Marcia and Tamara are running a race.

Marcia has run 4 kilometers.

Tamara has completed half of the race and is 2 kilometers ahead of Marcia.

Write an equation that represents the relationship between the distances each girl has run.

Let k represent the total length of the race in kilometers.

One efficient algorithm for finding the LCA in a tree is the Tarjan's off-line lowest common ancestor (LCA) algorithm. It uses DFS to preprocess the tree and build a table of ancestors for each node. The algorithm then uses this table to quickly find the LCA in constant time.

The running time of the algorithm is O(n) in both time and space, where n is the number of positions in the tree.

Here is a Java implementation of the Tarjan's off-line LCA algorithm:

class TreeNode {

 int val;

 List<TreeNode> children;

 public TreeNode(int val) {

   // constructor to initialize a node in the tree with a given value

   this.val = val;

   this.children = new ArrayList<>();

 }

}

public class LCA {

 private TreeNode root;

 private int[] depth;

 private int[] parent;

 private int[][] table;

 private int logN;

 public LCA(TreeNode root, int n) {

   // constructor to initialize the LCA class with a given root and number of nodes

   this.root = root;

   depth = new int[n];

   parent = new int[n];

   table = new int[n][20];

   logN = (int) (Math.log(n) / Math.log(2)) + 1;

   build();

 }

 private void build() {

   // preprocess the tree and build the ancestor table

   dfs(root, -1, 0);

   for (int i = 0; i < logN - 1; i++) {

     for (int j = 0; j < parent.length; j++) {

       table[j][i + 1] = table[table[j][i]][i];

     }

   }

 }

 private void dfs(TreeNode node, int p, int d) {

   // DFS to preprocess the tree

   parent[node.val] = p;

   depth[node.val] = d;

   for (TreeNode child : node.children) {

     dfs(child, node.val, d + 1);

   }

 }

 public int query(int p, int q) {

   // function to query the LCA of two nodes

   if (depth[p] < depth[q]) {

     int temp = p;

     p = q;

     q = temp;

   }

   int log = 1;

   while ((1 << log) <= depth[p]) {

     log++;

   }

   log--;

   for (int i = log; i >= 0; i--) {

     if (depth[p] - (1 << i) >= depth[q]) {

       p = table[p][i];

     }

   }

   if (p == q) {

     return p;

   }

   for (int i = log; i >= 0; i--) {

     if (table[p][i] != -1 && table[p][i] != table[q][i]) {

       p = table[p][i];

       q = table[q][i];

     }

   }

   return parent[p];

 }

}

And here is a test program:

public class Main {

 public static void main(String[] args) {

   TreeNode root = new TreeNode(0);

   TreeNode node1 = new TreeNode(1);

   TreeNode node2 = new TreeNode(2);

   Tree

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Answer:

Carmen's water bottle holds 1 8/10 liters of water, which can be written as 18/10 liters or 1.8 liters.

Jorge's water bottle holds 2 2/5 liters of water, which can be written as 22/5 liters or 4.4 liters.

Jorge's estimate of his water bottle holding about 1/2 liter more water than Carmen's is reasonable because:

4.4 liters (Jorge's bottle) - 1.8 liters (Carmen's bottle) = 2.6 liters

2.6 liters is close to the estimate of 1/2 liter, so Jorge's estimate is reasonable.

1st row

1. 2x + x + x, x = 0

Plug in 0 as x.

0 + 0 + 0 = 0

2. 2x + x + x, x = 1

Plug in 1 as x.

2 + 1 + 1 = 4

3. 2x + x + x, x = 2

Plug in 2 as x.

4 + 2 + 2 = 8

2nd row

1. 4x, x = 0

Plug in 0 as x.

4 * 0 = 0

2. 4x, x = 1

Plug in 1 as x.

4 * 1 = 4

3. 4x, x = 2

4 * 2 = 8

Notice how the answers are the same.

This is because 2x + x + x = 4x, meaning they are the same equation.

Answer:

Attached in file

Step-by-step explanation:

What does this mean by substitution?

The act of substituting in algebra / pre-algebra / etc is plugging in a number for 'x'

In this question, they are wanting you to plug in 0, 1, and 2 to two different equations (2x+x+x and 4x) to see if they are the same equations.

(I'm going to try and make a graph, as visual learning helps)

     | 2x+x+x       4x

x=0|      a             b

x=1 |       c             d

x=2|        e           f

Now I know that this is not perfect, but I inputted letters so that you know what I am solving for.

Solve for a:

Plug in x=0

2(0) + 0 + 0

2*0 = 0

0 + 0 + 0 = 0

So, a = 0 (What I am saying is the first blank is a, the blank next to it is b, etc)

Solve for b:

Plug in x=0

4(0) = 0

So, b = 0

Solve for c:

Plug in x=1

2(1) + 1 + 1

2 + 1 + 1

= 2  + 2

=4

b = 4

Solve for d:

Plug in x=1

4(1) = 4

d=4

Solve for e:

Plug in x=2

2(2) + 2 + 2

4 + 2 + 2

4 + 4

=8

e=8

Solve for f:

Plug in x=2

4(2)

f=8

There we go! I annotated this picture to have these values plugged into their respective spots!!

Hope this helped!!

The events with a probability of 1/2 regarding the letters of the word August are given as follows:

A probability is obtained by the division of the number of desired outcomes by the number of total outcomes in the context of a problem.

For the six letters in the word August, we have that:

Hence each event has a probability of 1/2, which is the desired probability for this problem.

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Answer:1/20

Step-by-step explanation:

trust

Answer:

1/20

Step-by-step explanation:

5 divided by 100 is =20

5divided by 5 is =1

=1/20

The area of a circle with a diameter of 16 meters is equal to: B. 64π square meters.

Mathematically, the area of a circle can be calculated by using this formula:

Area = πr²

Where:

r represents the radius of a circle.

Note: Radius = diameter/2 = 16/2 = 8 meters.

By substituting the radius into the formula for the area of a circle, we have the following;

Area of circle = π × 8²

Area of circle = 64π square meters.

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Answer:64 PI

Step-by-step explanation:

Answer: I'm sorry this question cannot be answered because of the lack of information, please resubmit your question with a picture of the page.

Step-by-step explanation:

We may multiply the y-values by -1 to flip the graph of y = (x - 2)^3 + 5 vertically. This will display the graph along the x-axis. The inverted graph's equation is y = -((x - 2)^3 + 5).

An equation is a formula in mathematics that expresses the equivalence of two expressions by linking them with the equals symbol =. In its most basic form, an equation is a mathematical statement that indicates that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical statement made up of two expressions joined by an equal sign is known as an equation. 3x - 5 = 16 is an example of an equation. We get the value of the variable x as x = 7 after solving this equation.

Here,

To flip the graph of y = (x - 2)^3 + 5 vertically, we can multiply the y-values by -1. This will reflect the graph across the x-axis. The equation for the flipped graph is: y = -((x - 2)^3 + 5).

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In the number 240.149, how does the value of the 4 in the hundredths place compare to the value of the 4 in the tens place, is ,the correct option is 1/1000.

Place value is the basis of our entire number system. This is the system in which the position of a digit in a number determines its value.

The number 42,316 is different from 61,432 because the digits are in different positions.

here,, we have,

i) the 4 in the hundredths place will give us the value 0.04 = 4/100

ii) the value of 4 in the tens place gives us the value = 4 × 10 = 40

iii) to compare the value of 4 in the hundredths place to the value of 4 in the tens place  we divide the value obtained in i) by the value obtained in ii).

Therefore we get the value of the comparison as = (4/100)/40

                                                                                  =1/1000

Therefore the correct option is 1/1000.

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The proof that the vertical angles provided in the image above are congruent is given below:

Statements                                                        Reasons

1. ∠2 and ∠4 are vert. angles                           (Given)

2. ∠2 and ∠3 are a linear pair                          (Definition of Linear Pair)

3. ∠3 and ∠4 are a linear pair                           (Definition of Linear Pair)

4. m∠2 + m∠3 = 180                                          Angle addition postulate

5. m∠3 + m∠4 = 180                                           Angle addition postulate

6. m∠2 + m∠3 = m∠3 + m∠4                             Subtraction property

7. m∠2 = m∠4                                                     Subtraction property

8. ∠2 ≅ 4                                                           Definition of Congruent Angles

When two lines connect, vertical angles are generated. The angles that are opposite each other in the four produced angles are vertical angles. They are also known as vertically opposed angles. These angles are never unequal.

Congruent angles are those of equal measure. Be a result, any angles of equal measure are referred to as congruent angles. They may be found in equilateral triangles, isosceles triangles, and where a transversal connects two parallel lines.

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Consider the simple given arrangement of three resistors, the total equivalent resistance of the arrangement will be smaller than the resistor C.

Please refer to the attached picture below for the give resistors arrangement.

The three resistors are arranged in a paralel circuit. Hence, the total equivalent resistance for the arrangement will be:

1/R total = 1/Ra + 1/Rb + 1/Rc

Where:

Ra = A

Rb = b

Rc = C

A < B < C

Then, the total resistance will be:

1/R total = 1/A + 1/B + 1/C

    1       = BC + AC + AB

R total              ABC

R total =           ABC      

                BC + AC + AB

Next, we will try to choose any random numbers that meet the resistors condition to proof this condition. This time, we choose:

A = 3

B = 2

C = 1

Then, we need to subtitute each number to its positions:

R total =          3 x 2 x 1      

                  2(1) + 3(1) + 3(2)

R total =          6    

               2 + 3 + 6

R total = 6/12

R total = 1/2 < C --> proven

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The domain of the given function would be -

- 5 ≤ x < 3.

An function in mathematics defines a relationship between one variable (the independent variable {x})  and another variable (the dependent variable {y}). Example → y = f{x} = ax + b.

Given is the graph of the linear equation as shown in the image.

Domain represents the set of values along the {x} - axis.

So, we can write -

- 5 ≤ x < 3

Therefore, the domain of the given function would be -

- 5 ≤ x < 3.

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Answer:

654.85:4

Step-by-step explanation:

just took the test!

Answer: y = -x

Step-by-step explanation:

slope = (-5-3)/(5+3) = -1

using the point (-3,3)

y-3 = -(x+3)

y-3 = -x - 3

y = -x

The value of the deposit after 7 years is given as follows:

$30,746.85.

The situation is modeled with an increasing exponential function, which has the format presented as follows:

y = a(1 + r)^x.

In which:

Considering the deposit and the yearly increase, the parameters are given as follows:

a = 25000, r = 0.03.

Hence the function is given as follows:

y = 25000(1.03)^x.

The value after 7 years is given as follows:

y = 25000(1.03)^7

y = $30,746.85.

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The equation 1 ≠ 2 is true because 1 and 2 do not have the same value

From the question, we have the following parameters that can be used in our computation:

The above statements are true

However, when both sides of the equation are multiplied by 0, we have

0 * 0/1 = 0 * 0/2

Evaluate

0 = 0

The result is 0 = 0 and not 1 = 2

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The simplest form of the square root is 14x⁷√2x.

To simplify simply means to make anything easier. In mathematics, simplifying an equation, fraction, or problem means taking it and making it simpler. Calculations and problem-solving techniques simplify the issue. By eliminating all common factors from the numerator and denominator and putting the fraction in its simplest/lowest form, we can simplify fractions.

Given the expression, [tex]\sqrt{(392x^{15} )}[/tex]

factors of 392 = 2*2*2*7*7

or factors of 392 = 2*14*14

and x¹⁵ = x¹⁴.x

substitute the values,

[tex]\sqrt{(2*14*14x^{14}x )}[/tex]

or √2x(√14²)(√x¹⁴)

= 14x⁷√2x

Hence option C is correct.

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Answer:

the equation of the line that contains the points (1, 1) and (3, 9) is y = 4x + 3.

Step-by-step explanation:

To find the equation of the line that contains the points (1, 1) and (3, 9), we can use the slope-point form of the equation of a line, which is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line, m is the slope of the line, and (x, y) are any other points on the line.

We can use the first point (1, 1) and the slope of the line to find the equation of the line:

m = (9 - 1) / (3 - 1) = 4

y - 1 = 4(x - 1)

y = 4x + 3

So, the equation of the line that contains the points (1, 1) and (3, 9) is y = 4x + 3.

Jen could send either a package of socks or the book.

Addition is one of the basic mathematical operations where two or more numbers is added to get a bigger number.

The process of doing addition is also called as finding the sum.

The table shows the items with weights which are already packed.

Total weight of the items already packed = 5.2 + 1.25 + 2.05 + 2.25

                                                                     = 10.75 pounds

But it is given that, package must weight less than 12 pounds.

Weight of the item that is being added must be less than weight = 12 - 10.75 = 1.25

Weight of sweatshirt is 1.25 pounds, so it can't be placed.

She could place either package of socks or book.

Hence Jen could any one of the following : a package of socks or book.

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The GCF of the two expressions is (y + 5).

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of each of the expressions. To find the GCF of two expressions, we need to factor out the common factors from each expression.

Given expressions: z(y + 5) - 3(y + 5)

We can start by factoring out (y + 5) from both terms:

z(y + 5) - 3(y + 5) = (y + 5)(z - 3)

So, the GCF of the two expressions is (y + 5).

The greatest common factor (GCF) is the largest positive integer that is a factor of two or more numbers. It is a useful concept in mathematics, especially in the study of numbers and their properties. The GCF helps us simplify expressions by cancelling out common factors and reducing the complexity of expressions. For example, the GCF of 12 and 18 is 6, meaning that 6 is the largest number that is a factor of both 12 and 18. To find the GCF, we can use prime factorization to identify the common factors and then multiply those factors to obtain the GCF.

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-√12 i is the complex conjugate of the complex number√-12

A complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit.

A conjugate of a complex number is another complex number which has the same real part as the original complex number and the imaginary part has the same magnitude but opposite sign.

It's important to remember that i²=-1

This means we can rewrite

√-12 as √12i² =√12i

For complex number z=x+iy the conjugate of complex number is z=x-iy

We have z=√-12 so conjugate is -√12 i

Hence, -√12 i is the complex conjugate of the complex number

√-12

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The slope of the line is 1/3.

Slope of a line is the ratio of the change in y coordinates to the change in the x coordinates of two points given.

Slope can also be defined as the tangent of the angle that the line is making with the positive X axis.

Slope of a line passing through (x, y) and (x', y') is given by,

Slope = (y' - y) / (x' - x)

Consider the points on the given line.

The line passes through (0, 2) and (3, 3).

Slope = (3 - 2) / (3 - 0)

         = 1/3

Hence the slope of the given line is 1/3.

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The perimeter of the figure is 4(π + 9)ft.

A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length.

Given is a 2 - D figure as shown in the figure attached.

We can write the perimeter as -

P = P{rectangle} + P{semicircle}

P = 2(10 + 8) + (4π)

P =  (4π) + (36)

P = 4(π + 9)

Therefore, the perimeter of the figure is 4(π + 9)ft.

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The graph of the function f(X)=2(3)^x is the graph a

From the question, we have the following parameters that can be used in our computation:

f(x) = 2(3)^x

The above equation is an exponential function

An exponential function is represented as

y = ab^x

Using the above as a guide, we have the following:

a = 2

b = 3

The graph that has this prperty is (a)

Hence, the graph is a

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Answer:

$38.4

Step-by-step explanation:

In other words, a 20% discount for a item with original price of $48 is equal to $9.6 (Amount Saved).  - Note that to find the amount saved, just multiply it by the percentage and divide by 100.

$48 - $9.6 (amount saved) = $38.4

Π(x) is asymptotic to x/ln(x) as x approaches infinity.

The prime counting function (denoted as Π(x)) is the number of prime numbers less than or equal to x. The prime number theorem states that Π(x) is asymptotic to x/ln(x) as x approaches infinity. This can be conceptually seen in the graph of Π(x) and x/ln(x), which converges as x increases.

Formally, the prime number theorem states that:

lim x→∞ Π(x)/(x/ln(x)) = 1

This can be shown by the following calculation:

lim x→∞ Π(x)/(x/ln(x))

= lim x→∞ Π(x) ln(x)/x

= lim x→∞ 1/x Σ ln(p)/p

= lim x→∞ 1/x Σ 1/p

= lim x→∞ 1/x ln(x)

= 1

Therefore, Π(x) is asymptotic to x/ln(x) as x approaches infinity.

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The scatterplot shows a linear relationship between left and right forearm lengths in cm, for 55 college students.

The regression line is the best fit line that describes the relationship between the two variables. The formula for a linear regression line is  y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of the line in the scatterplot is 0.95, which indicates that for every 1 cm increase in the left arm, there is a 0.95 cm increase in the right arm. The y-intercept of the regression line is 8.4, meaning that when the left arm is 0, the right arm is 8.4 cm long. The equation for the regression line in the scatterplot is y = 0.95x + 8.4. This equation can be used to predict the right arm length for any given left arm length.

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