The seats available to a baseball game come in four types: bleacher, box, club, and grandstand. There are 12,600 box seats and 5,400 club seats available. According to the graph, what is the total number of seats available?

Total of 3/5 can of black beans and  pinto beans used in preparing the chili by Ronald.

An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator either denominator of a complex fraction.

Fraction of beans used by Ronald to make chili.

black beans = 1/2  can

pinto beans = 7/10 can

Thus,

total beans = 1/2  can + 7/10 can

total beans = (1/2 + 7/10) can

Simplifying:

total beans = (5*1 + 7)/ 2*10  can

total beans = 12/20 can

total beans = 3/5 can

Thus, total of 3/5 can of black beans and  pinto beans used in preparing the chili by Ronald.

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Therefore , the solution of the given problem of domain comes out to be (a) Domain of fog: {1, 6} and (b) Range of fog: {1, 6} .

The range of potential values that a function can take is known as its domain. These numbers serve as a representation for the x-values of an equation like f. (x). The range of potential numbers on which a function can be used is known as its domain. The result that the function gives following the entry of the x value is this set.

Here,

To find the domain and range of the composition fog, we need to evaluate fog(x) for all x in the domain of g that maps to the domain of f.

Since g maps 1 to 5 and 6 to 4, we have:

fog(1) = f(g(1)) = f(5) = 6

fog(6) = f(g(6)) = f(4) = 1

Therefore, the domain of fog is {1, 6} and the range of fog is {1, 6}, written in set notation as:

(a) Domain of fog: {1, 6}

(b) Range of fog: {1, 6}

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The angle ∠5 and ∠6 are 135° and 45° respectively.

The study of points, lines, and shapes, as well as their characteristics and connections in two and three dimensions, is the subject of the mathematic branch known as geometry. It contains ideas like polygons, circles, triangles, angles, and solids.

For, Carla's adjustment,

angle ∠5 = 2 • angle ∠6

Then, ∠5 + ∠6 = 180° ( by linear pair)

2 × ∠6 + ∠6 = 180°

3 × ∠6 = 180°

∠6 = 60°

Then, ∠5 = 120°

For, Carla's assistant Joe adjustment

∠5 = 3 × ∠6

Again, ∠5 + ∠6 = 180° ( by linear pair)

3 ×∠6 + ∠6 = 180°

4 ×∠6 = 180°

∠6 = 45°   Then, ∠5 = 135°

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The equation 8(x-9)=6(2x-12)-4x (option C) has an infinite number of solutions.

Let's simplify the given equation and solve for x:

8(x-9) = 6(2x-12) - 4x

8x - 72 = 12x - 72 - 4x // Distributing 6

8x - 72 = 8x - 72 // Simplifying

We can see that both sides of the equation are equal, and there is no variable 'x' present. This means that the equation is an identity and it will hold true for any value of x.

Therefore, this equation has an infinite number of solutions, since any value of x will satisfy the equation.

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The values that completes the given function table is as attached.

For x ≥ 13, the table suggests that f(x) is always greater that g(x)

We are given the functions;

f(x) = 2ˣ and g(x) = 225x

Thus, to fill the function table, we have;

f(6) = 2⁶ = 64

f(7) = 2⁷ = 128

f(12) = 2¹² = 4096

f(13) = 2¹³ = 8192

f(14) = 2¹⁴ = 16384

g(6) = 225 * 6 = 1350

g(7) = 225 * 7 = 1575

g(12) = 225 * 12 = 2700

g(13) = 225 * 13 = 2925

g(14) = 225 * 14 = 3150

For x ≥ 13, the table suggests that f(x) is always greater that g(x)

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Answer:

[tex]\measuredangle \textsf{ BCA is Actue.}[/tex]

[tex]\measuredangle \textsf{ CAB is Actue.}[/tex]

[tex]\measuredangle \textsf{ ABC is Right}.[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked what type of angles are provided.}[/tex]

[tex]\textsf{First, the question is asking for the \underline{measure}, which \underline{affects} the types of angles.}[/tex]

[tex]\textsf{This Triangle is a \underline{Right Triangle}, according to the \underline{Angle Markings}.}[/tex]

[tex]\textsf{One angle must be \underline{Right}, and that angle is} \ \boxed{ \measuredangle \textsf{ABC.}}[/tex]

[tex]\textsf{Becuase a Triangle adds up to \underline{180}}^{\circ}\textsf{, the other \underline{2} angles are \underline{less than 90}}^{\circ}.[/tex]

[tex]\textsf{Meaning, that} \ \boxed{\measuredangle \textsf{BCA}} \ \textsf{and} \ \boxed{\measuredangle \ \textsf{CAB}} \ \textsf{are \underline{Acute}.}[/tex]

[tex]\fbox{\textsf{Notes:}}[/tex]

[tex]\textsf{An Acute Angle is \underline{less} than 90}^{\circ}.[/tex]

[tex]\textsf{A Right Angle is \underline{equal} to 90}^{\circ}.[/tex]

[tex]\textsf{An Obtuse Angle is \underline{greater} than 90}^{\circ} \textsf{, but \underline{less} than 180}^{\circ}.[/tex]

[tex]\textsf{A Straight Angle is \underline{equal} to 180}^{\circ}.[/tex]

[tex]\textsf{A Reflex Angle is \underline{greater} to 180}^{\circ} \textsf{, but \underline{less} than 360}^{\circ}.[/tex]

Consequently, 308 °is the main arc's length and minor arc is 52°

An angle whose apex is a circle's center is said to have a center angle.

The minor arc connects two locations on a circle in the shortest distance possible. The main arc connecting them is the longest arc.

. For instance, two locations A and B on a circle are connected by two arcs:

the major arc, which travels clockwise around the circle, and the minor arc, which travels anticlockwise (the minor arc) .

The minor arc's length is equivalent to the center angle's length in degrees.

Consequently, the minor arc is 52 degrees in length.

The major arc's measure is equivalent to 360 ° minus the minor arc's measurement.

The main arc's length is therefore 360 – 52 = 308 °.

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Hello !

Answer:

[tex] \large \boxed{ \sf y = - 3 + \frac{9}{2} x}[/tex]

Step-by-step explanation:

We want to isolate y in the following equality :

[tex] \sf9x - 2y = 6[/tex]

First, let's substract 9x from both sides of the equality :

[tex] \sf9x - 2y - 9x= 6 - 9x[/tex]

[tex] \sf - 2y = 6 - 9x[/tex]

Now let's divide both sides by -2 :

[tex] \sf \frac{ - 2y}{ - 2} = \frac{6 - 9x}{ - 2} [/tex]

[tex] \boxed{ \sf y = - 3 + \frac{9}{2} x}[/tex]

Have a nice day ;)

The polynomial which is as described in the task content is; P (x) = -0.2 (x - 1)² (x +4).

It follows from the task content that; P(x) has a root of multiplicity 2 at x = 1 and a root with multiplicity of 1 at (x - 4).

Therefore, the polynomial would take the form;

P(x) = a (x - 1)² (x + 4)

Therefore, since the y-intercept is; y = -0.8

Therefore, we have that!

-0.8 = a (0-1)² (0+4)

-0.8 = 4a

a = -0.8 / 4

a = -0.2

Ultimately, the formula for the polynomial P(x) as given in the task content is; P (x) = -0.2 (x - 1)² (x +4).

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a) The exponential decay function that models the situation whereby the value of houses in a city is decreasing at a rate of 3% per year is  f(x) = 450,000 x 0.97^x.

b) The value of this house that currently costs $450,000, based on the above exponential decay function, 4 years from now is $398,381.76.

c) It will take 13.3 years till the value of the house is $300,000.

An exponential decay function describes an exponential function whereby the value of an asset reduces consistently at a constant percentage rate over a period.

The formula for an exponential decay is given as y=a(1-b)^x.

In this exponential decay formula, y is the final amount, a is the original amount, b is the decay factor, and x is the amount of time that has elapsed.

Finally, we can use an online finance calculator to determine the number of years till the value of the house reduces to $300,000.

The current cost of a house in the city = $450,000

The decreasing rate per year = 3%

Continuous annual decreasing factor = 0.97 (100% - 3%)

Let the value of the house in x years = f(x)

a) f(x) = 450,000 x 0.97^x

b) Value of the house 4 years from now = 450,000 x 0.97^4

= $398,381.76

c) The number of years till the value is $300,000:

300,000 = 450,000 x 0.97^x

I/Y (Interest per year) = -3%

PV (Present Value) = $450,000

PMT (Periodic Payment) = $-0

FV (Future Value) = $-300000

Results:

Number of period (#) = 13.312

Thus, the total Decay after 13.312 years is $150,000.00 ($450,000 - $300,000).

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Answer:

x = -3 ; y = 5

Step-by-step explanation:

Hey there!

This question has given us information about both of the given variables, x, and y

It tells us the y is equal to x plus 8 and that x plus y is equal to 2

If y is equal to x plus 8, we can

substitute the y value into the second equation

This will give us the following equation:

x + (x + 8) = 2

Now we simplify to solve for x

2x + 8 = 2

2x = -6

x = -3

Now that we know the x value, we can substitute that value into any one of the two equations given to solve for y

I'm going to use the first equation

Since we know that x is -3, we can now input it into the first equation and then simplify;
y = -3 + 8

y = 5

Answer:

Step-by-step explanation:

[tex]\tt y=x+8\\x+y=2[/tex]

Substitute y= x+8, into x + y=2:-

[tex]\tt x+y=2[/tex]

Let y = x+8

[tex]\tt x+x+8=2[/tex]

[tex]\tt 2x+8=2[/tex]

Solve for x :-

[tex]\tt 2x+8=2[/tex]

Subtract both sides by 8:-

[tex]\tt 2x+8-8=2-8[/tex]

[tex]\tt 2x=-6[/tex]

Divide both sides by 2:-

[tex]\tt \cfrac{2x}{2} =\cfrac{-6}{2}[/tex]

[tex]\boxed{\bf x=-3}[/tex]

Now, to solve for y, Substitute x = -3 into y=x+8:-

[tex]\tt y=x+8[/tex]

Let x = -3

[tex]\tt y=-3+8[/tex]

Simplify:-

*-3 + 8 = 5*

[tex]\boxed{ \bf y=5}[/tex]

Therefore, the solutions to the given system are x= -3 & y =5.

______________________

Hope this helps!

Using probability,

a. In positive mammogram, the chance of having cancer is= 7.217%

b. If patient has cancer, chance of positive mammogram is = 90%

c. A woman with negative mammogram has cancer is = 0.114%

What do you mean by probability?

The chances of an event occurring is defined by probability. There are many instances in real life where we may need to make predictions about how something will turn out. We may be sure or not sure about the results of an event.

When this happens, we say that there is a chance that the event will happen or not happen. In general, probability has many wonderful uses in games, in business to make forecasts based on likelihood, and in this emerging branch of artificial intelligence.

Now in the question,

1247 women had a positive mammogram. Out of which 90 actually had cancer. So, in positive mammogram, the chance of having cancer is:

=90/1247

= 7.217%.

100 women who received mammogram cancer or which 90 had a positive mammogram. So, if patient has cancer, chance of positive mammogram is = 90/100

= 90%.

Now 8753 women had a negative mammogram of which 10 actually had cancer. So, a woman with negative mammogram has cancer is

= 10/8753

= 0.114%

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The complete question is:

he table to the right is based on the assumption that​ 1% of breast tumors are malignant. It also assumes that mammogram screening is ​90% accurate. Complete parts​ (a) through​ (c) below

Answer:

720

Explanation:

[tex]2:125[/tex]

[tex]x:45000[/tex]

to go from 125 to 45000 we need to multiply by 360

that's why we will multiply 2 with 360 to balance both ratios

when we do that we get 720

2:125

[tex]\bold{720:45000}[/tex]

Answer:

720

Step-by-step explanation:

Answer: 792,000

Step-by-step explanation: 330,000 x 2.4%by 20 years = 792,000

Cone B does not have volume 48bold pi cubic units with these dimensions, but it does have the same volume as Cone A because they both have volume 48bold pi cubic units.

A cone, also known as a vertex, is a three-dimensional polygonal object with a flat base and a soft tapering apex. In a plane with no vertices, a cone is formed by connecting a story arc of lines, half-lines, but rather routes those are common to all focuses on the outpost. A cone is a muti structure with just a peaceful transition from a flat, round or, base towards the vertex and apex, which represents as the base's axis. A three-dimensional polygonal shape with only an upwardly flat curved surface is known as a cone.

Because Cone A has a volume of 48bold pi cubic units, we can use the volume of a cone formula to calculate its height:

Cone volume = 1/3 * base area * height

1/3 * pi * 62 * 4 = 48bold pi

48pi = 48bold pi

As a result, Cone A's height is 4 units. The radius is also 6 units, as we can see.

Cone volume = 1/3 * base area * height

[tex]1/3 * pi * r2 * h = 48\\144 = r^2 * h\\144 = 8^2 * 3\\144 = 192[/tex]

Cone B does not have volume 48bold pi cubic units with these dimensions, but it does have the same volume as Cone A because they both have volume 48bold pi cubic units.

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The probability that a biscuit is made by machine J and is not broken is given as follows:

0.245 = 24.5%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

In this problem, we are given a probability tree, hence to obtain the desired probability we must just choose the correct node.

The desired node is given as follows:

Hence the probability that a biscuit is made by machine J and is not broken is given as follows:

0.25 x 0.98 = 0.245.

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Answer:

y=-3x+3

Step-by-step explanation:

y=mx+c

Substitute y = -3

x = -2

-3 = -3*-2+c

-3 = -6+c

-6+c = -3

c = -3+6

=3

y= -3x+3

The new balance is the sum of the unpaid balance, the finance charge, and any new purchases or fees:

New balance = Unpaid balance + Finance charge + Purchase

= $35.51 + $0.84 + $19.27

= $55.62.

We must first identify the balance remaining after the payment and purchase in order to calculate the outstanding balance:

Balance after payment = $50.51 - $15.00 = $35.51

Amount remaining after payment: $35.51 + $19.27 = $54.78

The balance remaining after the payment is made, which comes to $35.51, is the unpaid balance.

We must first determine the average daily balance before we can calculate the finance charge using the unpaid balance technique. Finding the daily balance and dividing the total by the number of days in the billing cycle will allow us to achieve this:

$50.51 times 14 days in April equals $707.

14 April 15–24: $35.51 multiplied by 10 days equals $355.

10 April 25–30: $54.78 multiplied by six days is $328.68

Balance total: $707.14 plus $355.10 plus $328.68 ($1,390.92)

Average daily balance: $1,390.92 divided by 30 equals $46.36.

Hence, the finance charge can be determined as follows:

Finance fee equals $46.36 * (0.18 / 365) * 30 = $0.84. Finance charge formula: Average daily balance * Daily periodic rate * Number of days in billing cycle

The total of the unpaid debt, the finance charge, and any additional purchases or fees is the new balance:

Purchase + Finance fee + New balance = $35.51 + $0.84 + $19.27 = $55.62.

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Answer:

The account balance on April 1st is $50.51. On April 15th a payment of $15.00 is made. On April 25th a purchase of $19.27 is made. The annual rate is 18%. What is the unpaid balance? What is the finance charge using the unpaid balance method? What is the new balance? Unpaid balance = $ Finance charge = $ New balance = $

Step-by-step explanation:

To calculate the unpaid balance, we first need to calculate the balance after the payment and the balance after the purchase:

Balance after payment on April 15th: $50.51 - $15.00 = $35.51

Balance after purchase on April 25th: $35.51 + $19.27 = $54.78

Next, we need to calculate the average daily balance for the billing period. The billing period runs from April 1st to April 30th, which is 30 days. We can split this into two parts:

April 1st to April 15th (15 days)

April 16th to April 30th (15 days)

For the first part of the billing period, the balance is $50.51 for 15 days. For the second part of the billing period, the balance is $54.78 for 15 days. The average daily balance is therefore:

($50.51 x 15 + $54.78 x 15) / 30 = $52.64

The unpaid balance is the average daily balance minus the payment, so:

$52.64 - $15.00 = $37.64

The finance charge using the unpaid balance method is the unpaid balance multiplied by the daily periodic rate and the number of days in the billing cycle. The daily periodic rate is the annual rate divided by 365 days, so:

Daily periodic rate = 0.18 / 365 = 0.000493

Finance charge = $37.64 x 0.000493 x 30 = $0.56

The new balance is the previous balance (after the purchase) plus the finance charge, so:

New balance = $54.78 + $0.56 = $55.34

Therefore, the unpaid balance is $37.64, the finance charge is $0.56, and the new balance is $55.34.

Answer:

Shorter section is (mm):
159

Longer section is (mm):
211

Step-by-step explanation:

Let x represent the longer section length and y the shorter section length

We are given x : y = 4:3

We can rewrite in fractional form as

[tex]\dfrac{x}{y} = \dfrac{4}{3}[/tex]

Multiply both sides by y to get
[tex]\dfrac{x}{y} \cdot y = \dfrac{4}{3}y\\\\x = \dfrac{4}{3}y[/tex]

We know that the combined length of both sections is 370 mm
x + y = 370

Substitute the expression for x in the above equation
[tex]\dfrac{4}{3}y + y = 370\\\\\dfrac{4}{3}y + y = y(\dfrac{4}{3} + 1})\\= y(\dfrac{4}{3} + \dfrac{3}{3})\\\\= y(\dfrac{7}{3})\\\\= \dfrac{7}{3}y[/tex]

Therefore we get
[tex]\dfrac{7}{3} y= 370\\\\[/tex]

Multiply both sides by [tex]\dfrac{3}{7}[/tex]

[tex]\dfrac{7}{3} y \times \dfrac{3}{7} = 370 \times \dfrac{3}{7} \\\\y = 158.57\; mm\\[/tex]

= 159 mm (to 3 significant figures)

Therefore the longer section = 370 - 159 = 211 mm

Check the ratio to verify

211/159 = 1.327 ≈ 1.33 which is 4/3

According to the given information, your share of the groceries was $39.95, your other roommate's share was $42.05, and Sarah's share was $0.

What is equation?

An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides separated by an equals sign (=). The expressions on both sides of the equation can contain numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

Let's call the amount of money Sarah spent on groceries "S".

Let's call the amount of money you spent on groceries "Y".

Let's call the amount of money your other roommate spent on groceries "O".

We know that:

S + Y + O = 82 (the total bill was $82)

Y = (S/2) - 0.05 (your groceries were $0.05 cheaper than half of Sarah's groceries)

O = Y + 2.10 (your other roommate's groceries were $2.10 more than your groceries)

We can use the second equation to substitute Y in the third equation:

O = Y + 2.10

O = (S/2) - 0.05 + 2.10

O = (S/2) + 2.05

We can now substitute both equations into the first equation:

S + Y + O = 82

S + ((S/2) - 0.05) + ((S/2) + 2.05) = 82

2S + 4 = 164

2S = 160

S = 80

Now that we know S, we can substitute it into the other equations to find Y and O:

Y = (S/2) - 0.05 = (80/2) - 0.05 = 39.95

O = Y + 2.10 = 39.95 + 2.10 = 42.05

So your share of the groceries was $39.95, your other roommate's share was $42.05, and Sarah's share was $80 - ($39.95 + $42.05) = $82 - $82 = $0.

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Answer:dear

Step-by-step explanation:

257,540.765

This is because congruent triangles' corresponding angles are identical, and since triangle ABC and triangle DEF are congruent, their corresponding angles are equal.

A triangle is a two-dimensional closed geometric object that consists of three line segments called sides that intersect at three places called vertices. Triangles can be distinguished by their angled sides. Triangles, depending on their sides, can be mutually perpendicular (all sides equal), ellipse, or scalene. Triangles are classed as acutely (all angles less than 90 degrees), equal (one angle equal to 90 degrees), or ambiguous (all angles more than 90 degrees) (all angles greater than 90 degrees). The area of a triangle may be determined by applying the formula A = (1/2)bh, wherein There is the area, b is the base of the triangle, and h is the height of the triangle.

In triangle ABC, we have:

In triangle DEF, we have:

This is because congruent triangles' corresponding angles are identical, and since triangle ABC and triangle DEF are congruent, their corresponding angles are equal.

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Answer:

Step-by-step explanation:

To prove the statement "A Set S with two or more vectors is linearly independent if and only if no vector in S is expressible as a linear combination of the other vectors in S," we must show that both directions of the statement are true. That is, we must show that if a set S is linearly independent, then no vector in S is expressible as a linear combination of the other vectors in S, and conversely, if no vector in S is expressible as a linear combination of the other vectors in S, then the set S is linearly independent.

First, let's assume that the set S is linearly independent. This means that for any vectors v1, v2, ..., vn in S, the equation a1v1 + a2v2 + ... + anvn = 0 has only the trivial solution a1 = a2 = ... = an = 0. We will prove that no vector in S is expressible as a linear combination of the other vectors in S.

Suppose, for the sake of contradiction, that there exists a vector v in S that can be expressed as a linear combination of the other vectors in S. That is, there exist vectors v1, v2, ..., vn-1 in S such that v = b1v1 + b2v2 + ... + bn-1vn-1, where not all of the bi's are zero. Without loss of generality, assume that b1 is nonzero. Then we can write v1 as a linear combination of the other vectors in S:

v1 = (1/b1)v - (b2/b1)v2 - ... - (bn-1/b1)vn-1

Substituting this expression for v1 in the equation a1v1 + a2v2 + ... + anvn = 0, we get:

a1[(1/b1)v - (b2/b1)v2 - ... - (bn-1/b1)vn-1] + a2v2 + ... + anvn = 0

Multiplying both sides by b1 and rearranging, we get:

(a1/b1)v + (-a1b2/b1)v2 + ... + (-a1bn-1/b1)vn-1 + a2v2 + ... + anvn = 0

This is a linear combination of vectors in S that equals the zero vector, and not all of the coefficients are zero, since a1 is nonzero. But this contradicts the assumption that S is linearly independent, so our assumption that there exists a vector in S that can be expressed as a linear combination of the other vectors in S must be false. Therefore, no vector in S is expressible as a linear combination of the other vectors in S.

Now let's assume that no vector in S is expressible as a linear combination of the other vectors in S. We will prove that the set S is linearly independent. Suppose, for the sake of contradiction, that there exist vectors v1, v2, ..., vn in S such that the equation a1v1 + a2v2 + ... + anvn = 0 has a nontrivial solution, where not all of the ai's are zero. Without loss of generality, assume that a1 is nonzero. Then we can write v1 as a linear combination of the other vectors in S:

v1 = (-a2/a1)v2 - ... - (an/a1)vn

Substituting this expression for v1 in the equation a1v1 + a2v2 + ... + anvn = 0, we get:

0 = a1(-a2/a1)v2 + ... + a

The required prove to show that ΔABC is similar to ΔMNC is explained below.

A given set of triangles are said to be similar if on comparing their corresponding properties, there exist some common relations among them. The similarity of two or more triangles do not imply that they are congruent.

Considering the given diagram, we have;

AM = MC    (Given)

BN = NC     (Given)

BC = BN + NC (addition property of a line segment)

AC = AM + MC (addition property of a line segment)

NC/ BC = MC/ AC = constant

<ACB = <MCN  (Reflexive property)

Therefore,

ΔABC ~ ΔMNC (Side-Angle-Side theorem)

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2

you divide 8 with 4 and the answer is 2

The values of x that would give a profit of $300 is 20 and 60.

Revenue is the total amount of money that can be made from selling x items while cost is the amount of money used to produce x items.

Profit is the difference between revenue and cost. Profit is given by

Profit = Revenue - Cost

Since Cost C = 60x + 300, Revenue R = 100x − 0.5x², hence:

Profit = Revenue - Cost

Profit = 100x − 0.5x² - (60x + 300)

Profit = 40x - 0.5x² - 300

For a profit of $300:

300 = 40x - 0.5x² - 300

0.5x² - 40x + 600 = 0

x = 20 and x = 60

Hence the values of x that would give a profit of $300 is 20 and 60.

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Step-by-step explanation:

that makes NOP and QRP similar triangles.

that means that there is one common scale factor for the lengths of all pairs of correlating sides.

that means

QR/NO = RP/OP

OP = OR + RP

20/32 = RP/(12 + RP)

20(12 + RP) = 32RP

240 + 20RP = 32RP

240 = 12RP

RP = 20

Answer: $400,000

Step-by-step explanation:

$500,000-20%=$400,000

By induction, we can conclude that in a semigroup, all products of x1, x2, ...... xn (in that order) are equal.

A semigroup is a mathematical structure consisting of a set of elements along with an associative binary operation. The operation must be associative, meaning that the result of the operation is independent of the order of the operands.

We can prove this by induction. Let S be a semigroup and x1, x2, x3, ..., xn be the elements of S.
Base Case: When n = 2, then the product of x1 and x2 is x1x2.
Inductive Step: Assume that the product of x1, x2, ..., xn (in that order) is equal. Then we can prove that the product of x1, x2, ..., xn+1 (in that order) is also equal.
Let a be the product of x1, x2, ..., xn (in that order). Then a = x1x2x3....xn.
Now the product of x1, x2, ..., xn+1 (in that order) is a x xn+1. Since S is a semigroup, a x xn+1 = x1x2x3....xn x xn+1.
Since the product of x1, x2, ..., xn (in that order) is equal, a = x1x2x3....xn. Therefore, the product of x1, x2, ..., xn+1 (in that order) is also equal.
By induction, we can conclude that in a semigroup, all products of x1, x2, ...... xn (in that order) are equal.

To learn more about semigroup
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