Find the length of YZ

Answer:

108 cm³

Step-by-step explanation:

V = a²(h/3) = 6²(9/3) = 36·3 = 108 cm³

The sum of the expressions -x² + 2 and 3x² - 7x + 8 is 2x² - 7x + 10

An equation is an expression that shows the relationship between two or more numbers and variables using mathematical operations. An equation can be linear, quadratic, cubic and so on, depending on the degree of the variable.

Find the sum of -x² + 2 and 3x² - 7x + 8. The sum of the expression is:

= (-x² + 2) + (3x² - 7x + 8)

Collecting like terms:

= -x² + 3x² - 7x + 2 + 8

Simplifying:

= 2x² - 7x + 10

The sum is 2x² - 7x + 10

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Answer:

Step-by-step explanation:

4) SinC = 30/34

C = sin^-1(30/34)

C = 61.93 degrees

A = 90-61.93

A = 28.07 degrees

5) SinA = 24/26

A = sin^-1(24/26)

A = 67.38

The measure of <D is 130 degree.

Angle sum property of triangle states that the sum of interior angles of a triangle is 180°.

Given:

<C= (x-4)

and, <D = 9x+ 4

and, <E = 2x+ 12

Now, In triangle CDE using Angle Sum Property

<D+ <E+ <F= 180

x - 4+ (9x+ 4) + 2x+ 12 = 180

12x + 12 = 180

12x = 168

x= 168/12

x = 14

So, the measure of <D = 9x+ 4 = 9(14) + 4 =  130 degree.

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The table of the statements and justifications used to prove that the quadrilateral GEOM is a parallelogram can be completed as follows;

            Statements           [tex]{}[/tex]                           Justifications

1.        [tex]{}[/tex]ΔGEO ≅ ΔOMG                                  1. Given

2. ∡EGO ≅ ∡GOM, ∡MGO ≅ ∡GOE [tex]{}[/tex]          2. CPCTC

3. [tex]\overline{GM}[/tex]║ [tex]\overline{EO}[/tex], [tex]\overline{GE}[/tex] ║ [tex]\overline{MO}[/tex]      [tex]{}[/tex]             3. Converse of alt. int. ∠s Theorem

4. GEOM is a parallelogram[tex]{}[/tex]                         4. Definition of a parallelogram

A parallelogram is a quadrilateral that have opposite parallel sides.

The details of the justifications in the table used to prove that quadrilateral GEOM is a parallelogram are presented as follows;

CPCTC

CPCTC is an acronym for Corresponding Parts of Congruent Triangles are Congruent, which indicates that if teo triangles are congruent, then the corresponding sides and angles in both triangles are also congruent.

Converse of the alt. int ∠s theorem

The converse of the alternate interior angles theorem states that if the alternate interior angles formed by two lines and their common transversal are congruent, then the lines are parallel.

The angles ∡EGO, and ∡GOM are alternate interior angles, similarly, the angles ∡MGO and ∡GOE are alternate interior angle.

Whereby ∡EGO ≅ ∡GOM and ∡MGO ≅ ∡GOE, according to the convderse of the alternate interior angles theorem, the lines [tex]\overline{GE}[/tex] is parallel to [tex]\overline{MO}[/tex], and [tex]\overline{GM}[/tex] is parallel to [tex]\overline{EO}[/tex],  [tex]\overline{GE}[/tex] ║ [tex]\overline{MO}[/tex], and [tex]\overline{GM}[/tex] ║ [tex]\overline{EO}[/tex]

Definition of a parallelogram

The definition of a parallelogram, indicates that, where  [tex]\overline{GE}[/tex] ║ [tex]\overline{MO}[/tex], and [tex]\overline{GM}[/tex] ║ [tex]\overline{EO}[/tex], then the quadrilateral GEOM is a parallelogram.

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(C) It will take him the most time to snowball his payments based on the highest interest rate.

Customers (cardholders) are given a credit card, a sort of payment card, to enable them to pay a business for goods and services based on the amount of debt they have accrued (i.e., promise to the card issuer to pay them for the amounts plus the other agreed charges).

A revolving account is set up by the card issuer, which is frequently a bank or credit union and gives the cardholder access to a line of credit from which they can borrow money to make purchases or get cash advances.

Consumer credit cards and business credit cards are the two types of credit cards.

Most cards are made of plastic, but others are made of metal (stainless steel, gold, palladium, titanium), and some have jewels inlaid in the metal.

As a result, Fred currently owes money on four separate credit cards.

Paying off your lowest obligations first, then going on to the next smallest, and so forth, is known as snowballing.

Debt avalanche, an alternative approach, recommends paying off debts with the highest interest rates first.

Therefore, (C) It will take him the most time to snowball his payments based on the highest interest rate.

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Correct question:
Fred has an outstanding balance on four different credit cards. Which method of paying off his credit card debt will take him the longest period of time?

A. Paying the minimum balance each month.

B. Making fixed payments.

C. Snowballing his payments according to the highest interest rate.

D. Snowballing his payments according to the lowest balance.

The inequality that represents the given situation is 8x + 10y ≥ 600

and, x ≥ 40.

Mathematical expressions with inequalities are those in which the two sides are not equal. Contrary to equations, we compare two values in inequality. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs are used in place of the equal sign.

Given:

The cost for one calendar = $8

The cost for one coffee mug = $10

Dominique must sell at least $600 of merchandise and it should include at least 40 calendars.

let x be the number of calendars and y be the number of coffee mugs;

then, the inequality is

8x + 10y ≥ 600

and, x ≥ 40

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Answer:

the last one, option D, 8x + 10y ≥ 600 and x ≥ 40

Step-by-step explanation:

took the test xx

Answer:

5609/7 = 801.3 (rounded)

Step-by-step explanation:

Answer:801.2857143

Step-by-step explanation: Remember to enter it in your calculator, if you get a fraction as an answer click shift and then = there you go

The volume of the pyramid is 400 cubic units.

A pyramid is a structure where outer surfaces are triangular and converge to a point at the top.

The volume of a pyramid with a square base is given as:

Volume = 1/3 x base area x height

We have,

Let's call the height of the pyramid "h".

We can use the Pythagorean theorem to find the slant height of the pyramid, which is the length of one of the edges from the apex to a corner of the base.

To do this, we can consider one of the right triangles formed by the apex, the midpoint of a side, and a corner of the base.

The hypotenuse of this right triangle is the segment connecting the apex to the midpoint of a side, which has a length of 13 units.

One leg of the right triangle is half of one side of the square base, which is 5 units.

The other leg of the right triangle is the height of the pyramid, which we want to find. So, we can write:

h² + 5² = 13²

Simplifying and solving for h, we get:

h²= 13² - 5²

h² = 169 - 25

h² = 144

h = 12

So, the height of the pyramid is 12 units.

To find the volume of the pyramid, we can use the formula:

Volume = (1/3) x Base Area x Height

The base of the pyramid is a square with sides of length 10 units, so its area is:

Base Area = 10² = 100 square units

Plugging in the values we know, we get:

Volume = (1/3) x 100 x 12

Volume = 400 cubic units

Therefore,

The volume of the pyramid is 400 cubic units.

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The number line that shows the solution set of the inequality is B

In math, an inequality is a statement that describes a relationship between two values or expressions that are not equal. An inequality is written using mathematical symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to).

-1/2x ≥ 4

we are to get the value of x

-x ≥ 4 ÷ 1/2

- x ≥ 4 x 2

-x ≥ 8

x ≤ -8

The provided inequality is represented by an arrow with a solid point at x = -8 and an arrow pointing in the direction of negative values.

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Answer:

20/29

Step-by-step explanation:

Recall that cosine of an angle is the side adjacent to it over the hypotenuse (SOH-CAH-TOA).
For this question the ratio is: 20/29

Answer:

  D.  1/2

Step-by-step explanation:

Given the side adjacent to an angle is 25 and the tangent of the angle is 1/5, you want to know the tangent of an angle with the same opposite side, but an adjacent side of 10.

The tangent of an angle is the ratio of the opposite side to the adjacent side.

  Tan = Opposite/Adjacent

Then the Opposite side is ...

  Opposite = Adjacent · Tan(x) = 25·1/5 = 5

For angle y, the opposite side is the same, but the adjacent side is only 10:

  Tan(y) = Opposite/Adjacent = 5/10

  tan(y) = 1/2

(a) We can represent Kristin's height above the ground (in meters) when she has traveled 24.9 meters around the Ferris wheel as f(24.9).

(b) We can represent Kristin's height above the ground (in meters) when she has traveled 49.71 meters around the Ferris wheel as f(49.71).

(a) We have to determine how can we represent Kristin's height above the ground (in meters) when she has traveled 24.9 meters around the Ferris wheel.

We can create the a table and function to describe that statement but F(24.9) is the best representation of Kristin's height above the ground (in meters) when she has traveled 24.9 meters around the Ferris wheel.

(b) We have to determine how can we represent Kristin's height above the ground (in meters) when she has traveled 49.71 meters around the Ferris wheel.

We can create the a table and function to describe that statement but F(49.71) is the best representation of Kristin's height above the ground (in meters) when she has traveled 49.71 meters around the Ferris wheel.

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The complete question is:

In the previous modules we developed formulas, tables, and graphs to represent how two varying quantities change together. For example, we considered how Kristin's height above the ground changed as her total distance traveled increased while riding a Ferris wheel.

We even assigned letters to represent the possible varying values each quantity can assume.

• Let d represent the distance Kristin has traveled (in meters) around the Ferris wheel starting at the bottom.

• Let h represent Kristin's height above the ground (in meters).

Note that the way we are thinking about this relationship is that we vary Kristin's distance traveled and observe what happens to her height above the ground.

What if you want to communicate information about her position after she traveled some number of feet? You could write it out in words, like "Kristin was 16.3 meters above the ground when she had traveled 23.7 meters around the Ferris wheel". However, that's a lot to have to write out, especially if you want to talk about multiple values. You could also draw a table or create a graph, but mathematicians developed a tool called function notation to help them represent values in a covarying relationship.

[Note: We will define the exact meaning of "function" later in this investigation. For now, just think of "function" as meaning the same thing as "relationship".]

First, we pick a letter to represent the name of the relationship. Let's call this relationship "f". Now, instead of saying "the relationship between Kristin's height above the ground (in meters) and her total distance traveled (in meters)" we can simply say "relationship f".

Also, we can use f(d) to represent values of Kristin's height above the ground (values of h). We read this as "f of d", and the parentheses here are just part of the notation. They do NOT indicate multiplication.

For example, f(13) represents Kristin's height above the ground (in meters) when she has traveled 13 meters around the Ferris wheel.

a. How can we represent Kristin's height above the ground (in meters) when she has traveled 24.9 meters around the Ferris wheel?

(Hint: you should use function notation.) Preview syntax error Enter an algebraic expression (more..]

b. How can we represent Kristin's height above the ground (in meters) when she has traveled 49.71 meters around the Ferris wheel? x Preview

The probability that the person reaches 10 before he reaches -5 is 0.475.

The probability of reaching 10 before reaching -5 in a random walk can be found using the concept of expected number of steps to reach a state. In this case, we can calculate the expected number of steps required to reach 10 and -5, and then compare the results.

E10 = 1 + 1/2 × E10 + 1/2 × E9

E9 = 1 + 1/2 × E10 + 1/2 × E8

Substituting the expression for E9 into the equation for E10, we get:

E10 = 1 + 1/2 × E10 + 1/2 × (1 + 1/2 × E10 + 1/2 × E8)

E10 = 1 + 3/4 × E10 + 1/4 × E8

Eliminating E10, we get:

E8 = 4 × (E10 - 1)

Substituting the expression for E8 into the equation for E10, we get:

E10 = 1 + 3/4 × E10 + 1/4 × 4 × (E10 - 1)

E10 = 20

So the expected number of steps to reach 10 is 20.

E-5 = 1 + 1/2 × E-4 + 1/2 × E-5

E-4 = 1 + 1/2 × E-5 + 1/2 × E-3

Substituting the expression for E-4 into the equation for E-5, we get:

E-5 = 1 + 1/2 × (1 + 1/2 × E-5 + 1/2 × E-3) + 1/2 × E-5

E-5 = 1 + 3/4 × E-5 + 1/4 * E-3

Eliminating E-5, we get:

E-3 = 4 × (E-5 - 1)

Substituting the expression for E-3 into the equation for E-5, we get:

E-5 = 1 + 3/4 × E-5 + 1/4 × 4 × (E-5 - 1)

E-5 = 26

So the expected number of steps to reach -5 is 26.

The probability of reaching 10 before reaching -5 is calculated by dividing the expected number of steps required to reach 9 (E9) by the expected number of steps required to reach 10 (E10). The reason for this is that the person will reach 9 after taking 19 steps, which is one step short of reaching 10. Hence, we can consider reaching 9 as an intermediate milestone before reaching 10.

P(10 before -5) = E9 / E10

Since the person has a 1/2 probability of moving one step in either direction, the expected number of steps required to reach 9 (E9) is given by:

E9 = 1 + 1/2 * E10 + 1/2 * E8

E9 = 1 + 1/2 * 20 + 1/2 * E8

E9 = 19/2

So, the probability of reaching 10 before reaching -5 is given by:

P(10 before -5) = E9 / E10

P(10 before -5) = 19/2 / 20

P(10 before -5) = 19/40

P(10 before -5) = 0.475

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The both groups are not same

What is Group?

In mathematics, a group is a set that contains an operation that allows any two members to be connected to form a third element in such a way that the operation is associative, an identity element will be determined, and every element has its inverse. These three requirements, which are group axioms, are valid for number systems as well as numerous other mathematical structures. For instance, the integer set and addition operation are two examples of groups. Consequently, a group is a mix of a set and a binary operation, to put it simply.

let group formed by the symmetries of a rectangle = 9

9 = {RO, R1201 H, V}

Ro rotation through O' H→ reflection in a horizontal line. Rigo Jutation through 180° V=> ref. in a vertical line.

Each group has 4 elements, they are not the same

Reason: - (Z4, +) has the features of order 4 which is one bar and 3 bar but R is empty when ordering 4.

So they are not the same.

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3/ \s7 \s + 2/ \s7 \s = 3 + 2/ \s7 \s = 5/ \s7

For the purposes of addition, subtracting, and comparing fractions, it is appropriate to convert both fractions to a common (equal, same) denominator. LCM(7, 7) = 7, which is the least common multiple of both denominators, can be used to calculate the common denominator. Adding the denominators together yields the common denominator, which need not be the lowest: 7 x 7 = 49. It cannot further simplify the fraction result by cancelling in the next intermediate step.

In other words, adding three and two sevenths results in five sevenths.

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The required △QST is similar to △URT by the AAA similarity postulate.

Similar triangles are those triangles that have similar properties,i.e. angles and proportionality of sides.

Here,

The question seems to be incomplete, assuming that triangles are equilateral triangles.

Consider the triangle,
△QST and △URT.
∠Q = ∠U     [as meaure of angle is 60°]
∠S = ∠R      [as meaure of angle is 60°]
∠T = ∠T      [as meaure of angle is 60°]

So,
△QST ~△URT by AAA similarity postulate.

So,
QS/UR = ST/RT  = QT/UT

Thus, the required △QST is similar to △URT by the AAA similarity postulate.

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Two events each have probability 0.2 of occurring. the probability that neither occur is 0.6. these events are: Mutually Exclusive Events. Thus, option C is correct.

Probability is the mathematical study of outcomes in uncertain situation. It is used to describe the likelihood of an event occurring and it can be expressed as number between 0 to 1. Probability is used to quantify the uncertainty associated with any given event it helps us understand how likely it is that something will happen and it can be used to predict future.

The two events in question are independent of one another, meaning that the occurrence of one event does not affect the probability of the other occurring. This means that the probability of neither event occurring is the product of the two probabilities, Therefore these events are Mutually Exclusive Events.

Thus, option C is correct.

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The temperature of the hot chocolate will decrease over time and after 4 minutes, it will be around 56 degrees Fahrenheit.

When you put something hot into a freezer, the temperature of the item will decrease over time until it reaches the temperature of the freezer. The rate of temperature decrease depends on the difference between the original temperature and the temperature of the freezer. In this case, the difference is 190 degrees Fahrenheit. The rate of temperature decrease should be roughly 1 degree Fahrenheit per second. After 4 minutes, the temperature of the hot chocolate should be around 56 degrees Fahrenheit. This is because, after 4 minutes, the hot chocolate would have lost around 190-56 = 134 degrees Fahrenheit of temperature. To be more precise, you would have to factor in how quickly the freezer can cool the hot chocolate, which is usually related to the size of the freezer, the size of the container the hot chocolate is in, and the amount of hot chocolate.

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There were 3.75g of sodium in the beans Kelley used and the estimated value for the sodium is 4g

weight of sodium in 1 cup of beans = 0.75

weight of sodium in 5 cups of beans = 0.75 × 5 = 3.75

estimate for the sodium used = 0.8g

therefore, ratio of the actual weight of sodium and the estimated value is

ratio = [tex]\frac{0.75}{0.8}[/tex]

lets assume ratio is x then,

[tex]\frac{3.75}{x}[/tex]   =  [tex]\frac{0.75}{0.8}[/tex]

x = 3.75 × [tex]\frac{0.8}{0.75}[/tex]

x = [tex]\frac{3.75}{0.75}[/tex] × 0.8

x = [tex]\frac{375}{75}[/tex] × 0.8

x = 5 × 0.8

x = 4.0 or 4

∴ the estimated answer is 4 and the weight of sodium used is 3.75g

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1968 of the $10 dollar ticket were sold by jazz & 6873  of the $20 dollar ticket were sold by jazz.

Jazz total amount received = $157,437

Total tickets sold = 8,856

Let the $10 tickets sold = x

The $20  tickets sold = 8,856 - x

Here the solution below :

$10(x) + 8,856 - x($20) = $157,437

$10x + $177120 - $20x = $157,437

-$10x = $157,437 - $177120

-$10x = - $19683

x = 1968.3

1968 of the $10 dollar ticket were sold by jazz & 6873  of the $20 dollar ticket were sold by jazz.

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The volume of the pyramid is 66.7 cm cube. Therefore, the answer is c.

The above diagram is a square base pyramid. The volume of the square base pyramid can be represented as follows:

volume of the pyramid = 1 / 3 BH

where

Therefore,

base area = l²

where

Hence,

base area = 5²

base area = 25 cm²

Therefore, let's find the volume of the pyramid.

Hence,

volume of the pyramid =  1 / 3 × 25 × 8

volume of the pyramid =  200 / 3

volume of the pyramid = 66.6666666667

Therefore,

volume of the pyramid = 66.67 cm³

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Answer:

30/57   29/54

Step-by-step explanation:

The numerator is the number on top of the fraction

the denominator is the number on the bottom

[tex]\frac{30}{57}[/tex]  30 is the numerator

    57 is the denominator

Answer:

[tex]\frac{30}{57}[/tex]

[tex]\frac{29}{54}[/tex]

Step-by-step explanation:

The numerator of a fraction is the number that is above the line and the denominator of a fraction is the number that is below the line. We can use this to help find fractions with only the numerator and denominator.

When 30 is the numerator and 57 is the denominator.

[tex]\frac{30}{57}[/tex]

When 29 is the numerator and 54 is the denominator.

[tex]\frac{29}{54}[/tex]

The four integers are 5, 5, 12 and 14.

A data set's mean (average) is calculated by summing all of the numbers in the set, then dividing by the total number of values in the set.

When a data collection is ranked from least to greatest, the median is the midpoint. The number that appears most frequently in a data set is called the mode.

Given:

Mean= 9

Median = 8.5

Mode = 5

Range = 9.

let the numbers arranged in ascending order are a,b,c,d

Median: b+ c /2 = 8.5

b+ c = 17 (number in the middle)

Mode: a = 5, b= 5 {because mode is the most repeating data}

Range: d- a = 9 (last one minus first one)

d= 9 + 5

d= 14

Now, b+c = 17

c= 17-5

c= 12

So, the data is 5 ,5, 12, 14.

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Rounding to 2 decimal places, the shaded area is 234 cm².

What is the radius of circle?

The radius of a circle is the distance from the center of the circle to any point on the circumference of the circle. It is half of the diameter of the circle.

The diameter of the circle is 0.2 meters, which means the radius of the circle is 0.2 / 2 = 0.1 meters.

The height of the square is a fifth of the diameter of the circle, so the height of the square is 0.2 / 5 = 0.04 meters.

To find the shaded area in cm, we need to find the area of the circle and subtract the area of the square. The area of the circle is given by the formula πr², where r is the radius of the circle. The area of the square is simply the height multiplied by the width, which is equal to the diameter of the circle.

So, the area of the circle is π * 0.1² = 0.0314 square meters. The area of the square is 0.2 * 0.04 = 0.008 square meters. The shaded area is the difference between these two, or 0.0314 - 0.008 = 0.0234 square meters.

To convert the area to cm², we need to multiply by 10⁴. The shaded area in cm² is 0.0234 * 10⁴ = 234 cm².

Rounding to 2 decimal places, the shaded area is 234 cm^2.

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As the firm develops a specific sort of sensor and ships them in boxes of four, each sensor weighed 10 grams at first.

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 phrase with two equal sides separated by an equal sign is called an equation. An example of an equation is 4 + 6 = 10.

Here,

Let's call the original weight of each sensor "w". The total weight of 4 sensors would be 4w.

With the new design, each sensor weighs 9 grams more, so each sensor weighs w + 9 grams. The total weight of 4 sensors is 76 grams, so we can write an equation:

4(w + 9) = 76

Expanding the left-hand side and solving for w, we have:

4w + 36 = 76

4w = 40

w = 10

So each sensor originally weighed 10 grams as company manufactures a special type of sensor, and packs them in boxes of 4 for shipment.

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The amount of fabric that Chloe cut is correct.

In general, the term "proportion" refers to a part, share, or amount that is compared to a total.

According to the concept of proportion, two ratios are in proportion when they are equal.

A mathematical comparison of two numbers is called a proportion. According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio. "::" or "=" are symbols used to indicate proportions.

Given:

The directions on a sewing pattern say to cut an extra 15% of fabric to account for an error.

and Chloe needs a 0.75 yard of fabric to make a skirt, so she cuts 0.1125 yard.

The amount of Fabric needed

0.75 x 1.15 = X

0.8625 = X

and, Chloe cut the fabric

= 0.8625 - 0.1125

= 0.75

Thus, the amount of fabric that Chloe cut is correct.

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The cost of the border that will go around the room is 104 dollars.

Khalil's living room is 5 yards wide and 8 yards long. He wants to put a border around the top of the room.  The cost of the border is $4.00 per yard. The cost to buy enough of the border to go around the room can be calculated as follows;

The border around the room is the perimeter of the room.

Therefore,

perimeter of the room = 2(l + w)

where

Hence,

perimeter of the room = 2(8 + 5)

perimeter of the room = 2(13)

perimeter of the room = 26 yards

Hence,

1 yard = 4 dollars

26 yards = ?

cross multiply

cost to border the room = 26 × 4

cost to border the room = 104 dollars

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the square root of the number equal or greater than 25 are 626 and 700.

What is inequality?

Mathematical expressions with inequalities on both sides are known as inequalities. In an inequality, we compare two values as opposed to equations. In between, the equal sign is changed to a less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

Olivia is chosen for the 12U softball team. What age is Olivia? Because it doesn't mention "equals," you are unaware of Olivia's age. But since you already know Olivia's age to be below or equal to 12, you can write it as Olivia's Age 12. This scenario relates to disparities in the real world.

The average slope over the longer intervals will be the average of the slopes in the (same length) intervals that make up the longer interval.

The given expression √x ≥ 25

So the square root of the number equal or greater than 25 are 626 and 700.

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LN is tangent to circle O at point M and QM is a diameter. The measure of ∠QML and ∠PMN are 90° and 63° respectively.

1) To Determine the degree measurement of ∠QML

Based on the Perpendicular Tangent Theorem, tangent lines are always perpendicular to a circle's radius at the point of intersection

So, the radius OM is perpendicular to line LN at point M.

Hence,

The value of ∠QML = 90°

2) To determine the degree measurement of angle PMN

∠QMN = ∠QMP + ∠PMN

By the angle addition postulate, we can calculate this measurement as follows:

∠QMN = 90°

∠QMP = 27°

Replace with and substitute the value of angle QML into the equation.

90° = 27° + ∠PMN

∠PMN = 90° - 27° = 63°

Therefore, the value of  ∠PMN is 63°

Learn more about perpendicular tangent theorem here brainly.com/question/30306156

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