How many ways can we put 7 adults and 3 kids in line (where order matters) so that no two kids are next to each other?

By setting up a proportion based on the similar triangles formed by Peter's line of sight and the analogous line from the top of the house to the mirror, we find that the house is 25 feet tall.

According to the Law of Reflection and the information provided, we have two similar triangles: one formed by Peter's eyes, the mirror, and the point on the ground directly below his eyes, and the other formed by the top of the house, the mirror, and the point on the ground directly below the top of the house.

The first triangle has a height of 5 feet (Peter's eye level) and a base of 6 feet (the distance from Peter to the mirror). The second triangle's height is what we are trying to find (height of the house), and its base is 30 feet (distance from the house to the mirror).

Because the triangles are similar, the ratios of their corresponding sides are equal. So, we can set up the following proportion:

Height of Peter's eyes / Distance to mirror = Height of the house / Distance to mirror from house

5 feet / 6 feet = Height of the house / 30 feet

Now, cross-multiply to solve for the height of the house:

5 feet * 30 feet = Height of the house * 6 feet

Height of the house = (5 * 30) / 6

Height of the house = 150 / 6

Height of the house = 25 feet

Therefore, the height of the house is 25 feet.

Answer:

B

Step-by-step explanation:

no

Answer:

Rectangle and Square.  

Step-by-step explanation:

To know if the angles of a quadrilateral is congruent you must overlap these angles one over the other, if all elements of the angles coincide, the angles are congruent; if the elements do not match, the angles are not congruent. In quadrilateral rectangle and square consecutive angles are always congruent.

In other words, two or more angles are said congruent if and only if they have the same measurement.

From use of trigonometric ratios, it is seen that the height of the tree is; 22.61 feet

We are told that you are standing 30 ft from the base of a tree.

Now, the distance from where you are standing to the top of the tree will be the hypotenuse of the triangle formed by you and the tree.

Thus, if the height of the tree is h and the angle the hypotenuse makes with the 30 ft base is 37°, then we have;

h/30 = tan 37

h = = 30 * tan 37

h = 22.61 ft

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Rational numbers are numbers that are written as a ratio of two integers.

For instance, given the rational number 10/15, the simplest form of the rational number will be expressed as (5*2)/(5*3) = 2/3

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Answer:

  the numbers are 4, 8, 14

Step-by-step explanation:

If we subtract 6 from the third number and from the total, we find that we now have 5 times the first number for a total of 20.

The first number is 4.

The second number is 2×4 = 8.

The third number is 8 + 6 = 14.

Sandy can install a maximum of 24 fixtures in 29 hours.

To calculate the number of fixtures Sandy can install in 29 hours, we need to determine the amount of time it takes for Sandy to complete one installation process (excluding assembly time).

Sandy takes 2/3 hour to hang the fixture and another 1/6 hour to wire it, totaling

2/3 + 1/6 = 7/6 hour.

Therefore, Sandy takes 7/6 hour per fixture.

Next, we can calculate the number of fixtures by dividing the total time available (29 hours) by the time it takes for one installation process.

Number of fixtures = Total time available / Time per fixture

Number of fixtures = 29 / (7/6) = 29 * (6/7) = 174/7 = 24.86

Since we can't have a fraction of a fixture, Sandy can install a maximum of 24 fixtures in 29 hours.

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the volume of the hole would be  3 * 6 * 4 =72 cubic feet.

 However, since it is a hole, there would be no dirt in it.

When the age of Newton will be 25 years , the age of his father will be reversed i.e., 52 yrs.

It is given that when Newton was 14 years old , his fathers age was 41 yrs.

We have to find out that at what age of Newton his fathers age will be reversed again ?

Solve for the value of x ;  10 x - 14 = 12x - 28 ?

The value of x will be 7.

Let's assume the tens place as x and one's place as y.

The actual number will be ; 10 x + y.

Reverse numbers pairs can be written as (10x+y) and (10y+x).

As the age of Newton's father is 41 yrs when he is 14 yrs old , so we have ;

(10x+y) - (10y+x) = 41–14 = 27

Hence , we can say that the age difference between Newton and his father is 27 years. This age difference will be same forever.

So ;

(10x+y) - (10y+x) = 27

and we have to find all values of x and y that will satisfy this equation.

⇒ (10x+y) - (10y+x) = 27

⇒ 9x - 9y = 27

⇒ x - y = 3

For all x’s and y’s who have their difference as 3 will answer the question. For example , (4,1) (5,2) (6,3) (7,4) (8,5) …. and so on.

Since Newton's father age today is 41 yrs.

The next age to satisfy the condition is 52 yrs.

This makes Newton’s age to be 52 - 27 = 25 which is the reverse of 52.

So , the answer of the question is that Newton should be of 25yrs , for his age to be the reverse of his father’s age.

Thus , when the age of Newton will be 25 years , the age of his father will be reversed i.e., 52 yrs.

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The transformation (x, y) -> (5x, 5y) is a scaling transformation that dilates a point by a factor of 5 in both the x and y directions, making the entire figure five times larger while preserving its shape.

The transformation described by (x, y) -> (5x, 5y) is a scaling transformation or dilation in both the x and y directions by a factor of 5. What this means is that any point on the plane, when subjected to this transformation, will move five times further away from the origin along both axes. For example, if you begin with the point (1, 1), applying this transformation will result in the point (5, 5). This dilation affects all points equally, meaning that the relative distances between points are preserved, but the entire figure is made larger by a factor of 5. Therefore, if you have a figure with dimensions of length and width, after the transformation, both dimensions will be increased by five times their original measure.

Final answer:

The transformation (x, y) -> (5x, 5y) represents a dilation, enlarging a figure by a factor of 5 along both the x-axis and y-axis, which alters size but not shape.

Explanation:

The transformation described by the rule (x, y) -> (5x, 5y) is a geometric transformation known as a dilation or scaling. This type of transformation alters the size of a figure without changing its shape. Specifically, the coordinates of each point in the figure are multiplied by 5, which means that the figure is enlarged by a factor of 5 in both the x-axis (east-west direction) and the y-axis (north-south direction). If, for example, we apply this transformation to a point (1, 2), the resulting coordinates will be (5, 10), and similarly, applying it to (3, 7) would yield (15, 35). This illustrates the dependence of y on x as each x-coordinate and y-coordinate are scaled by the same factor, maintaining the shape but increasing the size.

Consider the equation [tex] x^6+6x^3+5=0 [/tex].

First, you can use the substitution [tex] t=x^3 [/tex], then [tex] x^6=(x^3)^2=t^2 [/tex] and equation becomes [tex] t^2+6t+5=0 [/tex]. This equation is quadratic, so

[tex] D=6^2-4\cdot 5\cdot 1=36-20=16=4^2,\ \sqrt{D}=4,\\ \\ t_{1,2}=\dfrac{-6\pm 4}{2} =-5,-1 [/tex].

Then you can factor this equation:

[tex] (t+5)(t+1)=0 [/tex].

Use the made substitution again:

[tex] (x^3+5)(x^3+1)=0 [/tex].

You have in each brackets the expression like [tex] a^3+b^3 [/tex] that is equal to [tex] (a+b)(a^2-ab+b^2) [/tex]. Thus,

[tex] x^3+5=(x+\sqrt[3]{5})(x^2-\sqrt[3]{5}x+\sqrt[3]{25}) ,\\x^3+1=(x+1)(x^2-x+1) [/tex]

and the equation is

[tex] (x+\sqrt[3]{5})(x^2-\sqrt[3]{5}x+\sqrt[3]{25})(x+1)(x^2-x+1)=0 [/tex].

Here [tex] x_1=-\sqrt[3]{5} , x_2=-1 [/tex] and you can sheck whether quadratic trinomials have real roots:

1. [tex] D_1=(-\sqrt[3]{5}) ^2-4\cdot \sqrt[3]{25}=\sqrt[3]{25} -4\sqrt[3]{25} =-3\sqrt[3]{25} <0 [/tex].

2. [tex] D_2=(-1)^2-4\cdot 1=1-4=-3<0 [/tex].

This means that quadratic trinomials don't have real roots.

Answer: [tex] x_1=-\sqrt[3]{5} , x_2=-1 [/tex]

If you need complex roots, then

[tex] x_{3,4}=\dfrac{\sqrt[3]{5}\pm i\sqrt{3\sqrt[3]{25}}}{2} ,\\ \\x_{5,6}=\dfrac{1\pm i\sqrt{3}}{2} [/tex].