The figure below shows a circular courtyard.
Its diameter is 96 m.

(a) Use the calculator to find the circumference and area of the courtyard.
Use 3.14 for it in your calculations, and do not round your answers.
Make sure to include the correct units.

Circumference:
Area:

(b) The courtyard will be covered with gravel.
Which measure would be used in finding the amount of gravel needed?

- Circumference
- Area

(c) The courtyard will be paved.
Which measure would be used in finding the amount of pavement needed?

- Circumference
- Area

The parametric equations for the line through $(8,-9,9)$ parallel to the $-axis$ are $x = 8$, $y = -9 + t$, and $z = 9$.

Since the line is parallel to the $-axis$, we know that the direction vector of the line is $\langle 0, 1, 0 \rangle$. We can use this information to write the parametric equations of the line as:

x=8+0t+=8

y=-9+1t=-9+t

z=9+0t==9

where $t$ is a parameter. Therefore, the parametric equations for the line through $(8,-9,9)$ parallel to the $-axis$ are $x = 8$, $y = -9 + t$, and $z = 9$.

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The probability of randomly choosing a square and the letter "I" together is 0.0227.

Probability of choosing a given square:

The total number of possible outcomes = 16

The probability of choosing a particular square = 1/16

The probability of choosing a particular letter in MISSISSIPPI:

Let the letter be "I"

There are 4 "I"s in MISSISSIPPI out of a total of 11 letters

The probability of choosing an "I" = 4/11.

The probability of choosing a square and the letter "I" = (1/16) × (4/11)

The probability of choosing a square and the letter "I"  = 4/176 or 0.0227.

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The Number of tenths in 37.6 is 37.6 or 37 6/10 or 37 3/5 so the correct answer was given by Student 2

A decimal is simply another way of representing a fraction with a denominator of 10, 100, 1000, or any power of 10. In other words, the decimal point separates the whole number part from the fractional part, with each digit to the right of the decimal point representing a different power of 10.

The teacher asked how many tenths are equivalent to 37.6, which means we're looking for a fraction with a denominator of 10. To figure out the answer, we need to convert 37.6 into a fraction with a denominator of 10.

To do this, we look at the digit in the tenths place, which is 6. This tells us that 37.6 is equivalent to 37 and 6 tenths,

= 37 6/10.

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which in this case is 2.

= 37 3/5.

So, which student is correct? Student 2 answered 37.6 tenths, which is equivalent to 37 and 6 tenths, or 37 6/10. This means that Student 2's answer is correct.

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The number (up to isomorphism) of abelian groups of order n is different from the number (up to isomorphism) of abelian groups of order m, unless n and m are isomorphic.

To understand why, consider the fact that the number of abelian groups of a given order is determined by the prime factorization of that order. Specifically, the number of abelian groups of order p^n is equal to the number of partitions of n, where p is a prime number. Thus, the number of abelian groups of a given order is determined by the prime factorization of that order.

If two orders have different prime factorizations, then the numbers of abelian groups of those orders will be different. For example, the number of abelian groups of order 12 is different from the number of abelian groups of order 15, since 12 and 15 have different prime factorizations. On the other hand, if two orders have the same prime factorization, then the numbers of abelian groups of those orders will be the same (up to isomorphism), since the number of abelian groups of an order is determined solely by the prime factorization of that order.

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The quantity of the street that is being covered by the light would be = 226.87 m². That is option A.

To calculate the area of the circle, the formula that should be used is given as follows;

Area or circle = π r²

where r = Diameter/2

But diameter = 17/2

radius = 8.5

Area of circle = 3.14×8.5×8.5

= 226.87 m²

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The required measure of arc BD is 76°.

A figure of a circle is shown,
Where mCB is 136°  and subtended by points C and D at B is 74°.
The measure of the arc CD is given as,
= 2 * 74
= 148

Now, BD is given as,
mBD + mCD + mCB = 360
mBD + 148 + 136 = 360
mBD = 76

Thus, the required measure of arc BD is 76°.

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Answer:

x = 19

Step-by-step explanation:

Use the Intersecting Secants Theorem to solve this:

8(8 + x) = 9(9 + 15)

64 + 8x = 81 + 135

8x = 216 - 64 = 152

x = 152/8 = 19

The possible value of angle G are given as follows:

g = 10º.

Suppose we have a triangle in which:

The lengths and the sine of the angles are related as follows:

[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]

The relation for this problem is given as follows:

sin(126º)/83 = sin(g)/17

Hence the measure of angle g is obtained as follows:

sin(g) = 17 x sine of 126 degrees/83

sin(g) = 0.1657

g = arcsin(0.1657)

g = 10º. -> rounded to the nearest degree.

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The missing number of the set is 19 or 49.

We are given that;

The number series 21, 26, 33, 35, 44, 47

Now,

The range is the difference between the maximum and minimum values in the data set. Here are the steps to find the missing number:

First, we need to identify the maximum and minimum values in the data set. The maximum value is 47 and the minimum value is 21.

Next, we need to subtract the minimum value from the maximum value to find the range. This gives us 47 - 21 = 26.

Since we are given that the range is 28, we need to find a number that would make the range 28. This means that we need to either increase the maximum value or decrease the minimum value by 2.

One possible way to do this is to replace 21 with 19. This would make the minimum value 19 and the range 47 - 19 = 28.

Another possible way to do this is to replace 47 with 49. This would make the maximum value 49 and the range 49 - 21 = 28.

Therefore, by the given range the answer will be 19 or 49.

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Answer:50

Step-by-step explanation:

50 , middle school students preferred espresso-flavored frozen yogurt.

The circle was created so the percentage of the total circle is 100%

Therefore, the sum of percentages of frozen yogurt flavors is 100

Dutch chocolate + country vanilla + sweet coconut + espresso + cake batter =100

Given,

Dutch chocolate = 21.5 percent,

country vanilla = 28.5 percent,

sweet coconut = 13 percent,

espresso = x (say),

cake batter = 27 percent

substituting in above we get x

21.5+28.5+13+x+27=100

90+x=100

x=100-90

x=10

Therefore, espresso = 10 percent

Total students = 10% of 500

=50

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The approximate surface area available for the​ label is 26 in²

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

Radius, r = (3/2) meters

Height, h = 2 meters

See attachment for complete question

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

Area available for label = Area of cylinder - Circle area

So, we have

Area available for label = 2πr(r + h) - πr²

Substitute the known values in the above equation, so, we have the following representation

Surface area = 2π * (3/2) * (3/2 + 2) - π * (3/2)²

Evaluate

Surface area = 26

Hence, the approximate surface area available for the​ label is 26 in²

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Answer:

It is translated left 6 units.

To expand the function f(x) = 1/6 + x^8 into a power series with center c = 0, we can use Eq. (1) from the text, which states that:

f(x) = ∑[n=0 to ∞] (f^(n)(c)/n!)(x-c)^n

Plugging in c = 0 and f(x) = 1/6 + x^8, we get:

f(x) = ∑[n=0 to ∞] [(d^n/dx^n)(1/6) / n!] x^n + ∑[n=0 to ∞] [(d^n/dx^n)(x^8) / n!] x^n

The first term simplifies to (1/6) ∑[n=0 to ∞] (0 / n!) x^n = 1/6, while the second term simplifies to ∑[n=0 to ∞] (x^(n+8) / n!) = ∑[n=8 to ∞] (x^n / (n-8)!).

Therefore, the power series expansion of f(x) is:

f(x) = 1/6 + ∑[n=8 to ∞] (x^n / (n-8)!)

The interval of convergence can be determined using the ratio test, which gives:

lim[n→∞] |(x^(n+1) / ((n-7)!)) / (x^n / ((n-8)!))| = lim[n→∞] |x / (n-7)| = 0

This limit is less than 1 for all values of x, which means that the power series converges for all x. Therefore, the interval of convergence is (-∞, +∞)

To answer the question, we first need to use Eq. (1) from the text to expand the function f(x) = 1/6 + x^8 into a power series with center c = 0. We then simplify the two terms using the derivatives of 1/6 and x^8, respectively. Finally, we determine the interval of convergence using the ratio test.

The power series expansion of f(x) is 1/6 + ∑[n=8 to ∞] (x^n / (n-8)!), and it converges for all values of x, which means that the interval of convergence is (-∞, +∞).

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The expression 15+3(x-4) is equivalent to 3x +3 which is option B . I hope that helps!

Answer:

x = 10.1785714286 which rounds to 10.2

y = 15.25 which rounds to 15.3

Step-by-step explanation:

The 4 angles inside any quadrilateral = 360

We know that 1 angle is 105. So that means the other 3 angles are:

360-105 = 255

Also, any 2 adjacent angles in a quadrilateral = 180.

So 105 + (4y+14) = 180.

Let's solve for y.

105 + (4y+14) = 180

4y+14 = 75

4y=61

y=15.25

Now let's solve for X - - -

We know that the 3 angles OTHER than the 105 add to 255.

4y+14 + 7y+1 + 7x+1 = 255

11y+16+7x=255

11y+7x=239

If y = 15.25, plug that in and solve for x.

11y + 7x = 239

11(15.25) + 7x = 239

167.75 + 7x = 239

7x = 71.25

x = 10.1785714286

Let's double check that everything adds to 360:

105 + 4y+14 + 7y+1 + 7x+1 = 360

105 + 4(15.25) + 14 + 7(15.25) + 1 + 7(10.18) + 1 = 360

The power factor in a system is defined as the cosine of the angle between the voltage and current waveforms. The power factor in this system is 0.906, indicating a relatively efficient use of power.

In this case, the voltage waveform is given as V = 120V sin(377t + 20°) and the current waveform is given as I = 60A sin(377t + 45°). To find the power factor, we need to determine the angle between the voltage and current waveforms. First, let's convert the voltage and current waveforms to phasor form:
V = 120V ∠ 20°
I = 60A ∠ 45°
The angle between the voltage and current phasors is given by:
θ = θv - θi = 20° - 45° = -25°
The power factor is the cosine of this angle, so:
PF = cos(-25°) = 0.906

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The Taylor polynomials [tex]p_{4}[/tex] and [tex]p_{5}[/tex] centered at [tex]a = \frac{\pi}{6}[/tex] for f(x) = 5cos(x) are:

[tex]p_{4}(x) = \frac{ 5\sqrt{3}}{2} - \frac{5}{2} (x - \frac{\pi }{6}) - \frac{ 5\sqrt{3}}{4} (x - \frac{\pi }{6})^2+ \frac{5}{8}(x - \frac{\pi }{6})^3+ \frac{ 5\sqrt{3}}{48}(x - \frac{\pi }{6})^4[/tex][tex]p5(x) = p_{4}(x) = \frac{ 5\sqrt{3}}{2} - \frac{5}{2} (x - \frac{\pi }{6}) - \frac{ 5\sqrt{3}}{4} (x - \frac{\pi }{6})^2+ \frac{5}{8}(x - \frac{\pi }{6})^3+ \frac{ 5\sqrt{3}}{48}(x - \frac{\pi }{6})^4 - \frac{5}{384}(x - \frac{\pi }{6} )^6[/tex]

To find the Taylor polynomials centered at [tex]a = \frac{\pi}{6}[/tex] for f(x) = 5cos(x), we need to find the derivative of the function at [tex]x = \frac{\pi}{6}[/tex].  The first derivative of f(x) = 5cos(x) is -5sin(x), and the second derivative is -5cos(x).

Evaluating these derivatives at [tex]x = \frac{\pi}{6}[/tex] gives us

[tex]-5sin(\frac{\pi }{6}) = -\frac{5}{2}[/tex] and [tex]-5cos(\frac{\pi }{6}) = -\frac{5\sqrt{3} }{2}[/tex].

The Taylor polynomial [tex]p_{4}(x)[/tex] is then constructed using these derivatives and the powers of [tex]x - \frac{\pi}{6}[/tex] up to the fourth power.

Similarly, for [tex]p_{5}(x)[/tex], we add the fifth derivative term. Simplifying the expressions gives us the Taylor polynomials [tex]p_{5}(x)[/tex] and [tex]p_{4}(x)[/tex] center [tex]= \frac{\pi }{6}[/tex] for f(x) = 5cos(x).

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Since we need to round up to the next integer, the required sample size for this experiment is 284 people for the confidence level.

To find the required sample size for NASA's experiment, we can use the following formula for the sample size estimation in a proportion experiment:

[tex]n = (Z^2 * p * (1-p)) / E^2[/tex]

where:
- n is the sample size
- Z is the z-score corresponding to the desired confidence level (85% in this case)
- p is the estimated population proportion (0.33)
- E is the margin of error (0.04)

First, we need to find the z-score for an 85% confidence level. We can look this up in a z-table, or use an online calculator. The z-score for an 85% confidence level is approximately 1.44.

Next, we can plug the values into the formula:
[tex]n = (1.44^2 * 0.33 * (1-0.33)) / 0.04^2[/tex]
n ≈ (2.0736 * 0.33 * 0.67) / 0.0016
n ≈ 283.66

Since we need to round up to the next integer, the required sample size for this experiment is 284 people.

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The annual interest rate on the $5600 loan, with $1008 of interest accrued over 4 years, is 4.5%, as calculated using the formula for simple interest. Option B.

To find the annual interest rate, we can use the formula for simple interest: I = P * R * T, where I is the interest, P is the principal amount (loan amount), R is the interest rate, and T is the time in years.

Given that the loan amount is $5600 and the interest after 4 years is $1008, we can rearrange the formula to solve for R. In this case, R = (I / P) / T = (1008 / 5600) / 4 = 0.045 = 4.5%. Therefore, the annual interest rate on the loan is 4.5%. The correct answer is option (b) 4.5%.

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To find f(t) given ℒ−1{1/(s^2-6s+10)}, we can use partial fraction decomposition and complete the square to rewrite the denominator:

1/(s^2-6s+10) = A/(s-3+I) + B/(s-3-I)

where A and B are constants to be determined, and I is the imaginary unit.

Multiplying both sides by (s-3+I)(s-3-I) = s^2 - 6s + 10, we get:

1 = A(s-3-I) + B(s-3+I)

Expanding and collecting like terms, we get:

1 = (A+B)s - 6A + 10B - AI - BI

Since this equation must hold for all values of s, we can equate the coefficients of each power of s:

A + B = 0 (coefficient of s^1)
-6A + 10B - AI - BI = 1 (coefficient of s^0)

Solving these equations simultaneously, we get:

A = -1/2 + I/2, B = -1/2 - I/2

Substituting these values back into the partial fraction decomposition, we get:

1/(s^2-6s+10) = (-1/2 + I/2)/(s-3+I) + (-1/2 - I/2)/(s-3-I)

Taking the inverse Laplace transform of each term separately using standard formulas, we get:

f(t) = e^(3t/2) sin(t/2)

Therefore, the function f(t) is e^(3t/2) sin(t/2).

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Part A: The best graphical representation to display the data is histogram.

Part B : The histogram is represented below.

Part A

A histogram would be the ideal type of chart to display the provided data. The distribution and frequency of data can be shown using a histogram in various intervals or bins. It enables us to see how the grades are distributed and see any patterns or trends.

Part B

Follow these steps to make a histogram to represent the given data:

A visual representation of how the grades are distributed over the various intervals will be provided by the resulting histogram, making it simple to understand the class's overall performance.

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The change in radius is supposed to be 2.04 units to get 800 as volume.

To calculate the volume of a cylinder, we use the formula V = πr^2h, where V represents the volume, r represents the radius, and h represents the height.

Given that the height is fixed at 10 units, and the volume is desired to be 800 cubic units, we can rearrange the formula to solve for the radius:

V = πr^2h

800 = πr^2(10)

To isolate the radius, we divide both sides of the equation by π * h * 10:

800 / (π * 10) = r²

Simplifying further:

80 / π = r²

To find the value of the radius, we take the square root of both sides:

√(80 / π) = r

Using a calculator to approximate the square root of 80 divided by π, we find:

r ≈ 5.04

Therefore, to achieve a volume of 800 cubic units while keeping the height at 10 units, the radius would need to be approximately 5.04 units.

And change would be 5.04-3 = 2.04.

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The value of b such that p(70 ≤ x ≤ b) = 0.3 is 67.4 (rounded to one decimal place). The deviation from the mean for this value of b is about -2.12 times the standard deviation of 5.

To find b, we can use the z-score formula, where z = (x - μ) / σ. In this case, we want to find the value of b that corresponds to a probability of 0.3, which means that the area under the normal distribution curve between 70 and b is 0.3.
First, we need to find the z-score for x = 70. Using the formula, we get:
z = (70 - 70) / 5 =
Next, we need to find the z-score for the value of b that corresponds to a probability of 0.3. We can use a standard normal distribution table or a calculator to find this value. For example, using a calculator, we can input:
invNorm(0.3) = -0.5244
This means that the z-score for the value of b is -0.5244. We can use the z-score formula again to find the actual value of b:
-0.5244 = (b - 70) / 5
Solving for b, we get:
b = 67.38
Therefore, the value of b such that p(70 ≤ x ≤ b) = 0.3 is 67.4 (rounded to one decimal place).
In terms of deviation, we can see that the value of b is about 2.12 standard deviations below the mean (z = -0.5244 corresponds to an area of 0.3 under the normal distribution curve). This tells us that the value of b is relatively low compared to the mean of 70. The deviation from the mean for this value of b is about -2.12 times the standard deviation of 5.

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The product (k-1) (6k+5) is C, [tex]6k^2 - k - 5.[/tex] therefore, option C, [tex]6k^2 - k - 5.[/tex] is correct.

To find the product of (k-1) and (6k+5), we can use the distributive property of multiplication.

We can multiply each term in the first expression (k-1) by each term in the second expression (6k+5), and then simplify:

[tex](k-1)(6k+5) = k(6k+5) - 1(6k+5)\\(k-1)(6k+5) = 6k^2 + 5k - 6k - 5\\(k-1)(6k+5) = 6k^2 - k - 5[/tex]

Therefore, the answer is C, [tex]6k^2 - k - 5.[/tex]

We can check our answer by multiplying it out using the distributive property, and we should get the original expressions (k-1) and (6k+5) back.

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Answer:

a) AE and CD

b) AE and ED

c) AED and CDE

Answer:

answer below

Step-by-step explanation:

a) will be all of them

b)will be all of their unions, so the values they all have in common in this case 4, 6

c)will be the values in common with a and b and all of c,

d)will be all of the values of a and b and all of the values in common with c

sorry I csnnot give an actual answer at the moment, but i can explain what each question wants from you in literal word form.

Solving a system of equations we can see that she has 10 dimes and 50 pennies.

Let's define the variables:

x = number of pennies.

y = number of dimes.

With the given information we can write a system of equations:

x = 5*y

x*0.01 + y*0.10 = 1.50

We can replace the first equation into the second one:

5*y*0.01 + y*0.10 = 1.50

y*0.15 = 1.50

y = 1.50/0.15 = 10

So there are 10 dimes and 50 pennies.

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The given parabola 8(y-2) = (x + 2)², the focus is (-2, 4), the directrix is y = 6, the vertex is (-2, 2), and the axis of symmetry is the vertical line x = -2.

To find the focus, directrix, vertex, and axis of symmetry of a parabola in standard form, we can rewrite the given equation as y = (1/8)(x + 2)² + 2. Comparing this equation with the standard form y = a(x - h)² + k, we can determine the values of h, k, and a. From the equation, we can see that the vertex is given by (h, k), which in this case is (-2, 2). The vertex represents the point where the parabola reaches its minimum or maximum value.

The axis of symmetry is a vertical line passing through the vertex. Therefore, the axis of symmetry for this parabola is x = -2.

The focus of a parabola is a point that lies on the axis of symmetry and is equidistant from the directrix. The distance between the focus and the vertex is given by the equation |1/(4a)|, where a is the coefficient of the x-term. In this case, a = 1/8, so the distance between the focus and the vertex is |1/(4(1/8))| = |2| = 2. Since the vertex is at (-2, 2), the focus is located at (-2, 2+2) = (-2, 4).

The directrix of a parabola is a line perpendicular to the axis of symmetry and is equidistant from the focus. Since the vertex is at (h, k) = (-2, 2) and the focus is at (-2, 4), the directrix is a horizontal line located at y = 2 + 2 = 6.

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The probability of Z being less than 0.618 is approximately 0.7314. the probability of P(0.7 < X < 0.75) is approximately 0.7314.

a. The mean of X1 can be found by taking the expected value of the distribution:

E(X1) = ∫0^10 x f(x) dx

= ∫0^10 x[(3x^2)/2 + x] dx

= 78.75/4

= 19.6875

Therefore, the mean of X1 is 19.6875.

b. The variance of X1 can be found using the formula:

Var(X1) = E(X1^2) - [E(X1)]^2 E(X1^2) can be found by taking the second moment of the distribution:

E(X1^2) = ∫0^10 x^2 f(x) dx

= ∫0^10 x^2 [(3x^2)/2 + x] dx

= 1095/8

Therefore,

Var(X1) = 1095/8 - (78.75/4)^2

= 16.3203125

c. Using the central limit theorem, we can approximate the distribution of the sample mean with a normal distribution.

The mean of the sample mean is the same as the population mean, which we found to be 19.6875 in part a. The variance of the sample mean can be found by dividing the population variance by the sample size:

Var(X) = Var(X1)/n

= 16.3203125/100

= 0.163203125

Then, we can standardize the sample mean using the formula:

Z = (X - μ)/(σ/√n)

where μ is the population mean, σ is the population standard deviation (which we found to be √Var(X1) ≈ 4.0407), and n is the sample size.

Plugging in the values, we get:

Z = (0.725 - 0.7)/(4.0407/√100)

= 0.618

Using a standard normal distribution table or calculator, we can find that the probability of Z being less than 0.618 is approximately 0.7314. Therefore, the probability of P(0.7 < X < 0.75) is approximately 0.7314.

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Our quotient is 1482 R 4. We can also write this as 1482 and 4/5 or 1482.8.

To find the quotient of a division problem, we must divide the dividend by the divisor. In this case, our dividend is 7415 R 112 R 412 R 214 R 4 and our divisor is 5.

First, we divide 5 into 74 and get a quotient of 14 with a remainder of 4. We bring down the next digit, 1, and get a new dividend of 41. We divide 5 into 41 and get a quotient of 8 with a remainder of 1. We bring down the next digit, 2, and get a new dividend of 12.

We divide 5 into 12 and get a quotient of 2 with a remainder of 2. We bring down the next digit, 1, and get a new dividend of 21. We divide 5 into 21 and get a quotient of 4 with a remainder of 1.

Finally, we bring down the last digit, 4, and get a new dividend of 14. We divide 5 into 14 and get a quotient of 2 with a remainder of 4.

Therefore, our quotient is 1482 R 4. We can also write this as 1482 and 4/5 or 1482.8.

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