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Solve the system using elimination.
5x + 4y = 12
3x - 3y = 18
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The length of Mid segment QT is 57.56 units.

The incenter of a triangle can be discovered by drawing the triangle's angle bisectors and then locating the location of their intersection. This is possible with the aid of a compass.

The three line segments and the three lines that include those segments are equally spaced from the incenter and constitute each of the three sides of the triangle.

the single point where the three internal angle bisectors of a triangle meet and which also serves as the circumference of the circle around which it is inscribed.

The three line segments and the three lines that include those segments are equally spaced from the incenter and constitute each of the three sides of the triangle.  

ΔQUT is right triangle

QU = 55

UT= 17

Using Pythagoras Theorem

QT² = QU² + UT²

QT² = 55² + 17²

QT = √3314

QT = 57.56

Thus, The length of Mid segment QT is 57.56 units.

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By applying the rule of sin, it can be concluded that the length of the sides of the triangle is 15.48 cm, 39.68 cm, and 44.84 cm.

Triangle is a 2D shape that is bounded by three sides and has three vertices. The perimeter of a triangle is the sum of the lengths of the three sides of the triangle.

Perimeter = a + b + c, where a, b, and c are the sides of the triangle

One of the most important properties of a triangle is that the sum of the angles in a triangle equals 180°.

The rule of sin describes the ratio of the angles corresponding to the side lengths of the sine function.

a / sin A = b / sin B = c / sin C, where A, B, and C are the angles of the triangle

If the ratio of the angles is 1 : 3 : 5 and let x be the smallest angle, then we can put this information into the following equation:

Total angle = A + B + C

           180° = x + 3x + 5x

           180° = 9x

                x = 20°

Then we can calculate the other angles:

A = 20°

B = 3x = 3 * 20° = 60°

C = 5x = 5 * 20° = 100°

Now we put the value of A, B, and C into the rule of sin equation:

  a / sin A = b / sin B

a / sin 20° = b / sin 60°

   a / 0.34 = b / 0.87

              a = 0.34 * b / 0.87

              a = 0.39b

    c / sin C = b / sin B

c / sin 100° = b / sin 60°

    c / 0.98 = b / 0.87

               c = 0.98 * b / 0.87

               c = 1.13b

Then we go back to the perimeter formula:

perimeter = a + b + c

          100 = 0.39b + b + 1.13b

          100 = 2.52b

              b = 39.68 cm

Then we can calculate the length of the other sides, a and c:

a = 0.39b

  = 0.39 * 39.68

  = 15.48 cm

c = 1.13b

  = 1.13 * 39.68

  = 44.84 cm

Thus, the length of the sides of the triangle is 15.48 cm, 39.68 cm, and 44.84 cm.

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Answer: a. The independent variable in the equation C(a) = 5a is a. It represents the number of art classes Amir attends.

b. The dependent variable in the equation C(a) = 5a is C(a). It represents the cost Amir incurs as he attends art classes. The dependent variable is dependent on the independent variable because the cost will change as the number of classes attended changes.

The coordinate of the circumcentre of ΔABC is (-1, 0.15).

It is a two-dimensional figure which has three sides and the sum of the three angles is equal to 180 degrees.

We have,

Triangle ABC.

A = (-5, -4)

B = (3, -4)

C = (-1, 6)

The circumcentre of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect.

Now,

Let the circumcentre be M = (x, y).

The length from each side of the triangle to the circumcentre is equal.

So,

AO² = BO² = CO²

Now,

AO² = BO²

(x + 5)² + (y + 4)² = (x - 3)² + (y + 4)²

x² + 10x + 25 + y² + 8y + 16 = x² - 6x + 9 + y² + 8y + 16

10x + 25 = -6x + 9

16x + 16 = 0

x + 1 = 0

x = -1

BO² = CO²

(x - 3)² + (y + 4)² = (x + 1)² + (y - 6)²

x² - 6x + 9 + y² + 8y + 16 = x² + 2x + 1 + y² - 12y + 36

-6x + 9 + 8y + 16 = 2x + 1 - 12y + 36

20y - 8x + 25 - 36 = 0

20y - 8x - 11 = 0

20y = 11 + 8x

y = (11 + 8x)/20

y = (11 - 8)/20

y = 3/20

y = 0.15

Thus,

Circumcentre M = (-1, 0.15)

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Answer: To determine which loan(s) meet Jeremy's criteria of no more than an effective rate of 7.000% annually, we need to calculate the effective rate for each loan.

The effective rate is the true annual interest rate that a borrower will pay on a loan, taking into account the frequency of compounding.

Loan A: 6.785% normal rate compounded semi-annually

To calculate the effective rate, we can use the formula:

(1+ (nominal rate/2))^2 - 1

Effective rate = (1+ (6.785/2))^2 - 1 = 6.897%

Loan B: 6.795% nominal rate compounded, manually

Since the loan is compounded manually, the effective rate is the same as the nominal rate.

Effective rate = 6.795%

Loan C: 6.659% nominal rate, compounded weekly

To calculate the effective rate, we can use the formula:

(1+ (nominal rate/number of compounding periods per year))^(number of compounding periods per year) - 1

Effective rate = (1+ (6.659/52))^52 - 1 = 6.759%

Since 6.897% > 7.000% and 6.795% > 7.000% and 6.759% < 7.000%. So Loan C with 6.659% nominal rate, compounded weekly meets Jeremy's criteria.

Step-by-step explanation:

A.BC BD must be true in order to demonstrate that triangles ABD and ABC are congruent by SSS.

The SSS Congruence Theorem requires three pairs of corresponding sides to be congruent for two triangles to be considered congruent to one another.

We may infer the two triangles' characteristics from the provided image by looking at them:

AB ≅ AB (reflexive property) (reflexive property)

AC ≅ AD (provided) (given)

The two pairs of corresponding congruent sides are so present. As a result, for SSS to demonstrate that ABD ABC, the following must be true: A. BC BD.

A.BC BD must be true in order to demonstrate that triangles ABD and ABC are congruent by SSS.

The complete question is,

Have a look at the triangles below. In order to demonstrate that ABDABC, which of the following would also need to be true?

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The average rate of change of these function f(x), g(x), h(x) on interval [4, 7] from least to greatest is f(x)<g(x)<h(x) with values A1 = -7.5 < A2 = -6 < A3 = -3.

What is a function?

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.

Average rate of change of any function say S(x) on [a,b] is -

A = [S(b) - S(a)]/b-a....(1)

Now. for function f(x) on interval [4, 7].

Here, use values of f(4) and f(7) from graph and find value of f(x)=y for x=4 and value of f(x)=y for x=7.

f(4) = -10, f(7) = 5

Substitute the values in equation (1) -

A1 = [f(7) - f(4)]/(4 - 7)

A1 = [5 - (-10)]/-2

A1 = 15/-2

A1 = -7.5

Average rate of change for function g(x) on interval [4, 7].

Here, use the values of g(4) and g(7) from table.

g(4) = 7, g(7) = 19

Substitute the values in equation (1) -

A2 = [g(7) - g(4)]/(4 - 7)

A2 = [19 - 7]/-2

A2 = 12/-2

A2 = -6

Average rate of change for function h(x) = x² - 9x + 5 on interval [4, 7].

Here, use the values of h(4) and h(7) by putting x = 4 and x = 7 in function h(x).

h(4) = x² - 9x + 5

h(4) = (4)² - 9(4) + 5

h(4) = 16 - 36 + 5

h(4) = -15

h(7) = x² - 9x + 5

h(7) = (7)² - 9(7) + 5

h(7) = 49 - 63 + 5

h(7) = -9

Substitute the values in equation (1) -

A3 = [h(7) - h(4)]/(4 - 7)

A3 = [-9 - (-15)]/-2

A3 = [-9 + 15]/-2

A3 = 6/-2

A3 = -3

Now, average rate of change of these function f(x), g(x), h(x) from least to greatest is f(x)<g(x)<h(x).

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Answer: The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.

So, using the empirical rule, we can say that approximately 99.7% of housing prices in this small town fall within 3 standard deviations of the mean.

To find the range of prices that this corresponds to, we can use the following formula:

(mean - (3 x standard deviation), mean + (3 x standard deviation))

Plugging in the given values:

(157,000 - (3 x 8,000), 157,000 + (3 x 8,000))

(141,000, 173,000)

So, approximately 99.7% of housing prices in this small town are between $141,000 and $173,000.

Step-by-step explanation:

Christian is making a circular pizza. The diameter of the pizza is 12 inches. The pizza crust on the outside will be about 12 inches thick.

The formula for determining a pizza's circumference is C = 2πr, where C is the circumference, pi is a constant that is roughly equivalent to 3.14, and r is the pizza's radius.

We must determine the pizza's circumference in order to locate the crust on the pizza's outer edge. C = 2πr is the formula for a circle's circumference, where C is the circumference, pi is a constant that is roughly equivalent to 3.14, and r is the circle's radius.

The pizza has a diameter of 12 inches, so the radius, or 6 inches, is half of that. Thus, the pizza's circumference is:

                 C= 2π6 = 12π

The pizza's circumference represents the crust's outer edge. C = 26 inches = 12 inches As a result, the pizza's crust on the outside measures about 12 inches.

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Triangles are shown to be congruent by the statement that "Angles are congruent by vertical angles."

Triangles are said to be congruent if, under a correspondence, two of a triangle's sides and the angle that connects them are equal to two of another triangle's sides and the angle that connects them.

Given two triangles, Where the base of these triangles are same

And also given that the two angles are the same.

Since,

Two intersecting lines form a pair of vertical angles. The vertical angles are opposite angles with a common vertex (which is the point of intersection).

Thus, the third angle is also equal

From AAS:

AAS stands for Angle-Angle-Side. When two angles and a non-included side of a triangle are equal to the corresponding angles and sides of another triangle, then the triangles are said to be congruent

Therefore. "Angles are congruent by vertical angles" is used to show the triangles are congruent by AAS,

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The values of given differential equation  constant r are  = 1/5 and -3.

A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) dy/dx = f(x)

Here “x” is an independent variable and “y” is a dependent variable.

For example, dy/dx = 5x A differential equation contains derivatives which are either partial derivatives or ordinary derivatives.

The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity.

There are a lot of differential equations formulas to find the solution of the derivatives.

As y = e power rx

y¹= dy/dx= re power rx

and y'' = r²e power rx

Hence 5y'' +14y' – 3y = 0 can be written as

5r²e - 14re** - 3e¹² - 0

or 5r2 + 14r - 3 - 0

or 5r2 + 15r - r - 3 - 0

or (5r − 1)(r + 3) = 0

r=1/5 or r= -3

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In reality, only 55% of the calories produced by crops around the world are consumed directly by humans. Animal feed accounts for an additional 36% of usage.

The Complete Question.

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14.3 percentage of the trees were oaks, The total number of trees have increased by: 15/28.

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The composite translation for this problem is represented as follows:

<3, -9>

The four translation rules are defined as follows:

Another notation is the <x,y> notation, in which:

For the composition of translations, we add the movements, hence:

Meaning that the first option is the correct option.

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The number of M&M's that were in the bowl before Amelia arrived is; 12

Let the original amount in the bowl be x. Thus;

If Amelia grabbed 50%, then amount left = (100% - 50%)x = 50%x = 0.5x

Now, we are told that domicick took 50% of what was keft after amelia. Thus; New amount left = 100%x - (0.5x + 0.25x) = 0.25x

Now, we are told that there were finally 3 left after all the deductions. Thus;

0.25x = 3

x = 3/0.25

x = 12

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Answer: 7

Step-by-step explanation:

yarn = 4 1/5 = 21/5

measure of one piece = 3/5

no. of pieces of yarn = 21/5÷3/5

= 21/5×5/3

= 105/15 = 7

So the number of pieces of yarn cut will be 7

Answer:

7

Step-by-step explanation:

4.2' yarn

cut into 0.6' pieces

4.2/0.6 =7

The total average monthly expenditure in 2020 was £2400.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

The expenditure in 2020 will be calculated as:-

Average monthly expenditure in 2020 = 600 + 500 + 400 + 300 + 200 + 100 = £2400

So, the total average monthly expenditure in 2020 was £2400.

The complete question is given below:-

Average monthly expenditure (£)

600

500

400

300

200

100

The total average monthly expenditure in 2010 was £1200. What was it in 2020?

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The mean is 120 and the standard deviation is 20.

How to find the mean and standard deviation?

We must calculate the mean and standard deviation from the curve in the preceding graph.

As we can see from the normal distribution curve that the center is at 120.

As a result, 120 is the distribution's mean.

From the above curve:

Mean + Standard deviation = 140

We know that the Mean = 120.

120 + Standard Deviation = 140

Standard Deviation = 20

Therefore, 20 is the standard deviation.

Therefore, from the given data we can find that the Mean is 120 and the Standard deviation is 20.

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Answer:

Step-by-step explanation:

The number of selections that will contain no defective printers can be found using the nCr formula as well, but with n being the number of non-defective printers (34) and r being the number of items selected (2). So nCr = 34!/(2!(34-2)!) = 34!/(2!32!) = 5,832.

(a) There are 6,632 selections that can be made.

(b) There are 5,832 selections that will contain no defective printers.

(a)The number of selections can be found using the nCr formula: nCr = n!/(r!(n-r)!), where n is the total number of items (42) and r is the number of items selected (2). So nCr = 42!/(2!(42-2)!) = 42!/(2!40!) = 6,632.

(b) The number of selections that will contain no defective printers can be found using the nCr formula as well, but with n being the number of non-defective printers (34) and r being the number of items selected (2). So nCr = 34!/(2!(34-2)!) = 34!/(2!32!) = 5,832.

There are 6,632 selections that can be made when two printers are chosen from a shipment of 42, including 8 that are defective. Out of these selections, 5,832 will contain no defective printers.

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The coordinates of the rotated image are:

A' = (1, 1)

B' = (1, 3)

C' = (-2, 1)

Option A is the correct answer.

It is the movement of the shape in the left, right, up, and down directions.

The translated shape will have the same shape and shape.

There is a positive value when translated to the right and up.

There is a negative value when translated to the left and down.

We have,

Triangle ABC.

A = (1, 1)

B = (3, 1)

C = (1, 4)

When rotated 90 degrees counterclockwise around (1, 1).

A = (1, 1) will stay the same.

B = (3, 1) will change into B' = (1, 3)

C = (1, 4) will change into C' = (-2, 1)

Now,

The coordinates of the image after reflection are:

A' = (1, 1)

B' = (1, 3)

C' = (-2, 1)

This image is found in option A.

Thus,

The rotated image of the triangle is in option A.

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a. The total cost C as a function of the number of units produced is C = $12.13x + $92,000

b. The revenue R as a function of the number of units sold is R = $17.10x

c. The profit P as a function of the number of units sold is P = $5.97x - $92,000

(A) The total cost for a business is the sum of the variable cost and the fixed costs.

The variable cost per unit is $12.13, so the total variable cost for x units produced is $12.13x.

The fixed costs are $92,000 and do not change with the number of units produced, so the total cost C as a function of the number of units produced is given by -

C = $12.13x + $92,000

(B) The revenue is the total amount of money earned from selling the product. The product sells for $17.10 per unit, so the revenue R as a function of the number of units sold is given by -

R = $17.10x

(C) The profit is the difference between the revenue and the total cost. Using the equations from (A) and (B), the profit P as a function of the number of units sold is given by -

P = R - C = $17.10x - ($12.13x + $92,000) = $5.97x - $92,000

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Based on given data, Mean is 4, missing number can be 4,2,7., So the arrangement is c,4,6,8.

The mean of numbers is the average value of given numbers.

Given numbers are 4,2,5

Mean is 4

Mean = sum of numbers/ total number of values given

4 = (4+2+5+x) /4

16 = 11 +x

x = 5

Therefore the missing number is 5.

  Mode is the number which is most frequently repeated in a list of numbers.

Given numbers are 4,2,7 and x

Mode is given as 2

So x can be any number from the given values since any number can be repeated twice.

Therefore missing number can be 4,2,7.

 The median of the set is the middle number after the numbers are arranged from least to greatest.

There are four numbers given. So the median is the average of middle numbers.

To get the median of 5 the middle numbers should be 4 and 6.

So the arrangement is c,4,6,8

Therefore the missing number can be any number less than or equal to 4.

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Answer:

Step-by-step explanation:

Exercise 2:

1.To construct the triangle ABC, start by drawing a line segment BC of length 5cm. Using a protractor, measure and mark an angle of 80° at B and an angle of 40° at C. Connect the two marked angles with a line to form triangle ABC.

2.The bisector of angle ABC cuts the line segment AC at M.

a. To calculate MBC, we can use the Angle Bisector Theorem, which states that the ratio of the lengths of the bisector of a given angle of a triangle to the sides of the triangle is equal to the ratio of the other two sides of the triangle. So, MBC/AC = BC/AB. Since angle B=80° and angle C=40°, angle A is (180-(80+40))=60°. Therefore, AB/BC = MC/MB.

b. Using the Angle Bisector Theorem again, we can see that MBC/AC = BC/AB, which means that MBC/5 = 5/AB. And since AB/BC = MC/MB, we can deduce that MB = MC. Therefore, triangle MBC is an isosceles triangle.

c. As we know that MB=MC, we can say that triangle MBC is an isosceles triangle.

Note: To construct the triangle in a more accurate way, you may need a ruler and a compass, and the angles and sides should be measured and drawn with more precision.

The required length of a side of the rhombus is 10 cm.

A parallelogram is a particular instance of a rhombus. The opposing sides and angles in a rhombus are parallel and equal. A rhombus also has equal-length sides on each side, and its diagonals meet at right angles to form its shape. The rhombus is also referred to as a diamond or rhombus.

Let diagonals intersect each other at o,

In ΔROQ

RQ = [tex]\sqrt{OR^{2}+OQ^{2}}[/tex]

RQ =[tex]\sqrt{36+64} =\sqrt{100}[/tex]

RQ = 10 cm

Thus, required length of the side of rhombus is 10 cm.

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Complete question:

PQRS is a rhombus. The shorter diagonal PR measures 12 units and QS = 16 units. The measure of angle PQR = 60°. Find the length of a side of the rhombus.​

The authors argue that Cypher's decision to go back into the Matrix was a betrayal of his friends because it showed that he was willing to abandon his beliefs and betray those closest to him in order to return to a life of comfort and ease within the Matrix.

By doing so, he effectively gave up his freedom and became a tool of the machines once again, thereby betraying the ideals of the rebellion and those who had fought alongside him to achieve it. Hence, The authors argue that Cypher's decision to go back into the Matrix was a betrayal of his friends.

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In 2009, Florida would have 558 hospitals if it had the same hospital beds per million residents as Texas. This is a little more than what Florida's real number of hospitals was in 2009. (343).

A mathematical technique called proportional reasoning is used to compare the ratios of two or more identical quantities.

The proportionate reasoning theory is being used in this query.

In this instance, we're contrasting Florida's hospital density per million residents with Texas' hospital density per million residents.

We may predict the number of hospitals in Florida given the same ratio as Texas by comparing these two ratios.

1. Determine Texas's hospital density per million residents: 428 hospitals divided by 26.4 million residents equals 16.2 hospitals per million residents.

2. Determine Florida's hospital density per million residents: 343 hospitals / 19.5 million residents = 17.6 hospitals per million residents.

3. Multiply the population of Florida (19.5 million) by the number of hospitals per million of the population of Texas (16.2) to arrive at 315.3 hospitals: 19.5 million people x 16.2 hospitals per million of the population.

The result should be rounded to the next whole number: 315 hospitals are equal to 315.3 hospitals.

5. Contrast Florida's actual hospital count in 2009 with the expected number of hospitals there (315). (343).

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The percentage of scores below 112 is equal to 78.81.

Let X represent a test's score. X is regularly distributed, hence

P(X < 112)

= P[(X - 100)/15 < (112 - 100)/15]

= P[ SNV < 0.8] .8]

SNV stands for Standard Normal Variable,

A normal distribution with a mean of 0 and a standard deviation of 1 is the standard normal distribution. When referring to a random variable that has this typical normal distribution, the letter Z is frequently employed.

z = (X – μ) / σ

If X is a normal random variable, represents its mean, and represents its standard deviation. The normal distribution formula is also available here.

and the value is 0.7881.

Thus, the percentage of scores below 112 is equal to 78.81.

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The full question

If the scores for a test have a mean of 100 and a standard deviation of 15, what is the percentage of scores that will fall below 112? Assuming a normal distribution

Answer: y = 2-4x

Step-by-step explanation:

The area of the glacier at the beginning of the last year is given as follows:

56.31 km².

The area of the glacier is obtained applying the proportions in the context of this problem.

The current area is given as follows:

56.20 km².

The percentage of this area relative to the area at the beginning of the last year is given as follows:

100 - 0.2 = 99.8% = 0.998.

Hence the area at the beginning of the last year is obtained as follows:

0.998x = 56.20

x = 56.20/0.998

x = 56.31 km².

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