One factor of the function f(x) = x^3 − 9x^2 + 20x − 12 is (x − 6). Describe how to find the x-intercepts and the y-intercept of the graph of f(x) without using technology. Show your work and include all intercepts in your answer.

The discontinuity of the given function is at (−2, 1) and zero at (−3, 0).

The given function is:

f(x) =   [tex]\frac{x^{2} + 5x + 6}{x + 2}[/tex]

We will factorize the numerator and then reduce this function.

= [tex]\frac{x^{2} + 2x + 3x + 6}{x + 2}[/tex]

=  [tex]\frac{x(x + 2) +3 (x + 2)}{x + 2}[/tex]

= [tex]\frac{(x + 2) (x + 3)}{x + 2}[/tex]

If we take the value of x as -2, both the numerator and denominator will be 0. Note that for x = -2, both the numerator and denominator will be zero. When both the numerator and denominator of a rational function become zero for a given value of x we get a discontinuity at that point. which means there is a hole at x = -2.

Now, when we reduce this function by canceling the common factor from the numerator and denominator we get the expression f(x) = x + 3.  If we use the value of x = -2 in the previous expression we get;

f(x) = x + 3 =  = -2 + 3

f(x) = 1

Therefore, there is a discontinuity (hole) at (-2, 1).

If x = -3, the value of the function is equal to zero. This means x = -3 is a zero or root of the function.

Therefore, (-3, 0) is a zero of the function.

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Yes Kevin's comparison in the number line is correct because 10.9 is less than 11.5.

A number line is described as a picture of a graduated straight line that serves as visual representation of the real numbers.

We can agree with Kevin  after referencing the decimal value chart or decimal comparisons below.

o   t     h

10  9

11   5

A number line can also be referred to as a  pictorial representation of numbers on a straight line that is used for  comparing and ordering numbers.

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The original price of the pants be $40.

We have,

Grace bought a pair of pants on sale for $24.

This is 60% of original price.

let the original price be x.

So, 60% of x = 24

60/100 x =24

x = 2400/60

x = $40

Thus, the original price be $40.

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To find the equation of the line of intersection of two planes, we need to find the direction vector of the line. This can be done by taking the cross product of the normal vectors of the two planes.

Let the two planes be:
P1: 2x - y + 3z = 5
P2: x + 2y - 4z = -1
The normal vectors of these planes are:
n1 = <2, -1, 3>
n2 = <1, 2, -4>
Taking the cross product of these two vectors, we get:
n1 x n2 = <14, 10, 5>
This is the direction vector of the line of intersection.
To get the vector form of the equation, we need a point on the line. We can choose any point that lies on both planes. To make it easy, we can set z = 0 in both planes and solve for x and y.
From P1:
2x - y = 5
From P2:
x + 2y = -1
Solving these equations, we get:
x = -7/5
y = -3/5
So a point on the line is (-7/5, -3/5, 0).
Using this point and the direction vector, the vector form of the equation of the line of intersection is:
r = <-7/5, -3/5, 0> + t<14, 10, 5>
Here, t represents the parameter.

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The value of given function [tex]f(x) = \frac{4}{x+2}+2[/tex] at x = -4 is 0, and at x = 4 is 2.17. The graph and table is attached below.

To graph and create a table for f(x) = 4/(x+2) + 2, we can start by making a table of values. To calculate the value of F(x) at each given x, we substitute the value of x into the function and simplify as follows,

At x = -4

F(x) = (4/(-4+2)) + 2 = -2 + 2 = 0

Therefore, F(-4) = 0.

At x = -3

F(x) = (4/(-3+2)) + 2 = undefined (division by zero)

Therefore, F(-3) is undefined.

At x = -2

F(x) = (4/(-2+2)) + 2 = undefined (division by zero)

Therefore, F(-2) is undefined.

At x = -1

F(x) = (4/(-1+2)) + 2 = 6

Therefore, F(-1) = 6.

At x = 0

F(x) = (4/(0+2)) + 2 = 10

Therefore, F(0) = 10.

At x = 1

F(x) = (4/(1+2)) + 2 = 4

Therefore, F(1) = 4.

At x = 2

F(x) = (4/(2+2)) + 2 = 2.67

Therefore, F(2) = 2.67.

At x = 3

F(x) = (4/(3+2)) + 2 = 2.4

Therefore, F(3) = 2.4.

At x = 4

F(x) = (4/(4+2)) + 2 = 2.17

Therefore, F(4) = 2.17.

Now we can plot these points on a coordinate plane and connect them to create the graph.

A vertical asymptote is a vertical line on a graph that the function approaches but never touches or crosses. It occurs when the denominator of a rational function (a function with a fraction of polynomials) becomes zero and the function becomes undefined at that point.

A horizontal asymptote is a horizontal line on a graph that the function approaches as x approaches positive or negative infinity. It describes the long-term behavior of a function as x becomes very large or very small.

We can see from the graph that there is a vertical asymptote at x = -2, and a horizontal asymptote at y = 2.

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we can choose k = (m + Hn-1)/(O(n^3 log n^3)) to obtain the desired upper bound on Pr[Y ≥ k].

Assuming there are i non-empty bins so far, the probability that the next ball you throw will fall into an empty bin is (n-i)/n. This is because there are n bins in total, and i of them are already non-empty. Therefore, there are (n-i) empty bins left, and each has an equal probability of receiving the ball.

To find E[Y], we can use linearity of expectation. Let Yi be the number of balls we need to throw to hit the next empty bin after i bins have been filled. Then, we have:

Y = Y1 + Y2 + ... + Yn

We know that Y1 = m, since we need to throw m balls to fill the first bin. For i > 1, we have:

E[Yi] = 1/(n-i+1) + E[Yi-1]

The first term represents the probability of hitting an empty bin on the next throw, and the second term represents the expected number of throws we need after that happens. We can solve this recurrence relation by substitution:

E[Y2] = 1/(n-1) + E[Y1] = 1/(n-1) + m

E[Y3] = 1/(n-2) + E[Y2] = 1/(n-2) + 1/(n-1) + m

E[Y4] = 1/(n-3) + E[Y3] = 1/(n-3) + 1/(n-2) + 1/(n-1) + m

...

We can see a pattern emerging, where each term has an additional 1/(n-i) compared to the previous term. Therefore, we have:

E[Y] = m + 1/(n-1) + 1/(n-2) + ... + 1/n

= m + Hn-1

≈ m + ln(n) (where Hn-1 is the (n-1)th harmonic number)

To show that E[Y] = O(n log n), we can use the fact that the harmonic series diverges, but the sum of the first n terms is bounded by ln(n) + 1. Therefore, we have:

E[Y] ≈ m + ln(n) ≤ m + ln(n) + 1

= m + O(log n)

Since m is a constant, we can say that E[Y] = O(n log n).

Using Markov's inequality, we have:

Pr[Y ≥ k] ≤ E[Y]/k

Therefore, we want to find k such that:

E[Y]/k ≤ O(n^3 log n^3)

Substituting in the expression we found for E[Y], we have:

(m + Hn-1)/k ≤ O(n^3 log n^3)

Rearranging, we get:

k ≥ (m + Hn-1)/(O(n^3 log n^3))

Therefore, we can choose k = (m + Hn-1)/(O(n^3 log n^3)) to obtain the desired upper bound on Pr[Y ≥ k].

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A Total of 144 small cups can be filled with the water from the cooler before it's empty.

The cooler has 4 1/2 gallons of water, which can be written as 9/2 gallons. Each small cup holds 1/32 gallon. To find the number of small cups that can be filled with the water from the cooler, we need to divide the total amount of water by the amount of water in each cup.

9/2 ÷ 1/32 = (9/2) * (32/1) = 144 small cups.

Therefore, 144 small cups can be filled with the water from the cooler before it's empty.

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Complete Question:

A cooler is filled with 4 1/2 gallons of water. There are small cups that each hold 1/32 gallon.

How many small cups can be filled with the water from the cooler before it's empty?

As per the given volume, the new radius of the sphere is approximately 2.04 cm.

Let's first find the initial volume of the sphere. We know that the volume of a sphere is given by the formula V = (4/3)πr³, where V is the volume and r is the radius. Substituting the given values, we get:

V = (4/3)π(4³) = (4/3)π(64) = 268.08 cm³

Now, we can use the inverse proportionality between the volumes of the sphere and cone to find the new radius of the sphere. We know that the initial volume of the sphere (268.08 cm³) and the volume of the cone (48 cm³) are in inverse proportion to each other. This means that:

V(s) / V(c) = k

where k is a constant. We can find the value of k by substituting the initial volumes of the sphere and cone:

268.08 / 48 = k k = 5.584

Now, we can use this value of k to find the new radius of the sphere. We know that the new volume of the sphere is 6 cm³. This means that:

V(s) / V(c) = k

V(s) / 48 = 5.584

V(s) = 6 cm³

Substituting the values, we get:

6 / 48 = (4/3)πr³ / (4/3)π(4³)

Simplifying, we get:

r³ = 16/3

r = ∛(16/3)

r ≈ 2.04 cm

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The rectangular form of the complex number 5<30° can be found using the following formula:

a + bi = r(cosθ + i sinθ)

where a and b are the real and imaginary parts of the complex number, r is the modulus or magnitude of the complex number, and θ is the argument or angle of the complex number.

In this case, we have:

r = 5 (the modulus or magnitude)
θ = 30° (the argument or angle)

Using the formula, we can find the rectangular form as follows:

a + bi = 5(cos30° + i sin30°)

a + bi = 5(√3/2 + i/2)

a + bi = (5/2)√3 + (5/2)i

Therefore, the rectangular form of the complex number 5<30° is (5/2)√3 + (5/2)i.

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H. 0.270 is equal in value to 27% in the given case. Option 3 is correct

A percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is the percent sign (%).

To convert a percentage to a decimal, we divide the percentage by 100. For example, 50% is equal to 0.50 as a decimal.

To convert a percentage to a decimal, we divide the percentage by 100.

So to convert 27% to a decimal, we divide 27 by 100:

27/100 = 0.27

Therefore, 27% is equal to 0.27 as a decimal.

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The value of x that makes the photo area half of the entire area is: 1.12 in

The area of a rectangle is given by the formula:

A = L * w

where:

L is length

w is width

Thus:

Area of photo = 4 * 2 = 8 in²

We are told that this area is half of the entire ad. Thus:

¹/₂(4 + x)(2 + x) = 8

x² + 6x + 8 = 16

x² + 6x - 8 = 0

Solving using a quadratic calculator gives:

x = 1.12 in

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The probability that the mean speed of 20 randomly selected vehicles is between 35 and 40 mph is approximately 0.99997

A. To find the probability that the speed X of a randomly selected vehicle is between 35 and 40 mph, we need to standardize the values and use the standard normal distribution table.

We can standardize the values as follows:

[tex]z1 = \frac{(35 - 36.6)}{1.7} = -0.94[/tex]

[tex]z2 = \frac{(40 - 36.6)}{1.7} = 2.00[/tex]

Using the standard normal distribution table, we find the probability that a standard normal variable is between -0.94 and 2.00 to be approximately 0.7794.

B. To find the probability that the mean speed of 20 randomly selected vehicles is between 35 and 40 mph, we need to use the central limit theorem.

The central limit theorem tells us that the distribution of sample means will be approximately normal, with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.

Thus, the mean of the sampling distribution of the sample means is:

u-X=u=36.6

And the standard deviation of the sampling distribution of the sample means is:

[tex]\frac{ σ}{\sqrt{n} } = \frac{1.7}{\sqrt{20} } = 0.3808[/tex]

We can standardize the values using the formula:

[tex]z1 =\frac{35-36.6}{0.3808} = -4.21[/tex]

[tex]z2 =\frac{40-36.6}{0.3808} = 8.92[/tex]

we can find the probability that the standard normal variable is between -4.21 and 8.92 by finding the probability that it is greater than -8.92 (which is essentially 0) and subtracting the probability that it is greater than 4.21 from 1:

P(-4.21 < Z < 8.92) = 1 - P(Z > 4.21) = 1 - 0.00003 = 0.99997

Therefore, the probability that the mean speed of 20 randomly selected vehicles is between 35 and 40 mph is approximately 0.99997.

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There is only one point of inflection. Answer: d) 3

The second derivative of the function f(x) is:

[tex]f''(x) = 126x^5 + 40x^3 - 5[/tex]

The second derivative of a function is the derivative of its first derivative. It is denoted represents the rate of change of the slope of the function.

In other words, if the first derivative f'(x) represents the slope of the function, the second derivative f''(x) represents the rate at which the slope is changing.

The points of inflection occur where the concavity changes, that is where the second derivative changes sign or equals zero.

Setting f''(x) = 0, we have:

[tex]126x^5 + 40x^3 - 5 = 0[/tex]

This equation has only one real solution, which can be found numerically:

x ≈ 0.357

Therefore, there is only one point of inflection. Answer: d) 3

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a) The function are described as follows:

y= sin( x) - odd

y= cos (x) - even

y = tan (  x) - neither even or odd

b) y= sin (x) - symmetric about the origin

 y  = cos (x) - symmetric about tthe y-axis

y = tan (x) - does not have a point or line of symmetry

A) The following definitions can be used to  determine if a function is even or odd ...

If f(-x  ) = f (x ) for every x in the domain, the function is even.

If f(-x) =  -  f(x) for every x in the domain, t e function is odd

Using these definitions  , we can examine the following functions

y =  sin (x)

Because sin   (-x) = -sin(x) for any angle x, the function is odd.

y = cos  (x)

Cos  (-x) = cos (x) for any angle x, hence the function is even.

y = tan( x)

Tan(-x) = -tan(x) for every angle x.   Only if x does not equal ( n + 0.5), where n   is an integer, is the function even or odd.

B)

The line or  point of symmetry of a function is detrmined by whether it is even or odd...

The y -axis (x=0) is the line of symmetry for an even function.

The origin (0,0) is the point of symmetry for an odd function.

y = sin(x)     his is an odd function, so the point of symmetry is the origin (0,0).

y = cos(x) his is an even function, so the line of symmetry is the y-axis (x=0).

y = tan(x)

This is neither even nor odd, so it does not have a line or point of symmetry.

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Lauren should invite at least 25 people to her party in order to have at least 23 guests.

Option D is the correct answer.

We have,

Let x be the number of people Lauren should invite to her party.

Since 92% of the people she invites are expected to come, the actual number of guests she can expect is 0.92x.

We want to find the value of x that makes the expected number of guests at least 23.

Therefore, we can set up the following inequality:

0.92x ≥ 23

Solving for x, we divide both sides by 0.92:

x ≥ 25

Therefore,

Lauren should invite at least 25 people to her party in order to have at least 23 guests.

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The equation of a circle can be written as[tex](x - h)^2 + (y - k)^2 = r^2,[/tex]where (h, k) is the center of the circle and r is the radius.

Circle's basic equation is represented by:

[tex](x-h)^2 + (y-k)^2 = r^2[/tex]

where the radius is r and the circle's center's coordinates are (h,k).

Let's first consider what a circle is before determining its equation. A circle is a collection of all points in a plane that is uniformly distanced from a fixed point. The fixed point is referred to as the circle's center. The radius of a circle is the separation between the center and any point along its circumference. In this post, we'll go over what a circle equation in standard form is and how to calculate a circle's equation when the origin serves as its center.
Therefore, the equation of a circle whose center is (4, -5) and which has a radius of 3 is:

[tex](x - 4)^2 + (y + 5)^2 = 3^2[/tex]
Simplifying and expanding the equation gives:

[tex](x - 4)^2 + (y + 5)^2 = 9[/tex]

And that is the equation of the circle.

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The area of the 180° sector is  76.93 mi ²

The area of a circle of radius R is given by the formula:

A = pi*R²

Where pi = 3.14

Here the radius is 7 mi, and we want a section of 180°. Remember that the angle in a circle is 360°, then a section of 180° is half a circle, then the area is 0.5 times the one written above.

Then the area will be:

A = 0.5*3.14(7 mi)²

A = 76.93 mi ²

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The requried, value is less than 8.5, as Imran claimed. Therefore, he is correct and not wrong.

To verify this, we can use the formula given:

Petrol consumption = 100 × litres of petrol used / kilometres travelled

Substituting the given values, we get:

Petrol consumption = 100 × 21.3 / 250 = 8.52

Rounding this value to three significant figures, we get:

Petrol consumption = 8.52

This value is indeed less than 8.5, as Imran claimed. Therefore, he is correct and not wrong.

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Answer:

  [tex]A_n=\left[\begin{array}{cc}1&n\\0&1\end{array}\right][/tex]

Step-by-step explanation:

You want the general formula for the n-th power of matrix A, where ...

  [tex]A=\left[\begin{array}{cc}1&1\\0&1\end{array}\right][/tex]

The sequence of powers A, A², A³, A⁴ is shown in the attachment. It strongly suggests that the upper right element of the matrix is equal to the power.

The formula for An is ...

  [tex]\boxed{A_n=\left[\begin{array}{cc}1&n\\0&1\end{array}\right]}[/tex]

This indicates that the particle is accelerating due to gravity. At t=7, the rate at which the velocity is changing depends on the value of k, m, and g. This implies that the particle's acceleration may vary depending on these constants and time.

At time 0, the rate at which the velocity is changing can be found by taking the derivative of the velocity function with respect to time, t.

v(t) = (mg/k)(1-e^(-kt/m))

v'(t) = (mg/k)((ke^(-kt/m))/m)

Plugging in t=0 gives:

v'(0) = (mg/k)((k)/m) = g

Therefore, the rate at which the velocity is changing at time 0 is g.

At t=7, the rate at which the velocity is changing can also be found by taking the derivative of the velocity function with respect to time, t, and plugging in t=7:

v'(7) = (mg/k)((ke^(-7k/m))/m)

This value will depend on the specific values of k, g, and m that are not given in the question.

These rates of change tell us about the motion of the particle. If the rate of change of velocity is positive, the particle is accelerating. If it is negative, the particle is decelerating. If it is zero, the particle is moving at a constant velocity.

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To understand subtraction as an unknown-addend problem means to view subtraction as finding the missing number in an addition problem.

In the example given, subtracting 10-8 means finding the number that needs to be added to 8 to make 10. This is essentially an addition problem with an unknown addend.

To solve the problem, one needs to think about what number added to 8 will result in 10. This can be done by counting up from 8 until you reach 10, which is a difference of 2. Therefore, 10-8=2, and the missing number is 2.

In general, to subtract any two numbers using this method, you can start with the smaller number and count up until you reach the larger number. The difference between the two numbers is the missing addend. For example, to subtract 15-7, you would start with 7 and count up to 15, which is a difference of 8. So 15-7=8.

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To find the critical value for a two-tail test with a sample standard deviation, an n-value of 36, and an alpha of 0.05, you will need to use the t-distribution table (as the population standard deviation is not given). However, since you have only provided the sample standard deviation and not its actual value, I cannot calculate the specific critical value for you.

Here are the general steps to find the critical value:
1. Calculate the degrees of freedom: df = n - 1 = 36 - 1 = 35
2. Divide the alpha (0.05) by 2 for a two-tail test, resulting in 0.025 in each tail.
3. Look up the t-value corresponding to the degrees of freedom (35) and the given alpha level (0.025) in the t-distribution table.
4. The t-value you find in the table is your critical value.

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The product of the fractions is 275/20

A fraction can simply be defined as the part of a given whole variable, a whole number or a whole element.

In mathematics, there are different types of fractions, namely;

From the information given, we have that;

To determine the product of the fraction

6 1/4×2 1/5

convert to improper fractions, we get;

25/4 × 11/5

Multiply the numerators

275/4(5)

Multiply the denominator

275/20

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The accumulated balance after 17 years is $18,481.76.

What is compound interest?

Compound interest is when you earn interest on both the money you've saved and the interest you earn.

To solve this problem, we can use the formula for the future value of an annuity:

[tex]FV = PMT * [(1 + r/n)^{(n*t) - 1]} / (r/n)[/tex]

where FV is the future value, PMT is the periodic payment, r is the interest rate, n is the number of compounding periods per year, and t is the number of years.

In this case, PMT = $146.30, r = 5.3%, n = 4 (since interest is compounded quarterly), and t = 17. Plugging these values into the formula, we get:

[tex]FV = $146.30 * [(1 + 0.053/4)^{(4*17)} - 1] / (0.053/4)[/tex]

[tex]= $146.30 * (1.01325^{68} - 1) / 0.01325[/tex]

= $146.30 * 126.3584

= $18,481.76

Therefore, the accumulated balance after 17 years is $18,481.76.

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The area of each of the quadrilateral is calculated as:

11. 48 square meters;  12. 144.3 square centimeters;   13. 8.2 square yards.

14. 132 square yds;  

The area of each quadrilateral = height * base/width

11. Height = 6 m

Base = 8 m

Area= 6 * 8 = 48 square meters.

12. Height = 13 cm

Base = 11.1 cm

Area= 11.1 * 13 = 144.3 square centimeters.

13. Height = 4.1 yd

Base = 2 yd

Area= 2 * 4.1 = 8.2 square yards.

14. Height = 12 yd

Base = 11 yd

Area= 11 * 12 = 132 square yds

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The number of circles he will draw is 30 circles.

The branch of mathematics which involves the study and manipulation of mathematical symbols is known as Algebra. This field comprises the use of characters and signs to symbolize unknown values and their linkages.

How to solve:

On one card, we have 14 triangles and 6 circles

Therefore, if we have 70 triangles,

number of cards= 70/14

= 5 card

Thus, Total number of circles is 6 x 5 = 30 circles

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The expanded logarithmic expression of log_b (x²y)/z² is 2 log_b x + log_b y - 2 log_b z.

Expanding the given logarithmic expression using the properties of logarithms, we get:

log_b (x²y)/z² = log_b x²y - log_b z²

Using the power rule of logarithms, we can simplify log_b x²y as:

log_b x²y = log_b x² + log_b y

Then, using the quotient rule of logarithms, we can simplify log_b z² as:

log_b z² = 2log_b z

Substituting these simplifications in the original expression, we get:

log_b (x²y)/z² = log_b x² + log_b y - 2log_b z

This is the expanded form of the given logarithmic expression using the properties of logarithms.

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The question is -

Use properties of logarithms to expand the following logarithmic expression as much as possible.

log_b (x²y)/z²

The range will decrease by 3.

The range of a set of data is the difference between the largest and smallest values.

Set of numbers: 9, 5, 1, 7, 4, 4, 7, 9

Order: 1, 4, 4, 5, 7, 7, 9, 9

Range: 9 - 1 = 8

If the number 7 replaced the number 1 in the set

Order: 4, 4, 5, 7, 7, 7, 9, 9

Range: 9 - 4 = 5

How much would the range decrease if the number 7 replaced the number 1 in the set

8 - 5 = 3

So, the range decrease by 3 if the number 7 replaces the number 1 in the set.

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To find the probability of the event "the first die is 6 or the sum is 8," we need to count the number of outcomes that satisfy this event and divide it by the total number of possible outcomes.

To find the probability of the event "the first die is 6 or the sum is 8," we need to consider the total possible outcomes when rolling two dice and the favorable outcomes for this event.

Total possible outcomes: 6 sides on each die, so there are 6 x 6 = 36 possible outcomes.

Favorable outcomes:
1. First die is 6: There are 6 possible outcomes (6,1), (6,2), (6,3), (6,4), (6,5), and (6,6).
2. Sum is 8: There are 5 possible outcomes (2,6), (3,5), (4,4), (5,3), and (6,2).

However, (6,2) is counted twice, so we should subtract 1 from the total favorable outcomes: 6 + 5 - 1 = 10.

Probability = Favorable outcomes / Total possible outcomes = 10/36. Simplifying the fraction gives 5/18.

So, the probability of the event "the first die is 6 or the sum is 8" is 5/18.

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William has been making end-of-quarter contributions of $1200 for 11years 8months

What is quarter?

25 percent of anything is equal to one-quarter of it. This is how many athletic events are split up, and sports commentators frequently use phrases like "Here is the score at the end of the first quarter."

One-fourth is referred to as a quarter. Each of you will consume one-fourth of a pizza if it is divided into four pieces and shared with three buddies.

Given:

Final value (FV)= $74,385

Payment(PMT) = $1200

Present value(PV)= $0

So,

Interest rate(r)= 4.75/4= 1.1875%

By putting the value of each and using the financial calculator we get,

N= 46.729

Or, N≅46.73

So the number of years= 46.73/4= 11.68 years

Months= .68*12= 8.16months

Then the correct answer is 11years 8months

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