer:

The answer would be 1x10^2 since you move the decimal place to the left two times.

Step-by-step explanation:

The integral of (42³ - 2r + 7) dz is equal to (42³ - 2r + 7)z + C.

To integrate the function (42³ - 2r + 7) dz, we treat r as a constant and integrate with respect to z. The integral of a constant with respect to z is simply the constant multiplied by z:

∫ (42³ - 2r + 7) dz = (42³ - 2r + 7)z + C

where C is the constant of integration.

Note: The integral of a constant term (such as 7) with respect to any variable is simply the constant multiplied by the variable. In this case, the variable is z.

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The area of the parallelogram in terms of a, b, and c (the length of the diagonal) is:

(1/2) * (a * b * ✓(1 - ((a² + b² - c²) / (2ab)²

Using the formula Area = (1/2) * (a * b * sinθ)

In the case of a parallelogram, the opposite sides are parallel and equal in length. Therefore, the angle θ can be found using the Law of Cosines. The Law of Cosines states:

c² = a² + b² - 2ab * cosθ

Rearranging the equation, we get:

cosθ = (a² + b² - c²) / (2ab)

Area = (1/2) * (a * b * sinθ)

= (1/2) * (a * b * ✓(1 - cos²θ))

= (1/2) * (a * b * ✓(1 - ((a² + b² - c²) / (2ab))²))

= (1/2) * (a * b * ✓(1 - ((a² + b² - c²) / (2ab)

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