find the taylor series up to the degree 4 term at a = 8 for f(x) = ex.

a) The total cost of purchasing the car with the harmonized sales tax (HST) is $36,160.

b) If the customer leases the car instead of buying it, they will save $14,760.

c) After the customer returns the leased car, their lease options include:

A lease is a financing arrangement for capital assets, that enables the lessee to use the asset for a determined period while making periodic payments to the lessor.

Leases are classified into operating and finance (capital) leases.

Value of the car = $32,000

Harmonized Sales Tax (HST) = 13%

Cost of the car with HST = $36,160 ($32,000 x 1.13)

Down payment = $1,000

Financing part = $35,160 ($36,160 - $1,000)

Installment payments = 48

Periodic payments = $425

Total lease cost = $21,400 ($1,000 + 48 x $425)

Savings from leasing = $14,760 ($36,160 - $21,400)

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On solving the provided question we can say that the value of equation here will be as duration is 0<t<5 seconds.

An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions

Information given in the query:

Rooftop height is 70 feet.

Speed at start: 80 feet per second

s(t ) = - 16t² + 80t + 70

s(t) in feet here

now,

for the balls to touch the rooftop

[tex]s(t ) = - 16t^2+ 80t + 70 > 70\\ - 16t^2 + 80t + 70 > 70\\ - 16t^2 + 80t > 70 - 70\\ - 16t^2 + 80t > 0\\ 16t^2 - 80t < 0\\t( 16t - 80 ) < 0\\t < 0 and 16t - 80 < 0 or 16t < 80 or t < 5\\[/tex]

Hence,

since time cannot be negative, the time period is 0 t 5 seconds.

0<t<5

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Yes, the graph of linear equation is always a straight line.

Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.

Consider the equation y = mx + c to define a linear function with a straight line as its graph.

We are aware that the rate of change, m, for a linear function is constant. On the graph, shifting by 1 always causes you to go up by m as shown in attached image.

The graph thus resembles a staircase. It is a straight line because it always rises in equal-sized steps.

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The number of months it will take for both cities to have the same population is; 36 months

We are given;

Current population of city of rock hill = 70000

Rate at which people move into  city of rock hill = 100 people per month

Rate at which people move out of city of rock hill = 150 people per month

Net average increase a month = 100 - 150 = -50 people per month

Current population of City of Greenville = 61000

Rate at which people move into  City of Greenville = 200 people per month

If the number of months it will take for both cities to have the same population is x, then it means that;

70000 - 50x = 61000 + 200x

Rearrange to get;

200x + 50x = 70000 - 61000

250x = 9000

x = 9000/250

x = 36 months

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(-1, -2)is the best solution of  the system of equations.

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

A system of equations is graphed on this coordinate grid.

we need to find the ordered pair which is the best estimate of the solution of the system of equations.

(-1, -2) is the best solution of this graph because both the lines are intersecting at that point

Hence, (-1, -2)is the best solution of  the system of equations.

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Two positive numbers whose product is 192 and whose sum is a minimum are; 8√3 and 8√3.

The correct option is (D).

Minima is the minimum value of a function in given domain.

First Order Derivative Test

Let f be the function defined in an open interval I and f be continuous at critical point c in I such that f’(c) = 0.

If f’(x) changes sign from negative to positive as x increases through point c, then c is the point of local minima. And the f(c) is the minimum value.

Given,

Product of two positive numbers = 192

Sum should be minimum

Let x be the first positive number and y be the other positive number. So, the equation would be

x . y = 192

⇒ y = 192/x       --------(a)

Let S be the sum of the two positive numbers.

S = x + y

substituting value of y in terms of x from equation(a)

S = x + 192/x

⇒ S = x + 192 . x⁻¹

Differentiating with respect to x

S'(x) = 1 + (-1)192x⁻²

S'(x) = 1 - 192/x²

S'(x) = (x² - 192)/x²

To determine the minimum S, equating first derivative of S with respect to x to zero.

S'(x) = 0

(x²-192)/x² = 0

x² - 192 = 0

x² = 192

x = ±√192

Rejecting negative root, since the numbers are positive

x = √192

Substituting value of x in equation (a)

y = 192/√192

y = √192

x and y can be written as

x = √192 = 8√3

y = √192 = 8√3

Hence , 8√3  and 8√3 are the two positive numbers whose product is 192 and sum is a minimum.

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A summary of 200 factory workers' monthly sick days is provided. From this, we can tell that about 15 workers took more than 10 sick days per month.

The number of times an event or observation occurred during an experiment or research is referred to as its frequency in statistics. It can alternatively be described as a straightforward count of a specific occurrence. Relative frequency and cumulative frequency are the two main types of frequency seen in statistics.

The given table contains the number of sick days per month and the frequency of factory workers on sick days. Then, the number or frequency of workers who took more than 10 sick days per month is,

number of workers = frequency for 11-15 days + frequency for 16-20 days

=14+1

=15

The required answer is 15 workers.

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The correct rule of the transformation is,

⇒ Dilation of 0.5.

A transformation that occurs when a figure is moved from one location to another location without changing its size or shape is called translation.

We have to given that;

The coordinate of CDE are,

⇒ C (4, - 5), D (3, - 1), E (5, - 3)

And, After transformation the coordinate of image C'D'E' are,

⇒ C' (2, - 2.5) , D' (1.5, - 0.5), E' (2.5, - 1.5)

Now, We have;

⇒ C (4, - 5) = C' (2, - 2.5)

Let a dilation of the points = k

So, We can formulate;

⇒ 4 × k = 2

⇒ k = 2/4

⇒ k = 1/2

⇒k = 0.5

Hence, The correct rule of the transformation is,

⇒ Dilation of 0.5.

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Considering the coordinates of the preimage of line AB represented by A (6, -4), B (2, -8) and coordinates of the image of line A'B' represented by A' (9, -6), B'(3, -12) the dilation factor is 3/2

The transformation rule for dilation is as follows

(x, y) for a scale factor of r → (rx, ry)

following similar procedure for the given problem we have

A (6, -4) transformed to A' (9, -6)

B (2, -8) transformed to B' (3, -12)

hence 6r = 9, and -4r = -6

r = 9/6 = -6/-4

r = 3/2

The dilation factor of the transformation is 3/2

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The given differential equation by undetermined coefficients are y(x)=c1cos(31/2 x)+c2sin(31/2x)+(-4x2+4x-4/3)e3x

A differential equation in mathematics is an equation that includes one or more functions and their derivatives. The rate of change of a function at a place is determined by the derivatives of the function. It is mostly employed in disciplines like physics, engineering, biology, and others. The study of solutions that satisfy the equations and the characteristics of the solutions is the main goal of the differential equation.

y'' + 3y=0

λ2+3 =0

λ 1=-i31/2,

λ 2=i31/2.

y0(x)=c1cos(31/2 x)+c2sin(31/2x),

yp(x)=(Ax2+Bx+C)e3x

yp'(x)=(3Ax2+(3B+2A)x+3C+B)e3x

yp''(x)=(9Ax2+(9B+12A)x+2A+6B+9C)e3x

x2e3x:3A+9A=-48,

xe3x:3B+9B+12A=0,

e3x:3C+2A+6B+9C=0.

=>A=-4,B=4,C=-4/3.

yp(x)=(-4x2+4x-4/3)e3x,

y(x)=y0(x)+yp(x)

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The range of the given function y = 3x - 2 for the domain (-1, 2, 4) is

A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.

Domain of the function is the values of x for which the function is defined.

Range of the function is the value of the function when we put the values in the domain.

Given function,

y = 3x - 2

Given the domain (-1, 2, 4)

x = -1, then y = (3 × -1) - 2 = -5

x = 2, then y = (3 × 2) - 2 = 4

x = 4, then y = (3 × 4) - 2 = 10

Range is (-5, 4, 10).

Hence the range of the function is (-5, 4, 10).

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The rectangular box dimensions are 21m ,12m and 10m.

What is volume of the rectangular box?

The area of the base times the height of the rectangular prism equals the volume of the object. As a result, the formula for a rectangular prism's volume is given. A rectangular prism's volume is equal to its length, width, and height in cubic units.

Here the dimensions of a box is 2x+5 , x+4 and x+2 meters and volume of the box is 2520 [tex]m^3[/tex].

Volume of rectangular box = w×h×l [tex]unit^3[/tex]

=> 2520 = (2x+5)(x+4)(x+2)

=> 2520 = [tex](2x^2+13x+20)(x+2)[/tex]

=> 2520 = [tex]2x^3+4x^2+26x+13x^2+20x+40[/tex]

=> [tex]2x^3-16x^2+33x^2-264x+310x-2480=0[/tex]

=> [tex]2x^2(x-8)+33x(x-8)+310(x-8)=0[/tex]

=> [tex](x-8)(2x^2+33x+310)=0[/tex]

Here x-8=0 and [tex]2x^2+33x+310=0[/tex]

Then x=8 and x ∈ R.

Now the dimensions 2x+5=2(8)+5=16+5=21m.

=> x+4 = 8+4=12 m.

=> x+2= 8+2 = 10m.

Hence the rectangular box dimensions are 21m ,12m and 10m.

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Correct option is E, The roots of Auxiliary equation are √6 and -√6.

The solution of an nth-order differential equation or difference equation depends on an algebraic equation of degree n called the characteristic equation (or auxiliary equation) in mathematics.

The auxiliary equation, however, lacks true roots. Consider the simple harmonic equation y + y = 0, which has solutions and was given that name because of how it relates to the vibration of a musical tone. Y2(t) = cos t and Y1(t) = sin t. r2 + 1 = 0 is the auxiliary equation for the straightforward harmonic equation.

The Auxiliary form of the given equation is m² - 6 = 0

Using identity a² - b² = (a - b) (a + b),

⇒ The equation becomes (m - √6)(m+√6) = 0

⇒ either m = √6 or m = -√6

Hence, The roots of Auxiliary equation are √6 and -√6.

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The exact ratio of of the volume of the cone that is occupied by the sphere is 4/(1+√2)³.

What is a cone?

A cone is a three-dimensional geometric shape that narrows smoothly from a flat base (typically circular base) to a point called the apex or vertex (which creates an axis to the centre of base).

Given that the height and radius of the cone is the same.

Assume that the radius of the cone is x.

Thus the slant height of the cone is √(x² + x²) = x√2.

Assume the radius of the sphere is r.

△OEC and △BDC are similar triangles according to AAA rule.

Thus, BD/OE =BC/OC

x/r = x√2/(x -r)

Cross multiply:

x(x - r) = xr√2

x² - rx = xr√2

x² = xr√2 + rx

rx( 1+√2) = x²

r = x²/[x( 1+√2)]

r = x/ (1+√2)

The volume of the sphere is 4/3 ∏ r³ = 4/3 ∏ [ x/ (1+√2)]³

The volume of the cone is 1/3 ∏r²h =  1/3 ∏x³

The ratio of the volume of the sphere to the cone is

4/3 ∏ [ x/ (1+√2)]³  : 1/3 ∏r³

= 4 : (1+√2)³

= 4/(1+√2)³

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Distance traveled after toll payment is 1.2 miles on highway 40.

The distance may be calculated using a curved route. Displacement measurements can only be made along straight lines. Distance is path-dependent, meaning it varies depending on the direction followed. Displacement simply depends on the body's beginning and ending positions; it is independent of the route.

Distance is the sum of an object's movements, regardless of direction. Distance may be defined as the amount of space an item has covered, regardless of its beginning or finishing position.

The size or extent of the displacement between two points is referred to as distance. Keep in mind that the distance between two points and the distance traveled between them are not the same. The entire length of the journey taken between two points is known as the distance traveled. Travel distance is not a vector.

Distance traveled before toll payment =3/5 miles on highway 40

Distance traveled after toll payment =2*3/5  = 6/5 =

1.2 miles on highway 40.

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Answer:AC is equal to 88

Step-by-step explanation:

C=11

A=8

B=2

The average value of 3t^3-t^2 over the interval -1<=t<=2 is 11/4

Mean Value Theorem states if a function f(x) is continuous on the interval [a,b] and differentiable on (a,b), there is at least one value c in the interval (a<c<b) such that:

f '(c)=f(b)-f(a)/b-a

Average Value:

The average value of a function of a variable over a range represents the height of a rectangle of the same area as that defined by the function under the curve. This value can be calculated using a formula:

favg=1/b−a∫f(t)dt

consider f(x)=3t³-t² and limits are -1 to 2

Now, substitute these values in the above formula:

favg=1/2-(1)∫3t³-t²dt

favg=1/3∫(3/4t⁴-1/3t³)|₋₁²

favg=(12-8/3)-(3/4+1/3)

By solving the equation we get

favg=11/4

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Answer:

36

Step-by-step explanation:

p/9=4

p=4*9

p=36

Answer:

--------------------------------------

We observe that, the sequence is symmetric and all the terms are perfect squares:

This is neither arithmetic nor geometric.

The zero term is 16 and the fourth term is 0 so the nth term would be:

Answer:

a)  Neither

b)  t(n) = n² - 8n + 16

Step-by-step explanation:

An Arithmetic Sequence has a constant difference between each consecutive term.

A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.

As the sequence has neither a constant difference or a constant ratio, the sequence is neither arithmetic or geometric.

Work out the differences between the terms until the differences are the same:

First differences

[tex]16 \underset{-7}{\longrightarrow} 9 \underset{-5}{\longrightarrow} 4 \underset{-3}{\longrightarrow} 1 \underset{-1}{\longrightarrow} 0 \underset{+1}{\longrightarrow} 1 \underset{+3}{\longrightarrow} 4[/tex]

Second differences

[tex]-7 \underset{+2}{\longrightarrow} -5 \underset{+2}{\longrightarrow} -3\underset{+2}{\longrightarrow} -1\underset{+2}{\longrightarrow} 1\underset{+2}{\longrightarrow} 3[/tex]

As the second differences are the same, the sequence is quadratic and will contain an n² term.  The coefficient of n² is always half of the second difference.  Therefore, the coefficient of n² = 1.

Write out the numbers in the sequence n² and determine the operation that takes n² to the given sequence:

[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} n&0& 1& 2&3 &4 &5 &6 \\\cline{1-8}n^2 &0& 1& 4& 9&16 & 25&36 \\\cline{1-8} \sf operation&+16& +8&+0&-8&-16&-24&-32\\\cline{1-8} \sf sequence & 16&9 &4 & 1& 0& 1& 4\\\cline{1-8}\end{array}[/tex]

As the operation is not constant, work out the differences between the operations:

[tex]16\underset{-8}{\longrightarrow} 8\underset{-8}{\longrightarrow} 0\underset{-8}{\longrightarrow} -8\underset{-8}{\longrightarrow} -16\underset{-8}{\longrightarrow} -24\underset{-8}{\longrightarrow} -32[/tex]

As the differences are the same, the second operation in the sequence is -8n.  Write out the numbers in the sequence with both operations and and determine the operation that takes (n² - 8n) to the given sequence:

[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} n&0& 1& 2&3 &4 &5 &6 \\\cline{1-8}n^2 -8n&-0&-7&-12&-15&-16&-15&-12\\\cline{1-8}\sf operation &+16&+16&+16&+16&+16&+16&+16\\\cline{1-8} \sf sequence & 16&9 &4 & 1& 0& 1& 4\\\cline{1-8}\end{array}[/tex]

As the operation is constant, the final operation in the sequence is +16.

So the equation for the nth term is:

[tex]\implies t(n)=n^2-8n+16[/tex]

Answer:Well, we already see that the number of girls is greater than the number of boys, to find the result of girls, just do the division that is 30:2= 15 and boys is 30:3= 10

i hope i have helped you

Step-by-step explanation:

Answer:

8 boys, 22 girls

To solve, you must isolate the variable. This means moving the variable and it's coefficient to the other side of the equation with only the same variable on the same side with all the constants on the other side.

If you multiply, divide, add, subtract, square root, exponent both sides of the equation by the same value, that end value stays the same.

Q)

Lets set the amount of boys to x.

The amount of girls is 3x-2 as the amount of boy is x.

Set up an equation.

x+3x-2 = 30

30 is the amount of members.

Add 2 on both sides.

x+3x+2-2=30+2

Simplify by combining like terms.

4x = 32

Divide 8 on both sides.

[tex]\frac{4x}{4} =\frac{32}{4}[/tex]

x = 8

There are 8 boys in the class.

Plug back into the equation or subtract from 30 to solve for the amount of girls.

30 - 8 = 22

8(3) - 2 = 22

Answer:

Step-by-step explanation:

The vertex (h, k) is always in the format of "x-" so if we have a positive h value for the vertex, it would be written as "x-(h)" and if we have a negative h value for the vertex, it would be written as "x-(-h)" which is the same as "x+h". Since our sign inside the absolute value symbols is negative, the h value for the the vertex is 9. The k value is always what's outside the absolute value symbols, including the sign. Ours is a positive 7. Thus the vertex is (9, 7).

Note that in the above scenario, the original dimensions of the piece of metal that is rectangular are:

Let x equal the width of the box and x + 15 equal the length of the box.  

Subtract 6 (the amount cut out) from both of these to get the following terms: (x+9) and (x-6).  

These are the length and width of the box respectively.  We also know that the height of the box is 3.  Plugging these into the formula for the volume of a box gives you:

3(x+9)(x-6)=858

If the above is expanded, we have:

3x²+9x-162=858

Subtract 858 from both sides to make the quadratic equation equal zero.  Plug the a, b and c coefficients into the quadratic formula (or factor) to find x.

3x²+9x-162- 858 = 0

⇒ 3x²+9x -1020= 0

At the stage we solve for x:

3x2+9x−1020 = 0

Factor left side of the equation.

3(x−17)(x+20)=0

Set factors equal to 0.

x−17=0 or x+20=0

x=17 or x=−20

We can't get a negative side, so the answer that makes sense is x =17 recall that the metal is 15 inches longer than its width.

So if the Width is x and the Lenght is x +15

Then Width is 17 while the length is 17 + 15 = 32inces.

Thus the original dimensions of the metal are:

Length: 32 Inches

Width: 15 Inches

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The surface area of the capsule is calculated by adding the surface area of the cylinder and the surface area of the hemispherical bowl.

A hemispherical bowl is a type of bowl with a rounded bottom that is shaped like half of a sphere. It is often used in laboratories, kitchens, and dining rooms. The bowl’s curved surface prevents the contents from spilling, which makes it an ideal choice for mixing, swirling, and scooping. Additionally, the bowl’s shape makes it look aesthetically pleasing, so it can be used as a decorative piece as well.

The surface area of the cylinder = 2πr2 + 2πrh

Where r is the radius of the cylinder and h is the height of the cylinder.

The radius of the cylinder = 12 cm

The height of the cylinder = 16 cm

Therefore, the surface area of the cylinder = 2π(12)2 + 2π(12)(16) = 1536π cm2

The surface area of the hemispherical bowl = 2πr2

Where r is the radius of the hemispherical bowl.

The radius of the hemispherical bowl = 12 cm

Therefore, the surface area of the hemispherical bowl = 2π(12)2 = 144π cm2

The total surface area of the capsule = 1536π cm2 + 144π cm2 = 1680π cm2

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The events Grade 5 and Science"  are independent events,  P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) .Option B is the correct option.

What is frequency?

The frequency (f) of a specific value is the number of occurrences of the value in the data. The frequency distribution of a variable is the collection of all possible values and the frequencies associated with these values. Frequency distributions are represented graphically as frequency tables or charts.

Given table is

                   English      Science   Total

Grade 5           24              36           60

Grade 8           16               24           40

Total                40              60           100

Total students is 100.

Total student in grade 5 is 60.

The number of students who prefer science is 60.

The number of students who are in grade 5 and prefer science is 36.

The probability of grade 5 is P(grade 5) = 60/100 =0.6

The probability of science is P(science) = 60/100 = 0.6

The probability of Grade 5 ∩ Science is P(Grade 5 ∩ Science) = 36/100 =0.36.

Thus P(Grade 5 ∩ Science) =P(Grade 5) ⋅ P(Science).

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Answer: 40 dollars

Step-by-step explanation:

i hope the explanation is right:)

If he invites 5 friends for 40 bucks each then one friend that he knows will go is 40 dollars. So the only thing you have to do for this problem is multiply 40 by the number of friends which in this case i believe is 1.

Answer:

2,4,6,8,0

Step-by-step explanation:

A number is divisible by 2 if it ends in 2, 4, 6, 8 or 0. A number is divisible by 5 if it ends in 5 or 0. A number is divisible by 10 if it ends in a 0.

Answer:

B,C

Step-by-step explanation:

B. All waves have both an amplitude and frequency.

C. All waves have a repeating pattern.

A statement A is incorrect, Waves are a disturbance that transfer energy through space without transferring matter.

B is true, All types of waves have an amplitude, which is the measure of the maximum displacement of a point on the wave from its rest position and a frequency, which is the number of complete oscillations of a point on the wave per unit of time.

C is also true, All types of waves have a repeating pattern, which is known as a waveform. The shape of the waveform determines the type of wave. For example, sine waves have a smooth, curved shape, while square waves have sharp, angular edges.

On solving the provided question we can say that in the circle the radius, r = 4 and center, c = -3, -1, so area of the circle will be = [tex]\pi r^2[/tex] = 3.14*4*4 = 50.24 cm sq.

Every point in the plane that is a certain distance away from a certain point forms a circle (center). It is, thus, a curve formed by points moving in the plane at a fixed distance from a point. At every angle, it is also rotationally symmetric about the center. A circle is a closed two-dimensional object where every pair of points in the plane are equally spaced out from the "center." A line that goes through the circle creates a specular symmetry line. At every angle, it is also rotationally symmetric about the center.

here,

in the circle

the radius, r = 4

and center, c = -3, -1

area of the circle will be = [tex]\pi r^2[/tex] = 3.14*4*4 = 50.24 cm sq.

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The number of red counters in the bag is given as follows:

8.

A probability is obtained by the division of the number of desired outcomes by the number of total outcomes.

The probabilities for this problem are given as follows:

The sum of the probabilities is of one, hence the value of x is obtained as follows:

2x + x + 0.45 + 0.25 = 1

3x = 0.3

x = 0.1.

Hence the proportion of red is given as follows:

0.2.

There are 18 blue counters in the bag, which is of 0.45 of the total amount, hence the total amount is obtained as follows:

0.45n = 18

n = 18/0.45

n = 40.

Meaning that the number of red counters is given as follows:

0.2 x 40 = 8.

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