the measures of the legs of a right triangle can be represented by the expressions 6x^(2)y 9x^(2)y. Use the Pythagorean Theorem to find a simplified expression for the hypotenuse.

The distance between the point P( 8 , 0 , -6 ) to the plane x + y + z = 6 is

d = ( 4 / √3 )

What is the distance between a point and a plane?

The distance between a point and a plane is the smallest perpendicular distance between the point and the plane in question. In other words, the length of the perpendicular parallel to the normal vector dropped from the given location to the provided plane is the shortest distance between two points.

Let the point be P( x₀ , y₀ , z₀ )

Let the plane be Ax + By + Cz + D = 0

The distance between point P and plane is

d = | Ax₀ + By₀ + Cz₀ + D | / √ ( A² + B² + C² )

Given data ,

Let the point be P( x₀ , y₀ , z₀ ) = P ( 8 , 0 , -6 )

Let the equation of plane be x + y + z = 6

A = 1

B = 1

C = 1

D = -6

Now , the distance between the point and the plane is given by ,

d = | Ax₀ + By₀ + Cz₀ + D | / √ ( A² + B² + C² )

Substituting the values in the equation , we get

d = | 8 + 0 + ( -6 ) + ( -6 ) | / √ ( 1 + 1 + 1 )

d = | 8 - 6 - 6 | / √3

d = | -4 | / √3

d = 4 / √3

Therefore , the value of d = 4 / √3

Hence , The distance between the point P( 8 , 0 , -6 ) to the plane
x + y + z = 6 is d = ( 4 / √3 )

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The angles of the rhombus determines the inclination of the sides of the

rhombus.

The altitude of the rhombus bisects the opposite side.

The angle of the rhombus at the vertex point of the altitude = An obtuse angle

The measure of the angles of the rhombus

The length of the sides of a rhombus are equal.

The opposite angles of a rhombus are equal.

Let x represent the length of a side of the rhombus, we have;

The bisector from the vertex of the obtuse angle = An altitude

The angle the altitude forms with the side it bisects = 90°

Therefore;

The triangle formed by the altitude, half the bisected side and the side of the rhombus adjacent to the bisected side = A right triangle

The hypotenuse side to the right triangle = The side of the rhombus

Let θ represent the angle opposite the altitude, and between the sides of

the rhombus, in the right triangle by trigonometric ratios, we have;

[tex]\displaystyle cos(\theta) = \frac{0.5 \cdot x}{x} = 0.5[/tex]

θ = arccos(0.5) = 60°

Therefore, two acute (opposite) angles of the rhombus are 60°

The two obtuse angles of the rhombus are each = (360° - 2×60°) ÷ 2 = 120°

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Perimeter of a polygon is equal to sum of all sides of the polygon

Sides of the polygons are 11 ft, 7 ft, 18 ft, 22 ft, 14 ft and 23.5 ft

Sum of all sides of the polygon is given by = 11 ft + 7 ft + 18 ft + 22 ft + 14 ft + 23.5 ft

= 95.5 ft

Perimeter of a polygon = 95.5 ft

Hope this helps..!!

Thank you :)

Answer:

x = -3

y = -8

Step-by-step explanation:

I took the test

You would multiply 60 times 10.

Hope it helped!:)

Answer:

a) Expected amount of winnings = -$0.084

Standard deviation = $3.00

b) Expected amount of winnings = -$0.084

Standard deviation = $1.73

c) The expected winnings for both cases is exactly the same, but the spread of the wins about the expected value/mean is low in the case (b) compared to case (a).

So, with the expected winnings using the method in (a) and (b) the same, the risk associated with betting $3 dollar per game is higher because it has a higher standard deviation. This points to higher spread of losses and wins betting $3 once than betting $1 three different times.

Step-by-step explanation:

(a) Suppose you play roulette and bet $3 on a single round. What is the expected value and standard deviation of your total winnings?

It's either one wins or not with a bet of $3.

We can then obtain the probability mass function.

Random variable X represents the amount of winnings.

- If one bets $3 and picks the right colour, the person wins $6.

X = amount of winnings = 6 - 3 = $3

probability of winning = (18/37) = 0.486

- If one bets $3 and picks the wrong colour, then the person wins $0.

X = amount of winnings = 0 - 3 = -$3

Probability of losing = (19/37) = 0.514

The probability mass function is then

X | P(X)

3 | 0.486

-3 | 0.514

E(X) = Σ xᵢpᵢ

xᵢ = each variable

pᵢ = probability of each variable

E(X) = (3×0.486) + (-3×0.514) = -$0.084

Standard deviation = √(variance)

Variance = Var(X) = Σx²p − μ²

μ = E(X) = -0.084

Σx²p = (3²×0.486) + [(-3)² × 0.514] = 9

Variance = 9 - (-0.084)² = 8.993

Standard deviation = √8.993 = 2.999 = $3

(b) Suppose you bet $1 in three different rounds. What is the expected value and standard deviation of your total winnings?

With a bet of $1,

It's either one wins or not with a bet of $1.

We can then obtain the probability mass function.

Random variable X represents the amount of winnings.

- If one bets $1 and picks the right colour, the person wins $2.

X = amount of winnings = 2 - 1 = $1

probability of winning = (18/37) = 0.486

- If one bets $1 and picks the wrong colour, then the person wins $0.

X = amount of winnings = 0 - 1 = -$1

Probability of losing = (19/37) = 0.514

The probability mass function is then

X | P(X)

1 | 0.486

-1 | 0.514

E(X) = Σ xᵢpᵢ

xᵢ = each variable

pᵢ = probability of each variable

E(X) = (1×0.486) + (-1×0.514) = -$0.028

For 3 rounds of $1, the expected amount of winnings = 3 × -0.028 = -$0.084

Variance = Var(X) = Σx²p − μ²

μ = E(X) = -0.028

Σx²p = (1²×0.486) + [(-1)² × 0.514] = 1

Variance = 1 - (-0.028)² = 0.999216

Variance for combining 3 independent distributions = Σ λᵢ²σᵢ²

Variance of 3 rounds of $1 bets = 3[1²× var(X)] = 3 × var(X) = 3 × 0.999216 = 2.997648

Standard deviation = √(variance) = √(2.997648) = $1.73

(c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness of the two games?

The expected winnings for both cases is exactly the same, but the spread of the wins about the expected value/mean is low in the case (b) compared to case (a).

So, with the expected winnings using the method in (a) and (b) the same, the risk associated with betting $3 dollar per game is higher because it has a higher standard deviation. This points to higher spread of losses and wins betting $3 once than betting $1 three different times.

Hope this Helps!!!

Prior to the major issue, the woodlot had 72 more maple trees than it had after.

To solve this problem

Assuming x is a positive constant, we can represent the initial number of oak trees as 5x and the initial number of maple trees as 9x.

Prior to the problem, the ratio of maple trees to oak trees was 9:5, which produced the following equation:

[tex](9x)/(5x) = 9/5[/tex]

Later, the new ratio changed to 3:11 after new trees were planted to replace the original amount of trees. After planting, 132 oak trees were reported to have been planted. We so established the equation:

11y = 132

Solving for y:

y = 132/11 = 12

Now, we can find the new number of maple trees:

3y = 3 * 12 = 36

There were 36 maple trees when new trees were planted.

Before the problem, there were 9x as many maple trees as there were.

9x = 9 * 12 = 108

The difference between the original and new number of maple trees is:

108 - 36 = 72

So, Prior to the major issue, the woodlot had 72 more maple trees than it had after.

In addition to linear, quadratic, rational, and radical functions, there are exponential functions. Exponential functions have the form f(x) = bx, where b > 0 and b ≠ 1. Just as in any exponential expression, b is called the base and x is called the exponent.

An example of an exponential function is the growth of bacteria. Some bacteria double every hour. If you start with 1 bacterium and it doubles every hour, you will have 2x bacteria after x hours. This can be written as f(x) = 2x.

Before you start,   f(0) = 20 = 1

After 1 hour           f(1) = 21 = 2

In 2 hours              f(2) = 22 = 4

In 3 hours              f(3) = 23 = 8

and so on.

With the definition f(x) = bx and the restrictions that b > 0 and that b ≠ 1, the domain of an exponential function is the set of all real numbers. The range is the set of all positive real numbers. The following graph shows f(x) = 2x.

To find the dimensions of a rectangle with length '8 cm more than 3 times the width' and a perimeter of 64 cm, form the equation 2(3W+8) + 2W = 64. Solving for W gives the width, which can be substituted into the equation 3W+8 to give the length.

The subject of the question is algebra, specifically how to form equations based on word problems involving geometric shapes. The rectangle problem you've given can be solved using the formulas for the perimeter of a rectangle.

The perimeter 'P' of a rectangle is given by P=2(length) + 2(width). In the problem, the length 'L' is given as 8 cm more than 3 times the width 'W', or L=3W+8. The perimeter 'P' is stated as 64 cm, so substituting our equation for length into the perimeter equation, we get 2(3W+8) + 2W = 64. This equation will yield the width of the rectangle. To get the length, you simply substitute the width into the length equation.

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General Idea:

A cyclic quadrilateral has all its vertices on the circumference of the circle. The opposite angles of a cyclic quadrilateral are supplementary, that is it add up to get 180 degrees.

Applying the Concept:

Opposite angles B and D add up to give 180 degrees.

[tex] \angle B + \angle D=180\\ \\ 4x-20+x=180\\ Grouping \; like \; terms\\ \\ 4x+x-20=180\\ Combining \; like\; terms\\\\ 5x-20 =180\\ Adding \; 20 \; on\; both\; sides\\ \\ 5x-20+20=180+20\\ Combine\; like\; terms\\ \\ 5x=200\\ Divide\; 5\; on\; both\; sides\\ \\ \frac{5x}{5} =\frac{200}{5} \\ Simplify \; fraction\; on\; both\; sides\\ \\ x=40\\ \\ Substitute \; 40\; for\; x\; in\; the\; expression\; for\; \angle B \\ \angle B=4x-20=4(40)-20=160-20\\\\\angle B=140 [/tex]

Conclusion:

When Quadrilateral ABCD ​ is inscribed in this circle, the measure of angle B for the given diagram is [tex] 140^{\circ} [/tex]

Erika has three pieces of ribbon, each of 25 yards' length.

Then in total she has 3·25=75 yards of ribbon.

Since 1 yd = 36 in, then 75 yd = 75·36=2700 in.

Now she needs to cut pieces that are 22 inches long, thus

[tex]2700:22=122\dfrac{16}{22}[/tex] pieces.

Answer: she can cut from the three pieces of ribbon 122 pieces that are 22 inches long.

Answer:

The answer is : -15 (C)

Step-by-step explanation:

–19 = x – 4

x - 4 = -19

x = -19 + 4

x = -15

Answer:

The constant of variation for the given quadratic equation is, 30

Step-by-step explanation:

One of the form of a quadratic equation is written as:

[tex]y = kx^2[/tex]     ....[1]

where k is the coefficient and for this case the constant of variation.

In order to obtain the answer for the given equation, we write the given equation to the form above.

[tex]0.1y=3x^2[/tex]  or

[tex]y=(\frac{3}{0.1})x^2[/tex] or

[tex]y=30x^2[/tex]

Comparing this equation with equation [1], to get the value of k;

k=30.

therefore, the  constant of variation is, 30.

Answer:

-3,2

Step-by-step explanation:

just took the test edge 2020

If the resort makes snow while it is snowing naturally then for 9 hours it will take until the run can open.

The four basic mathematical operations are the addition, subtraction, multiplication, and division of two or even more integers. Among them is the examination of integers, particularly the order of actions, which is crucial for all other mathematical topics, including algebra, data organization, and geometry.

As per the given information in the question,

The pace at which a resort creates snow is t/12,

the rate at which snow falls naturally is t/36.

(1/12) + (1/36) = (1/t)

Take LCM,

(3/36) + (1/36) = (1/t)

1/t = 4/36

4t = 36

t = 9 hours.

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The linear equation is defined as the equation, which can be represented in the form of y=mx+b.

The cost of one pizza is $ 8.99.

Given that,

Now, using the linear equation formula:

Total cost = Cost of three pizzas + Cost of Salad

Substituting the values, the equation becomes:

$ 38.95 = 3x + $ 11.98

$ 38.95 - $ 11.98 = 3x

$ 26.97 = 3x

x = [tex]\dfrac{26.97}{3}[/tex]

x = $8.99

Therefore, the value of each pizza is $8.99.

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Answer:

1.  the width of the carpet in the living room

Ans. The monomial, x, a factor of the expression x2 + 9x

2  the length of the carpet in the living room

Ans . the binomial, (x + 9), a factor of the expression x2 + 9x

3  the area of the carpet in the bedroom

Ans. the second-degree term of the expression x2 + 9x

4.  the increase in the area of carpet needed for the living room

Ans. the first-degree term of the expression x2 + 9x

Step-by-step explanation:

Jennifer has carpet in her square bedroom.

Formula of area of square = [tex]a^2[/tex]

Where a is the side length

Side length of her bedroom is x

Area of the bedroom is [tex]x^2[/tex]

So, area of carpet for bedroom is [tex]x^2[/tex]

She decides to also purchase carpet for her living room which is rectangular in shape and 9 feet longer than her bedroom. (no change in width)

So, length of living room = 9+x

Width = x

Area of living room = [tex]x*(9+x)[/tex]

                                 [tex]=x^{2} +9x[/tex]

So, increase in area of carpet for living room in comparison to the area of carpet for bedroom is 9x

1.  the width of the carpet in the living room

Ans. The monomial, x, a factor of the expression x2 + 9x

2  the length of the carpet in the living room

Ans . the binomial, (x + 9), a factor of the expression x2 + 9x

3  the area of the carpet in the bedroom

Ans. the second-degree term of the expression x2 + 9x

4.  the increase in the area of carpet needed for the living room

Ans. the first-degree term of the expression x2 + 9x