state the zeros of the polynomial (include multiplicity): f(x) = (x+9)(x-1)³(2x + 5)​.

Probabilities associated with this distribution are

(a) P(Z < 0.5) = 0.6915

(b) P(Z = 0.5) = 0

(c) P(Z ≥ 2.3) = 0.0107.

(d)  P(-1.4 ≤ Z ≤ 0.6) = 0.6449.

(e) The value of z0 such that P(|Z| ≤ z0) = 0.32 is 0.9945.

The statement "~(0,1)" refers to a standard normal distribution with mean 0 and standard deviation 1.

We can use the standard normal distribution table or a calculator to find probabilities associated with this distribution. Here are the solutions to the given problems:

(a) P(Z < 0.5) = 0.6915, where Z is a standard normal random variable.

(b) P(Z = 0.5) = 0, since the probability of a continuous random variable taking any specific value is always zero.

(c) P(Z ≥ 2.3) = 0.0107.

(d) P(-1.4 ≤ Z ≤ 0.6) = P(Z ≤ 0.6) - P(Z ≤ -1.4) = 0.7257 - 0.0808 = 0.6449.

(e) The value of z0 such that P(|Z| ≤ z0) = 0.32 is the 0.16th percentile of the standard normal distribution.

From the standard normal distribution table, we can find that the 0.16th percentile is approximately -0.9945. Therefore, z0 = 0.9945.

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To eliminate the parameter t from the given parametric equations x = 4sin(3t) and y = 4cos(3t), we can use the trigonometric identity sin^2(t) + cos^2(t) = 1.

Squaring both equations, we get: x^2 = 16sin^2(3t) y^2 = 16cos^2(3t) Adding these two equations and using the trigonometric identity, we get: x^2 + y^2 = 16(sin^2(3t) + cos^2(3t)) x^2 + y^2 = 16 Taking the square root of both sides, we get: sqrt(x^2 + y^2) = 4 .

Therefore, the equation that represents the given parametric equations as a single equation in x and y is: x^2 + y^2 = 16 This is the equation of a circle with center at the origin and radius 4.  the equation becomes: (x/4)² + (y/4)² = 1 Finally, we can write the single equation in x and y as: x² + y² = 16.

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Answer:

answer this question in this photo.

The demand equation is  b. D = -25x + 225.

Since the demand equation is linear and involves "x" as the price, and "D" as the number of units, we can use the two points given to determine the equation.

At a price of $1, 200 units are demanded: (1, 200)
At a price of $9, 0 units are demanded: (9, 0)

Now, we can find the slope (m) using the formula:

m = (y2 - y1) / (x2 - x1)

In this case:

m = (0 - 200) / (9 - 1)
m = -200 / 8
m = -25

Now, we can use one of the points (either point will give the same result) to find the y-intercept (b) by plugging the values into the linear equation:

D = m * x + b

Using the point (1, 200):

200 = -25 * 1 + b
b = 200 + 25
b = 225

Now, we have the demand equation:

D = -25x + 225

So the correct answer is: b. D = -25x + 225.

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The line of best fit for the scatter plot is given as follows:

y = 2x.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

The graph touches the y-axis at the origin, hence the intercept b is given as follows:

b = 0.

Hence:

y = mx.

When x = 20, y = 40, hence the slope m is given as follows:

20m = 40

m = 40/20

m = 2.

Hence the equation is:

y = 2x.

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The t(20) distribution is wider than the t(40) distribution, For the same value of t, the t(40) distribution has the smallest tail area and for the same middle area C, the t(20) distribution has the largest t* critical value.

a. Which distribution is wider?
The t(20) distribution is wider than the t(40) distribution. As the degrees of freedom increase, the t distribution approaches the standard normal distribution, and its width decreases.
b. For the same value of t, which distribution has the smallest tail area?
For the same value of t, the t(40) distribution has the smallest tail area. As the degrees of freedom increase, the distribution becomes more concentrated around the mean, and the tails become smaller.
c. For the same middle area C, which distribution has the largest t* critical value?
For the same middle area C, the t(20) distribution has the largest t* critical value. With fewer degrees of freedom, the distribution is wider and requires a larger t* value to cover the same middle area as compared to the t(40) distribution.

a. The t(40) distribution is wider than the t(20) distribution. This is because as the degrees of freedom increase, the t-distribution approaches a standard normal distribution, which has a smaller variance than the t-distribution with fewer degrees of freedom.
b. For the same value of t, the t(40) distribution has the smallest tail area. This is because as the degrees of freedom increase, the t-distribution approaches a standard normal distribution, which has smaller tail areas than the t-distribution with fewer degrees of freedom.
c. For the same middle area C, the t(20) distribution has the largest t* critical value. This is because as the degrees of freedom decrease, the t-distribution has heavier tails, which require larger t* values to maintain the same middle area C.

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Nothing happened for real.

i think its not a big deal for you but it will be a big deal for your parents ;))))

Use your knowledge of genetic biology and lecture them. Maybe they don't understand.

Just kidding =))))

In short, you may have an optical problem =))))

P/s: don't be furious :'))) it's gonna easy to get old

ok done. Thank to me >:333

The possible number of different passwords in scientific notation with two digits of accuracy after the decimal point for the computer is 3.42 x 10^14.

To find the number of different passwords possible, given that, each character may be any lowercase letter or digit, we must first determine the total number of available characters.

There are 26 lowercase letters and 10 digits, so there are a total of 26 + 10 = 36 available characters.

Since replications are allowed and the password consists of 11 characters, we can use the formula:

Number of different passwords = (Total number of available characters) ^ (Password length)

Number of different passwords = 36 ^ 11

Calculating this value, we get 341,821,345,910,986. To represent this number in scientific notation with two digits of accuracy after the decimal point, we divide by 10 raised to the power of the number of digits minus 1:

341,821,345,910,986 / 10^14 = 3.42 x 10^14

So, the possible number of different passwords for a computer consisting of eleven characters is 3.42 x 10^14.

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(a) ∑n=1[infinity]8n7n is a geometric series and it diverges. Answer: DIV.

(b) ∑n=2[infinity]13n is a geometric series and it diverges. Answer: DIV.

(c) ∑n=0[infinity]3n92n+1 is a geometric series and it converges. Sum = 2.452.

(d) ∑n=5[infinity]7n8n is a geometric series and it converges. Sum = 0.0954.

(e) ∑n=1[infinity]7n7n+4 is a geometric series and it converges. Sum = 3.5

(f) ∑n=1[infinity] 7/8*[tex](7/8)^{(n-1)[/tex] + [tex]3/8*(3/8)^{(n-1)[/tex]. Both of these are geometric series and the series converges. Sum= 1.6.

(a) This series can be rewritten as ∑n=1[infinity][tex](8/7)^n[/tex]. This is a geometric series with ratio r=8/7 which is greater than 1. Hence, the series diverges. Answer: DIV.

(b) This is a geometric series with first term a=13 and common ratio r=13. Since |r|>1, the series diverges. Answer: DIV.

(c) This series can be written as ∑n=0[infinity] [tex]3^n/(9^2)^n[/tex] * [tex]9^{(1/(2n+1))[/tex]. The first part of the series is a geometric series with a=1 and r=3/81<1. The second part of the series is also a geometric series with a=[tex]9^{(1/3)[/tex] and r=[tex](9^{(1/3)})^2=9^{(2/3)[/tex]<1. Therefore, the series converges. To find the sum, we use the formula for the sum of an infinite geometric series:

sum = a/(1-r) + b/(1-c)

where a and r are the first term and common ratio of the first geometric series, and b and c are the first term and common ratio of the second geometric series. Substituting the values, we get:

sum = 1/(1-3/81) + [tex]9^{(1/3)}/(1-9^{(2/3))[/tex]

= 1.01 + 1.442

= 2.452

Answer: 2.452.

(d) This series can be written as ∑n=5[infinity] [tex](7/8)^n[/tex]. This is a geometric series with ratio r=7/8 which is less than 1. Hence, the series converges. To find the sum, we use the formula for the sum of an infinite geometric series:

sum = a/(1-r)

where a and r are the first term and common ratio of the series. Substituting the values, we get:

sum = [tex](7/8)^5/(1-7/8)[/tex]

= [tex]7/8^4[/tex]

= 0.0954

Answer: 0.0954.

(e) This series can be rewritten as ∑n=1[infinity] [tex](7/7.4)^n[/tex]. This is a geometric series with ratio r=7/7.4<1. Hence, the series converges. To find the sum, we use the formula for the sum of an infinite geometric series:

sum = a/(1-r)

where a and r are the first term and common ratio of the series. Substituting the values, we get:

sum = 1/(1-7/7.4)

= 3.5

Answer: 3.5.

(f) This series can be rewritten as ∑n=1[infinity] [tex]7/8*(7/8)^{(n-1)[/tex] + [tex]3/8*(3/8)^{(n-1)[/tex]. Both of these are geometric series with ratios less than 1, so the series converges. To find the sum, we add the sums of the two geometric series:

sum = 7/8/(1-7/8) + 3/8/(1-3/8)

= 1 + 3/5

= 1.6

Answer: 1.6.

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Step-by-step explanation:

A trapezoid bases are the parallel sides

   the AVERAGE of the bases  X  the height    is the area

First one :   Height = 6cm    average of bases = (9.3+4.1) / 2 = 6.7 cm

    area =   6.7 cm * 6 cm = 40.2 c^2

The other three are similar...just different numbers...Using this method, you should be able to do them now .....

Answer:

A trapezoid bases are the parallel sides

  the AVERAGE of the bases  X  the height    is the area

First one :   Height = 6cm    average of bases = (9.3+4.1) / 2 = 6.7 cm

   area =   6.7 cm * 6 cm = 40.2 c^2

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Step-by-step explanation:

Using the Tangent-Secant Theorem, the value of x, is calculated in teh figure as: x = 16.

The Tangent-Secant Theorem states that if a tangent and a secant are drawn from an external point to a circle, then the square of the length of the tangent is equal to the product of the lengths of the secant and its external part.

Applying the theorem:

60² = (2x - 5 + 48)(48)

3,600 = (2x + 43)(48)

3,600 = 96x + 2,064

3,600 - 2,064 = 96x

1,536 = 96x

1,536/96 = 96x/96

16 = x

x = 16

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The value of L(2) is for recursive definition of Lucas numbers is 3.

According to the given recursive definition of the Lucas numbers, L(2) = 3 since n=2 is the second term in the sequence and its value is defined as 3.
Based on the recursive definition of Lucas numbers L(n) given, let's determine the value of L(2):

L(n) = 1 if n = 1
L(n) = 3 if n = 2
L(n) = L(n - 1) + L(n - 2) if n > 2

Since we're looking for L(2), we can use the second condition in the definition:

L(2) = 3

So, the value of L(2) is 3.

A set of numbers called the Lucas numbers resembles the Fibonacci sequence. The series was researched in the late 19th century by the French mathematician François Édouard Anatole Lucas, who gave it its name.

This is how the Lucas sequence is described:

For n > 1, L(0) = 2 L(1) = 1 L(n) = L(n-1) + L(n-2)

The sequence's initial few numerals are thus:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ...

The Lucas sequence, like the Fibonacci sequence, offers a variety of intriguing mathematical characteristics and linkages to different branches of mathematics. For instance, exactly like in the Fibonacci sequence, the ratio of successive Lucas numbers converges to the golden ratio[tex](1 + \sqrt{5})/2[/tex].

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14 packages of nets of six were taken to the village. If there are 82 people in the village, then 82 nets are needed to provide one net to each person.

However, we also know that there are 2 nets remaining, which means that a total of 82 + 2 = 84 nets were provided.

To determine how many packages of nets of six were taken to the village, we can divide the total number of nets by 6, and round up to the nearest whole number since we can't have a partial package of nets:

84 nets / 6 nets per package = 14 packages (rounded up)

Therefore, 14 packages of nets of six were taken to the village.

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Let V be a vector space, and T:V→V a linear transformation then the value of T(v⃗ 1) = -v⃗ 1, T(v⃗ 2) = v⃗ 2 and T(4v⃗ 1 − 4v⃗ 2) = -4v⃗ 1 - 4v⃗ 2.

We can use the given information to find the value of T for various vectors in V and T:V→V a linear transformation such that T(5v⃗ 1+3v⃗ 2)=−5v⃗ 1+5v⃗ 2 and T(3v⃗ 1+2v⃗ 2)=−5v⃗ 1+2v⃗ 2.

For 5v⃗ 1 + 3v⃗ 2, we have:

T(5v⃗ 1+3v⃗ 2) = −5v⃗ 1+5v⃗ 2

5T(v⃗ 1) + 3T(v⃗ 2) = -5v⃗ 1 + 5v⃗ 2

Similarly, for 3v⃗ 1 + 2v⃗ 2, we have

T(3v⃗ 1+2v⃗ 2) = −5v⃗ 1+2v⃗ 2

3T(v⃗ 1) + 2T(v⃗ 2) = -5v⃗ 1 + 2v⃗ 2

Solving these equations for T(v⃗ 1) and T(v⃗ 2), we get

T(v⃗ 1) = -v⃗ 1

T(v⃗ 2) = v⃗ 2

Now, we can use these values to find T(4v⃗ 1 − 4v⃗ 2)

T(4v⃗ 1 − 4v⃗ 2) = 4T(v⃗ 1) - 4T(v⃗ 2)

= 4(-v⃗ 1) - 4(v⃗ 2)

= -4v⃗ 1 - 4v⃗ 2

Therefore, T(4v⃗ 1 − 4v⃗ 2) = -4v⃗ 1 - 4v⃗ 2.

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Answer:

See below

Step-by-step explanation:

The answer is yes, y + z ∈ S2.

We need to determine whether the sum of two vectors in S2, y + z, is also in S2.

Let y = (y1, y2, y3) and z = (z1, z2, z3) be two vectors in S2. Then, we know that:

y1y2 + 2y3 = 0 (since y ∈ S2)

z1z2 + 2z3 = 0 (since z ∈ S2)

To show that y + z ∈ S2, we need to show that:

(y + z)1(y + z)2 + 2(y + z)3 = 0

Expanding the left-hand side, we have:

(y1 + z1)(y2 + z2) + 2(y3 + z3) = y1y2 + y1z2 + z1y2 + z1z2 + 2y3 + 2z3

Substituting the expressions for y1y2 + 2y3 and z1z2 + 2z3 from above, we get:

(y1y2 + 2y3) + (z1z2 + 2z3) + y1z2 + z1y2 = 0

Since y1y2 + 2y3 = 0 and z1z2 + 2z3 = 0, we have:

y1z2 + z1y2 = 0

Therefore, we have shown that (y + z)1(y + z)2 + 2(y + z)3 = 0, which implies that y + z ∈ S2.

Hence, the answer is yes, y + z ∈ S2.

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For X following a χ²-distribution with 20 degrees of freedom, the point 'a' such that P(X < a) = 0.025 is approximately 8.26.

To find the point 'a' such that P(X < a) = 0.025, where X follows a χ²-distribution with 20 degrees of freedom, you need to use the inverse chi-square distribution function (also called the chi-square quantile function).

Here's the step-by-step explanation:

1. Identify the given parameters: X follows a χ²-distribution with 20 degrees of freedom, and we need to find a point 'a' such that P(X < a) = 0.025.

2. Use the inverse chi-square distribution function (quantile function) with the given probability and degrees of freedom. This function will give you the value of 'a' corresponding to the specified probability.

In most statistical software or calculators, you can find this function. For example, in R programming, you can use the "qchisq()" function:

a = qchisq(0.025, df = 20)

3. Calculate the value of 'a'.

In this case, a ≈ 8.26.

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The median of the sample data collected by the researcher in the following sample data 512685675124 is 5, which is option (d).

We need to arrange the numbers in ascending or descending order.

1 1 2 2 4 5 5 5 6 6 7 8 (arranged in ascending order)

The middle number is the median. Since there are 12 digits in the given sample data, the median is the average of the 6th and 7th digits.

So, the median is (5 + 5) / 2 = 5

Therefore, median of the sample data collected by the researcher is 5 (Option d).

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The complete formulation of the integer programming problem can be written as:

Minimize Z = Σi=1n Σj=1m cij xij

subject to:

Σj=1m xij = 1, for i = 1, 2, ..., n

Σi=1n xij = 1, for j = 1, 2, ..., m

xij ∈ {0, 1}, for i = 1, 2, ..., n and j = 1, 2, ..., m.

To formulate an integer programming problem for assigning each route to one bidder (and each bidder to only one route), we can follow these steps:

Define decision variables: Let xij be a binary variable, where xij=1 if bidder i is assigned to route j, and xij=0 otherwise. Here i = 1, 2, ..., n is the index for bidders, and j = 1, 2, ..., m is the index for routes.

Define the objective function: The objective is to minimize the total cost of assignment, which can be represented as the sum of the cost of each assignment, given by cij. Therefore, the objective function can be formulated as:

Minimize Z = Σi=1n Σj=1m cij xij

Define the constraints:

Each bidder can only be assigned to one route: Σj=1m xij = 1, for i = 1, 2, ..., n.

Each route can only be assigned to one bidder: Σi=1n xij = 1, for j = 1, 2, ..., m.

The decision variables are binary: xij ∈ {0, 1}, for i = 1, 2, ..., n and j = 1, 2, ..., m.

These constraints ensure that each bidder is assigned to only one route, and each route is assigned to only one bidder.

The complete formulation of the integer programming problem can be written as:

Minimize Z = Σi=1n Σj=1m cij xij

subject to:

Σj=1m xij = 1, for i = 1, 2, ..., n

Σi=1n xij = 1, for j = 1, 2, ..., m

xij ∈ {0, 1}, for i = 1, 2, ..., n and j = 1, 2, ..., m.

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Full Question ;

formulate an ip that assigns each route to one bidder (and each bidder must be assigned to only one route)

The given set contains the digits of the number 457,636 and can be written in roster notation as {4, 5, 7, 6, 3}.

list the elements of the set in roster notation. (enter empty or ∅ for the empty set.) {x | x is a digit in the number 457,636.  

Sure, I can provide some additional information on sets and roster notation.

In mathematics, a set is a collection of distinct objects, called elements or members of the set. One way to represent a set is through roster notation, which lists the elements of the set inside braces { } separated by commas. For example, the set of even numbers less than 10 can be written in roster notation as {2, 4, 6, 8}.

In the given problem, we are asked to list the elements of the set in roster notation where the set contains the digits of the number 457,636. The digits of the number are 4, 5, 7, 6, and 3, so the set can be written in roster notation as {4, 5, 7, 6, 3}.

It is worth noting that sets can be empty, denoted by the symbol ∅ or by the word "empty". An empty set contains no elements. For example, the set of integers greater than 10 and less than 0 is an empty set, which can be represented in roster notation as ∅ or {}.

In summary, roster notation is a way to represent sets by listing their elements inside braces. The given set contains the digits of the number 457,636 and can be written in roster notation as {4, 5, 7, 6, 3}.

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In the two functions as the value of V(x) increases, the value of W(x) also increases.

The value of functions, V(x) and W(x) is determined as follows;

for h(-2, 1/4); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2⁻²⁺³ = 2¹ = 2

w(x) = 2ˣ ⁻ ³ = 2⁻²⁻³ = 2⁻⁵ = 1/32

for h(-1, 1/2); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2² = 4

w(x) = 2ˣ ⁻ ³ = 2⁻⁴ = 1/16

for h(0, 1); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2³ = 8

w(x) = 2ˣ ⁻ ³ = 2⁻³ = 1/8

for h(1, 2); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2⁴ = 16

w(x) = 2ˣ ⁻ ³ = 2⁻² = 1/4

for h(2, 4); the value of the functions is calculated as follows;

v(x) = 2ˣ ⁺ ³ = 2⁵ = 32

w(x) = 2ˣ ⁻ ³ = 2⁻¹ = 1/2

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a = 4 and b = 1. To find a and b, we need to first find the conditional PDF fX|Y(x|0.5), which represents the distribution of X given that Y = 0.5.

We can use Bayes' rule to find the conditional PDF:

fX|Y(x|0.5) = fX,Y(x,0.5) / fY(0.5)

where fY(0.5) is the marginal PDF of Y evaluated at 0.5, and can be found by integrating fX,Y over all possible values of X:

fY(0.5) = ∫ fX,Y(x,0.5) dx

= ∫ cxy dx (from x=0 to x=0.5)

= c(0.5)²

= c/8

Now, we can find fX,Y(x,0.5) by evaluating the joint PDF at x and y=0.5:

fX,Y(x,0.5) = cxy

= c(0.5)x

So, we have:

fX|Y(x|0.5) = (c(0.5)x) / (c/8)

= 4x

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To find the area under the standard normal curve to the left of z=1.43, you will need to use a standard normal (Z) table or an online calculator. Here's a step-by-step explanation:

1. Identify the given value of z: z=1.43
2. Look up the value in a standard normal (Z) table or use an online calculator to find the corresponding area to the left of z=1.43.
3. The table or calculator will provide the area under the curve to the left of z=1.43.
4. Round the answer to four decimal places, if necessary.

Using a standard normal table or calculator, the area under the standard normal curve to the left of z=1.43 is approximately 0.9236 when rounded to four decimal places.

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a)The quantity at time t=0 is  110.

b)The quantity is increasing over time .

c) The percent per unit time growth or decay rate is 11.1% .

d)Yes, the growth rate continuous.

a. At time t=0, the quantity Q can be found using the given formula Q = 110(1.137[tex])^{2}[/tex].

  Plugging in t=0, we get.

  Q = 110(1.137[tex])^{0}[/tex]

      = 110(1) = 110.

b. The quantity is increasing over time because the base (1.137) in the formula is greater than 1, which means that Q grows as time (t) increases.

c. The percent per unit time growth rate can be found by taking the derivative of the function and dividing by the initial quantity:

dQ/dt = 110(1.137[tex])^{t}[/tex]* ln(1.137)
dQ/dt at t=0 = 110(1.137[tex])^{0}[/tex] * ln(1.137) = 12.2

% growth per unit time = (dQ/dt)/Q * 100% = 12.2/110 * 100% = 11.1%

d. The growth rate is continuous, as it follows an exponential growth pattern described by the formula Q = 110(1.137)^t, where the base is constant and the time variable is continuous.

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The Volume of Remaining block is  (l w h - 1125) cm³

We have,

Edge of cube = 15 cm

So, Volume of cube

= 15 x 15 x 15

= 1125 cm³

Now, Volume of Remaining block

= Volume of cuboid - Volume of cube

= l w h - 1125 cm³

Here we just have to put the dimension of cuboid in place of l w h.

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Spearman's rank correlation coefficient would be an appropriate statistic  for an associational research question.

When the two variables of interest are non-normally distributed and skewed, Spearman's rank correlation coefficient would be an appropriate statistic to use for an associational research question involving the correlation between two non-normally distributed continuous variables.

Spearman's rank correlation coefficient is a nonparametric measure of correlation that is used to assess the strength and direction of association between two ranked variables. It measures the degree to which the rank order of one variable is related to the rank order of another variable, regardless of their actual values.

Unlike Pearson's correlation coefficient, which assumes a linear relationship between the variables and normality of data, Spearman's correlation coefficient is robust to outliers, non-linear relationships, and non-normality of data. It works by converting the data into ranks, which can be used to compute the correlation coefficient.

Therefore, if we have two non-normally distributed, skewed continuous variables and want to examine the association between them, Spearman's rank correlation coefficient would be an appropriate statistic to use.

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Answer:

C. A set of objects chosen from a larger set in which the order of the objects doesn't matter.

Step-by-step explanation:

That is the definition of a combination. Order does not matter

For example, selecting 3 students from a total of 10 students. The set selected is a smaller set from a larger set

Based on the information provided, Railway Cabooses paid an annual dividend of $2.50 per share. However, the company has been reducing its dividends by 11.7 percent each year.

To calculate the current annual dividend, we can use the formula: current dividend = previous dividend * (1 - dividend reduction rate).
So, the current annual dividend would be $2.50 * (1 - 0.117) = $2.21 per share. To determine how much you should pay to purchase stock in this company, we need to use the dividend discount model.
The formula for this model is stock price = annual dividend / (required rate of return - dividend growth rate).
Plugging in the values from the problem, we get:
Stock price = $2.21 / (0.13 - 0.117) = $34.15 per share.
Therefore, if your required rate of return is 13 percent, you should be willing to pay $34.15 per share to purchase stock in Railway Cabooses.

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d) Least squares line ŷ = 521.2542 - 0.2349x.

e) Correlation coefficient r ≈ -0.9869

f) Estimated gold medal time ŷ for 1924 ≈ 71.26 sec

g) Estimated gold medal time ŷ for 1992 ≈ 53.14 sec

h) Estimate the gold medal time for the 2012 ŷ ≈ 52.12 sec

d) To find the least squares line, we need to calculate the mean and standard deviation of the year (x) and the time (y):

mean of x = (1912 + 1932 + 1952 + 1968 + 1976 + 1984 + 2000 + 2012)/8 = 1972

mean of y = (82.2 + 72.4 + 66.8 + 66.8 + 61.2 + 60.0 + 55.6 + 55.9 + 54.6 + 53.8 + 53.1)/11 = 63.2

standard deviation of x = √(((1912-1972)² + (1932-1972)² + ... + (2012-1972)²)/8) ≈ 44.54

standard deviation of y = √(((82.2-63.2)² + (72.4-63.2)² + ... + (53.1-63.2)²)/10) ≈ 10.53

Then, we can calculate the correlation coefficient r:

r = (1/10) * (((1912-1972)/44.54)(82.2-63.2)/10.53 + ((1932-1972)/44.54)(72.4-63.2)/10.53 + ... + ((2012-1972)/44.54)*(53.1-63.2)/10.53) ≈ -0.9869

Using the formula for the least squares regression line, we have:

b = r * (standard deviation of y / standard deviation of x) ≈ -0.2349

a = mean of y - b * mean of x ≈ 521.2542

Therefore, the least squares line is ŷ = 521.2542 - 0.2349x.

f) To estimate the gold medal time for 1924, we substitute x = 1924 into the equation for the least squares line:

ŷ = 521.2542 - 0.2349(1924) ≈ 71.26 sec

g) To estimate the gold medal time for 1992, we substitute x = 1992 into the equation for the least squares line:

ŷ = 521.2542 - 0.2349(1992) ≈ 53.14 sec

h) To estimate the gold medal time for 2012, we substitute x = 2012 into the equation for the least squares line:

ŷ = 521.2542 - 0.2349(2012) ≈ 52.12 sec

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We have computed the Taylor polynomials of the given function f (x) = cos (4x), using around 6 decimals for approximation. These polynomials can then be used to approximate the given function.

Function is a block of code that performs a specific task. It can accept input parameters and return a value or a set of values. Functions are used to break down a complex problem into simple, manageable tasks. They also help improve code readability and re-usability. By using functions, you can write code more efficiently and easily maintain your program.

The Taylor series of a given function is a polynomial approximation of that function, derived using derivatives. In this case, we are asked to compute the Taylor polynomial for the function f (x) = cos (4x).

The Taylor polynomials of f are as follows:

p0(x) = 1

p1(x) = 1 - 8x2

p2(x) = 1 - 8x2 + 32x4

p3(x) = 1 - 8x2 + 32x4 - 128x6

p4(x) = 1 - 8x2 + 32x4 - 128x6 + 512x8

For any approximations, we can use around 6 decimals. For instance, if x = 0.5, then p4(0.5) = 0.988377, which is an approximation of the actual value of f (0.5), which is 0.98879958.

In conclusion, we have computed the Taylor polynomials of the given function f (x) = cos (4x), using around 6 decimals for approximation. These polynomials can then be used to approximate the given function.

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