10 How many ways can a company select 3 candidates to interview from a shortlist of 15? (2 Points) a. 555 b. 455 c. 444 d. None of them

a) The probability that a randomly selected box contains at least one defective bulb is approximately 0.099. b) The probability that a randomly selected box contains at most one defective bulb is approximately 0.9998.

a) To find the probability that a randomly selected box contains at least one defective bulb, we can use the complement rule: the probability of the complement of the event (no defective bulbs) is easier to calculate. The probability that a bulb is not defective is 199/200, so the probability that all 20 bulbs in a box are not defective is (199/200)²⁰. Therefore, the probability that at least one bulb is defective is:

1 - (199/200)²⁰ ≈ 0.099

So the probability that a randomly selected box contains at least one defective bulb is approximately 0.099.

b) To find the probability that a randomly selected box contains at most one defective bulb, we can calculate the probability of each possible outcome (0, 1, or 2 defective bulbs) and add them up.

The probability of no defective bulbs is (199/200)²⁰ ≈ 0.9048.

The probability of exactly one defective bulb is:

20C1 * (1/200) * (199/200)¹⁹ ≈ 0.0950

where 20C1 is the number of ways to choose one defective bulb out of 20.

The probability of two defective bulbs is:

20C2 * (1/200)² * (199/200)¹⁸ ≈ 0.0002

where 20C2 is the number of ways to choose two defective bulbs out of 20.

Therefore, the probability of at most one defective bulb is:

0.9048 + 0.0950 ≈ 0.9998

So the probability that a randomly selected box contains at most one defective bulb is approximately 0.9998.

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Answer:

Option A.

Step-by-step explanation:

A 30° - 60° - 90° triangle is a Right Triangle that has special side measures.

Let's summarize the sides in a ratio.

[tex]Short \ Leg: Long \ Leg: Hypotenuse\\x:x \sqrt{3} : 2x[/tex]

The short leg is just x.

The long leg is multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is double of the short leg.

For example, if the short leg is 2;

[tex]Short \ Leg = 2\\Long \ Leg = 2\sqrt{3} \\Hypotenuse = 4[/tex]

Let's look at the 4 options provided. We should check if the values of the sides match with a 30° - 60° - 90° triangle.

Option A has a short leg with the value of 5.

The long leg is correct, it's multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is correct, it's double of the short leg.

Option A is correct!

Option B has a short leg with the value of 5.

The long leg is correct, it's multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is incorrect, it's triple of the short leg.

Option B is incorrect.

Option C has a short leg with the value of 5.

The long leg is incorrect, it's multiplied by [tex]2\sqrt{3}[/tex].

The hypotenuse is correct, it's double of the short leg.

Option C is incorrect.

Option D has a short leg with the value of 10.

The long leg is incorrect, it's multiplied by [tex]\frac{1}{2} \sqrt{3}[/tex].

The hypotenuse is incorrect, it's [tex]1 \frac{1}{2}[/tex] of the short leg.

Option D is incorrect.

Our only 30° - 60° - 90° triangle is Option A.

The probability of the number of defective bulbs in the sample greater than 15 is approximately 0.9177.Hence option A is correct.

Suppose that a factory produces light bulbs, and the percentage of defective bulbs is 3.5%. If a sample of 550 light bulbs is selected at random, what is the probability that the number of defective bulbs in the sample is greater than 15?We are required to calculate the probability of the number of defective bulbs in the sample greater than 15. Percentage of defective bulbs = 3.5%Number of light bulbs in the sample = 550Let X be the number of defective bulbs in the sample. We know that the probability distribution of X is a binomial probability distribution because the sample is selected randomly and the sample size is less than 10% of the total population,

which is considered as large. Let P (X > 15) be the probability of the number of defective bulbs greater than 15.Now, mean μ = np = 550 × 0.035 = 19.25 and standard deviation σ = √npq Where q = 1 − p = 1 − 0.035 = 0.965∴ σ = √npq = √550 × 0.035 × 0.965 = 3.05Therefore, Z = (15 − μ) / σ = (15 − 19.25) / 3.05 ≈ −1.39Using normal distribution, P (Z > −1.39) = 1 − P (Z ≤ −1.39)We get P (Z ≤ −1.39) = 0.0823Using the standard normal table or calculator, we get P (Z > −1.39) = 1 − 0.0823 = 0.9177Therefore, the probability of the number of defective bulbs in the sample greater than 15 is approximately 0.9177. Hence, option A is correct.

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The mass of 5600 paper clips is 5.6 kilograms.

Here we use the unitary method to solve the problem. The unitary method is a mathematical technique used to solve problems.

It involves finding the value of one unit of a given quantity and then using that value to determine the value of other units of the same or different quantities.

Here we have

One paperclip has a mass of 1 gram.  

Mass of 1000 paperclips = 1 kilogram  

The mass of 1 paperclip in kilogram = 1/1000 = 0.001 kg

Similarly

Mass of 5600 paperclips = 0.001 kg × 5600 = 5.6 kg

Therefore,

The mass of 5600 paper clips is 5.6 kilograms.

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The cost of the fabric needed to make the square table cloth is $3.75

Firstly, it's essential to understand that the cost of the fabric is calculated per square foot. This means that the more square feet of fabric you use, the higher the cost will be. Therefore, the first step is to measure the dimensions of the tablecloth.

For instance, if the tablecloth is 5 feet long and 3 feet wide, the total area would be 15 square feet.

Next, you can multiply the total square footage of the tablecloth by the cost of the fabric per square foot.

In this case, if the cost of the fabric is $0.25 per square foot, the cost of the fabric needed for the tablecloth would be

=> 15 (total square feet) x 0.25 (cost per square foot) = $3.75.

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The new circle equation is: (x - 4)² + y² = 25

The x-intercepts are:  (9, 0) and (-1, 0)

You already have the answer for the first question, so let's look at b and c.

Here we have a circle equation:

x² + y² = 25

And now this circle is translated by the vector (4, 0) to make a new circle, so the new center will be at the point (4, 0).

That means that the new coordinates of the circle are:

(x - 4)² + (y - 0)² = 25

(x - 4)² + y² = 25

Now we want to get the intercepts of the x-axis, to get these we need to take y = 0.

then:

(x - 4)²  = 25

(x - 4) = ±5

x = 4 ± 5

The x-intercepts are (9, 0) and (-1, 0)

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The probabilities regarding a person spending are given as follows:

a) More than 28: 0.2033 = 20.33%.

b) Between 9 and 21: 0.3608 = 36.08%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for this problem are given as follows:

[tex]\mu = 23, \sigma = 6[/tex]

The probability of a person spending more than 28 is one subtracted by the p-value of Z when X = 28, hence:

Z = (28 - 23)/6

Z = 0.83

Z = 0.83 has a p-value of 0.7967.

1 - 0.7967 = 0.2033 = 20.33%.

The probability of spending between 9 and 21 is the p-value of Z when X = 21 subtracted by the p-value of Z when X = 9, hence:

Z = (21 - 23)/6

Z = -0.33

Z = -0.33 has a p-value of 0.3707.

Z = (9 - 23)/6

Z = -2.33

Z = -2.33 has a p-value of 0.0099.

Hence:

0.3707 - 0.0099 = 0.3608 = 36.08%.

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The length of XZ is 5.5 m.

The length of side XZ is calculated by applying the following cosine and sine rule.

If the length of WY is 7 m, then ∠WYZ is calculated as follows;

cos Y = (z² + w² - y² ) / (2zw)

where;

cos Y = ( 7² + 5.1² - 3² ) / ( 2 x 7 x 5.1 )

cos Y = 0.9245

Y = cos⁻¹ (0.9245)

Y = 22.4⁰

The value of ∠WYX is calculated as follows;

cos Y = (x² + w² - y² ) / (2xw)

cos Y = ( 7² + 5² - 4.8² ) / ( 2 x 7 x 5)

cos Y = 0.728

Y = cos⁻¹ (0.728)

Y = 43.28⁰

The value of ∠ZYX = 43.28⁰ + 22.4⁰ = 65.68⁰

The length of XZ is calculated by using the following cosine rule.

|XZ|² = |XY|² + |ZY|² - (2 x |XY| x |XY|) cos Y

|XZ|² = 5² + 5.1²  -  (2 x 5 x 5.1 ) x cos (65.68)

|XZ|² = 30

|XZ| = √30

|XZ| = 5.5 m

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The evaluation of the expression "twelve more than the product of 5 and a number x" evaluated at x=20 is equal to 112.

The expression "the product of 5 and a number x" can be written as 5x. Then, "twelve more than the product of 5 and a number x" is 5x + 12.

The expression "12 more than the product of 5 and x" can be written as "5x + 12." If x=20, then the expression evaluates to 5(20) + 12 = 100 + 12 = 112. Therefore, when x=20, "twelve more than the product of 5 and a number x" is equal to 112.

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Complete Question:

twelve more than the product of 5 and a number x? x is 20.

The most likely measurement of the distance from the earth to the moon would be 1.3 x 10^8 ft.

Therefore the answer is A)1. 3 x 10⁸ ft.

This is because the distance from the earth to the moon is approximately 238,900 miles, which is equivalent to approximately 1.3 x 10^8 feet. Options B, C, and D are all significantly larger or smaller than this value and do not reflect the actual distance from the earth to the moon.

In general, measurements of distance can be expressed in a variety of units, such as feet, meters, or miles. It's important to use the correct unit when making calculations or comparisons to ensure that the results are accurate and meaningful. When dealing with very large or very small distances, scientific notation can be a useful way to express the measurement in a compact and standardized form.

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--The question is incomplete, answering to the question below--

"which value would be the most likely measurement of the distance from the earth to the moon?

A)1. 3 x 10⁸ ft.

B)1. 3 x 10⁻⁹ ft.

C)1. 3 x 10¹⁰⁰ ft.

D)1. 3 x 10² ft."

Answer:

Not remain

Step-by-step explanation:

The two events are independent when one event has no effect on another event.

Question:2 ( 15 marks) An urn contains seven black balls and five white balls. Three balls are randomly awn from the urn without replacement and let X denote the number of black balls drawn. Then e three balls are put back to the urn. Next, four balls are randomly drawn from the urn without placement and let Y denote the number of black balls drawn. Let A=X+Y and B=X−Y . a. Find Cov(A,B) . b. Find Var(A+B) . c. Are A and B independent? Why?Solution:Given,An urn contains seven black balls and five white balls. Three balls are randomly drawn from the urn without replacement and let X denote the number of black balls drawn.Then these three balls are put back to the urn.Next, four balls are randomly drawn from the urn without placement and let Y denote the number of black balls drawn.Let A=X+Y and B=X−Y.a. Cov(A,B)Cov(A, B) = Cov(X+Y, X-Y)= Cov(X,X) - Cov(X,Y) + Cov(Y,X) - Cov(Y,Y) = Var(X) - Cov(X,Y) - Cov(Y,X) + Var(Y)Also, Var(X) = E(X^2) - [E(X)]^2=E(X)(E(X)+1)-E(X)^2 = E(X) + E(X)^2 - E(X)^2 = E(X) = 3.5Simillarly, Var(Y) = E(Y) + E(Y)^2 - E(Y)^2 = E(Y) = 2The probability of choosing a black ball from the urn P(X=1) = (7/12), The probability of choosing a black ball from the urn P(Y=1) = (7/12)Cov(X, Y) = E(XY) - E(X)E(Y) E(XY) = P(X=1,Y=1) + P(X=0,Y=0) = (7/12) * (6/11) * (5/10) + (5/12) * (4/11) * (3/10) = 21/110Cov(X, Y) = E(XY) - E(X)E(Y) = (21/110) - 3.5*2 = -14/110=-0.1272Thus, Cov(A, B) = 3.5 + 2 + 2 - 14/110 = 54/55=0.9818b. Var(A+B)Var(A+B) = Var(X+Y+X-Y) = Var(2X) + Var(2Y) = 4Var(X) + 4Var(Y) = 4(3.5) + 4(2) = 16Thus, Var(A+B) = 16.c. Are A and B independent? Why?A and B are not independent because Cov(A,B) ≠ 0, where Cov(A,B) = 54/55 ≠ 0.Note: In statistics, independence is a condition in which two events (A and B) are unrelated to each other. If the probability of an event A is not affected by the occurrence of another event B, these events are independent. That means, two events are independent when one event has no effect on another event.

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Value of the given expression when x = 0 and y = -1 is 3/8.

The given expression is (3x^2 y - 2x - 6y - 9)/(-11y + 5). We need to evaluate it when x = 0 and y = -1.Let's substitute the given values of x and y in the expression.(3x^2 y - 2x - 6y - 9)/(-11y + 5) = (3(0)^2 (-1) - 2(0) - 6(-1) - 9)/(-11(-1) + 5)= (-3 + 6 - 9)/(-5 - 11)= -6/-16= 3/8Therefore, the value of the given expression when x = 0 and y = -1 is 3/8.An HTML formatted answer is given below.

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Sure! CER stands for Claim, Evidence, and Reasoning. It is a great way to organize our thoughts and show our work.

Claim: Melody had X amount of money after making the deposit.

Evidence: Melody's account was overdrawn by $32. She made a deposit of $100.

Reasoning: To find out how much money Melody has after making the deposit, we need to add the amount of the deposit to the amount of money she had before the deposit.

So, let's start by finding out how much money Melody had before the deposit. Since her account was overdrawn by $32, we can assume that she had a negative balance of $32. To find out the positive balance, we need to add $32 to $100 (the amount of the deposit):

$32 + $100 = $132

Therefore, Melody had $132 after making the deposit.

Claim: Melody had $132 after making the deposit.

Evidence: Melody's account was overdrawn by $32. She made a deposit of $100.

Reasoning: We added the amount of the deposit to the amount of money she had before the deposit to find out how much money she had after making the deposit.

Rational numbers can be represented by the amount (not the %) of two integers. A fraction with a numerator that is not zero is considered fair. In addition to   [tex]1/2[/tex] and [tex]1/5[/tex], rational numbers also include  [tex]3/4[/tex]  .

To rationalize the denominator of 1/√10, we need to multiply both the numerator and denominator by √10. This will give us:

Furthermore, "0" could be stated in a number of ways as a real purpose, such as [tex]0/1, 0/2[/tex]  , as well as 0/3. But also [tex]1/0, 2/0, 3/0[/tex]  , and others. Seven is a respectable number. Two variables can be divided to produce a rational number.

[tex]1/[/tex]√[tex]10[/tex] × √10/√10 = √10/10

We can simplify this fraction by dividing both the numerator and denominator by 2, which gives us:√10/5 This is the simplest form of the expression with a rational denominator.

Therefore, the solution of the given problem of rational numbers comes out to be [tex]1/10[/tex] can be expressed as  [tex]10/10[/tex] When the denominator is logical.

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Answer: 56

Step-by-step explanation:

The results of the composition of two functions are listed below:

Case 1:  

a) f[g(2)] = 3

b) f[g(x)] = 18 · x² - 60 · x + 51

c) g[f(x)] = 6 · x² - 2

d) g[g(x)] = 9 · x - 20

e) f[f(- 2)] = 163

Case 13: g[f(x)] = (√x + 2)² + 3, Dom {g[f(x)]} = [0, + ∞)

Case 15: g[f(x)] = (3√x + 1) / [27 · (√x)³], Dom {g[f(x)]} = (0, + ∞)

Case 17: g[f(x)] = 1 / [[(2 / x) + 4] - 4] = x / 2, Dom {g[f(x)]} = All real numbers.

Case 21: g[f(x)] = - 3 / (√2 - 4 · x), Dom {g[f(x)]} = All real numbers except x = √2 / 4.

In this problem we must determine, analyze and evaluate the composition of two functions, whose definition is shown below:

f ° g (x) = f [g (x)]

Now we proceed to determine the composition of functions:

Case 1: f(x) = 2 · x² + 1, g(x) = 3 · x - 5

a) f[g(2)] = 2 · (3 · 2 - 5)² + 1 = 2 · 1² + 1 = 2 + 1 = 3

b) f[g(x)] = 2 · (3 · x - 5)² + 1 = 2 · (9 · x² - 30 · x + 25) + 1 = 18 · x² - 60 · x + 51

c) g[f(x)] = 3 · (2 · x² + 1) - 5 = 6 · x² - 2

d) g[g(x)] = 3 · (3 · x - 5) - 5 = 9 · x - 20

e) f[f(- 2)] = 2 · [2 · (- 2)² + 1]² + 1 = 2 · (2 · 4 + 1)² + 1 = 2 · 9² + 1 = 162 + 1 = 163

Case 13: f(x) = √x + 2, g(x) = x² + 3

g[f(x)] = (√x + 2)² + 3

Dom {g[f(x)]} = [0, + ∞)

Case 15: f(x) = 3√x, g(x) = (x + 1) / x³

g[f(x)] = (3√x + 1) / (3√x)³ = (3√x + 1) / [27 · (√x)³]

Dom {g[f(x)]} = (0, + ∞)

Case 17: f(x) = 1 / (x - 4), g(x) = (2 / x) + 4

g[f(x)] = 1 / [[(2 / x) + 4] - 4] = x / 2

Dom {g[f(x)]} = All real numbers.

Case 21: f(x) = √2 - 4 · x, g(x) = - 3 / x

g[f(x)] = - 3 / (√2 - 4 · x)

Dom {g[f(x)]} = All real numbers except x = √2 / 4.

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Answer:

250 cubic cm

Step-by-step explanation:

Side length = x

[tex]2x^{2} + 4x(10) = 120[/tex]

[tex]x^{2} +2x - 30 =0[/tex]
After factorization, we will get (x+6) ( x-5) = 0

side length should be positive, so we take x to be 5.
Dimensions will be 5 x 5 x 10 = 250 cubic centimeters.

Answer:

The price of an ice cream sundae is $4.75.

Step-by-step explanation:

Let m be the cost of one milkshake and s be the cost of one ice cream sundae.

4m + 2s = 23.50 -------> 8m + 4s = 47.00

8m + 6s = 56.50 -------> 8m + 6s = 56.50

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

2s = 9.50

s = 4.75

The value of y for x ranging from -4 to 4 are 34, 19, 6, -3, -2, 1, 10, 23 and 38, respectively.

To find the values of y for the given range of x, we can substitute each value of x into the equation y = 2x² - x - 2 and simplify. The completed table of values is:

(The table is attached below)

The given relation is y = 2x² - x - 2. To create a table of values for this relation, we can substitute values of x in the given range of -4 to 4 and calculate the corresponding values of y using the equation. In the first table, we substitute values of x ranging from -4 to 0 and calculate the corresponding values of y. In the second table, we substitute values of x ranging from 0 to 4 and calculate the corresponding values of y. These tables show how the values of y change as x changes within the given range, and provide a way to graph the relation as well.

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After addressing the issue at hand, we can state that This is an unusual angles result because it implies that point A is on line segment BC, and thus triangle ABC is a straight line.

An angle is a shape in Euclidean geometry that is made up of two photons, known as the tone's sides, that connect at their center at a point known as the angle's vertex. Two rays may combine to form an angular velocity there in plane where they are positioned. When two planes make contact, an angle is formed. These are started referring to as wing angles. In plane geometry, an angle is really the plausible configuration of multiple rays or rows that express a termination. The English word "angle" originates from the Latin word "angulus," that means "horn." The vertex is the point at which the two rays, also renowned as the angle's sides, intersect.

We can use the property that the sum of angles in a triangle is 180 degrees to find the value of x in the given figure.

We can begin by calculating the value of angle ABC using the following information:

ABC angle = 180 - angle ABD - ACD angle = 180 - 35 - 58 = 87 degrees

angle ABE stands for angle. Angle ACD = 58 degrees Angle = CDE 35 degrees CBE

angle BAC = angle - 180 angle - ABC ABE stands for angle. CBE = 180 - 87 - 58 - 35 = 0°

This is an unusual result because it implies that point A is on line segment BC, and thus triangle ABC is a straight line.

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Answer:

D

Step-by-step explanation:

every term of one expression gets multiplied with every term of the other expression.

(6x² - 3x - 6)(4x² + 5x + 4) =

= 6×4x²×x² + 6×5x²×x + 6×4x² - 3×4x×x² - 3×5x×x -

3×4x - 6×4x² - 6×5x - 6×4

3 terms × 3 terms = 9 terms.

now we combine similar factors for the 9 terms

24x⁴ + 30x³ + 24x² - 12x³ - 15x² - 12x - 24x² - 30x - 24

and now we combine similar terms

24x⁴ + 18x³ - 15x² - 42x - 24

The graph of the compressed function is graph A.

Remember that for any function f(x), we define a horizontal compression of scale factor k (whre we must have k > 1) is defined as:

f(k*x)

Here we have f(4x), so this is an horizontal compression of scale factor 4.

Notice that the original function goes from x = -4 to x = 4

Then the compressed function must go from x = -4/4 = -1 to x = 4/4 = 1

Which is what we can see in graph A, so that is the correct option.

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Answer:

Step-by-step explanation:

3/70

The probability that the next customer pays with the debit card is 3/70 or 0.04.

Probability:

The probability of an event is a number that indicates the probability of the event occurring. Expressed as a number between 0 and 1 or as a percent sign between 0% and 100%.

Given that:

Last week there are 32 customer who paid cash

03 customers used debit card

35 customer used credit card.

Now,

The Total Number is = 32 + 3 + 35

                                   = 70

For cash buyer = 32/70 = 16/35

                         = 0.45

For Debit Card = 3/70

                         = 0.04

For Credit Card = 35/70 = 1/2

                          = 0.5

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The expression 8* is equivalent to both 32 and 32².Expression 8* is an algebraic expression that can be expanded to 8*1 = 8. Thus, 8* is equivalent to 8.

An algebraic expression is a combination of terms, variables and operators that when simplified, evaluates to a numerical value. It is usually written in the form of equations and inequalities, where the terms represent numbers and variables represent unknown values.

When multiplied by 1, 8* is equivalent to 32. This is because 8*1 = 8 and 32*1 = 32. So, 8* = 32.

Expression 8* is also equivalent to 32². This is because 8*1 = 8 and 32² is 8 multiplied by itself. So 8* = 32².

The expressions A. (²/²), B. 8.8x+1, and F. 8-8-1 are not equivalent to 8*.

Expression A. (²/²) is not equivalent to 8*. This is because (²/²) is an undefined expression, while 8* is a defined expression.

Expression B. 8.8x+1 is not equivalent to 8*. This is because 8.8x+1 is an algebraic expression that includes the variable x, while 8* does not include any variables.

Expression F. 8-8-1 is not equivalent to 8*. This is because 8-8-1 is a subtraction expression, while 8* is a multiplication expression.

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If it will be 20 feet from the ground then the time will be 3.75sec.

The speed οf an οbject, alsο knοwn as v in cοmmοn parlance and kinematics, is the size οf the change in pοsitiοn per unit οf time οr the size οf the change in pοsitiοn οver time; as such, it is a scalar quantity.

The average speed οf an οbject in a given periοd οf time is equal tο the distance travelled by the οbject divided by the length οf the interval the instantaneοus speed is the upper limit οf the average speed as the length οf Velοcity and speed are different cοncepts.

The rοcket was launched 200 feet up a cliff, which was its height.

u = 120 feet per secοnd is the rοcket's initial speed.

h(t) = -16t2 + 120t + 200 represents the height οf the rοcket with respect tο time t.

By inputting t = 1 h(1) = -16 (1)2 + 120 (1) + 200 = 304, yοu can calculate the rοcket's height after 1 secοnd.

304 feet b after οne secοnd, which is the rοcket's height. The value οf the maximum pοint οf the height-related equatiοn's curve can be determined by using the fοrmula belοw tο determine the maximum height.

At the greatest pοint, h'(t) = d(-16t2 + 120t + 200)/dt = -32t + 120 = 0 t = 120/32 = 3.75.

Hence the time will be 3.75sec.

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The x-coordinate where f'(x) = 0 and x is in the interval [0, pi] is:

x = π/6

What is derivative?

In calculus, the derivative of a function is a measure of how much the function changes as its input variable changes. More specifically, the derivative of a function f(x) at a particular value of x, denoted by f'(x), is defined as the limit of the ratio of the change in the function value to the change in the input variable as the change in the input variable approaches zero.

To find the x-coordinates where f'(x) = 0 for f(x) = 2x + sin(4x) in the interval [0, pi], we need to find the derivative of f(x) and set it equal to 0.

f(x) = 2x + sin(4x)

f'(x) = 2 + 4cos(4x)

Setting f'(x) equal to 0, we get:

2 + 4cos(4x) = 0

cos(4x) = -1/2

We know that cos(4x) = -1/2 has solutions at 4x = 2π/3 and 4x = 4π/3 (plus any multiple of 2π), because these are the solutions to cosθ = -1/2 in the interval [0,2π). So, we can write:

4x = 2π/3 or 4x = 4π/3

Solving for x in each equation, we get:

x = π/6 or x = π/3

However, we need to check that these solutions are in the interval [0, pi].

π/6 is in the interval [0, pi], but π/3 is not.

Therefore, the only x-coordinate where f'(x) = 0 and x is in the interval [0, pi] is:

x = π/6

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To make a geometric shape with 100 points, there are many options depending on the desired shape.

There are many possible geometric shapes you can make with 100 points, depending on the specific constraints and requirements you have. Few examples are,

Circle: One way to create a circle with 100 points is to evenly distribute the points around the circumference of a circle with a given radius.

Square: Another option is to create a square with 100 points. To do this, you can divide the sides of the square into 25 segments, and then place 4 points at each segment endpoint.

Regular polygon: You can create a regular polygon with 100 sides by following a similar method to the circle.

Spiral: You can create a spiral shape with 100 points by placing the points along a logarithmic spiral.

Fractal: Finally, you can create a fractal shape with 100 points by applying a recursive algorithm to divide and subdivide segments of an initial shape.

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On solving the inequalities 2(5-x) > 6 and 22 > 2(9+x) the correct comparison is "The inequalities have the same solutions."

What is an inequality?

In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.

To solve the inequality 2(5-x) > 6, we can simplify as follows -

2(5-x) > 6

10 - 2x > 6

-2x > 6 - 10

-2x > -4

x < 2

So the solution to this inequality is x < 2.

To solve the inequality 22 > 2(9+x), we can simplify as follows -

22 > 2(9+x)

22 > 18 + 2x

4 > 2x

2 > x

So the solution to this inequality is x < 2.

Both inequalities have the same solution, x < 2.

Therefore, the correct comparison of solutions is option D.

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