Based on a random sample of 25 units of product X, the average weight is 102 lb and the sample standard deviation is 10 lb. We would like to decide whether there is enough evidence to establish that the average weight for the population of product X is greater than 100 lb. Assume the population is normally distributed. Using the critical value rule, at α =. 01, we can reject the null hypothesis

7) Given the extreme values in a set of data as 5 and 19, the truth about the data set is C. The range is 14.

8) The mean of the data set 5, 9, 15, 18, 22 is B. 14.

9) The mode for the following set of data, 4, 5, 5, 6, 7, 7, 8, 12, is A. there is none.

The range is the difference between the extreme values of a data set.

This difference is computed by subtracting the minimum value from the maximum value.

The mean represents the average value of a data set, computed by dividing the total value by the number of items.

The mode is one value in the data set that occurs most.  There cannot be more than one mode in a data set.

Range between 5 and 19 = 14 (19 - 5)

Mean of 5, 9, 15, 18, 22 = 13.8 (69 ÷ 5) = 14

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The difference in the account balances is $138,435.93.

We have,

We can solve this problem by using the formula for the future value of an annuity:

[tex]FV = PMT \times [(1 + r)^n - 1] / r[/tex]

where FV is the future value of the annuity, PMT is the yearly contribution, r is the annual interest rate, and n is the number of years.

Using the given information, we can find the future value of the annuity if the person starts at age 35:

FV1

= $5,000 x [(1 + 0.065)^30 - 1] / 0.065

= $431,874.32

Now we can find the future value of the annuity if the person starts at age 40:

FV2 = $5,000 x [(1 + 0.065)^25 - 1] / 0.065

= $293,438.39

The difference in the account balances is:

FV1 - FV2

= $431,874.32 - $293,438.39

= $138,435.93

Therefore,

The difference in the account balances is $138,435.93.

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The probability that the experiment ends before the 6th toss and an even number of tosses are required is 5/16.

The given experiment involves tossing a coin until the first time the same result appears twice in succession. This means that the experiment could end after two tosses if both tosses yield the same result (e.g., heads-heads or tails-tails) or it could continue for many more tosses until this condition is met.

The sample space for this experiment can be represented as a binary tree where the root node represents the first toss and the two branches from the root represent the two possible outcomes (heads or tails). The next level of the tree represents the second toss, with two branches emanating from each branch of the root (one for heads and one for tails). This process continues until the experiment ends with two successive outcomes being the same.

The probability of each outcome in the sample space can be computed by multiplying the probabilities of each individual toss. Since each toss has a probability of 1/2 of resulting in heads or tails, the probability of any particular outcome requiring n tosses is 1/2^n.

(a) A = The experiment ends before the 6th toss.

To calculate the probability of this event, we need to sum the probabilities of all outcomes that end before the 6th toss. This includes outcomes that end after the second, third, fourth, or fifth toss. Thus, we have:

P(A) = P(outcome ends after 2 tosses) + P(outcome ends after 3 tosses) + P(outcome ends after 4 tosses) + P(outcome ends after 5 tosses)

= (1/2^2) + (1/2^3) + (1/2^4) + (1/2^5)

= 15/32

Therefore, the probability that the experiment ends before the 6th toss is 15/32.

(b) B = An even number of tosses are required.

An even number of tosses are required if the experiment ends after the second, fourth, sixth, etc. toss. The probability of this event can be calculated as follows:

P(B) = P(outcome ends after 2 tosses) + P(outcome ends after 4 tosses) + P(outcome ends after 6 tosses) + ...

= (1/2^2) + (1/2^4) + (1/2^6) + ...

This is a geometric series with first term a = 1/4 and common ratio r = 1/16. Using the formula for the sum of an infinite geometric series, we have:

P(B) = a/(1-r) = (1/4)/(1-1/16) = 4/15

Therefore, the probability that an even number of tosses are required is 4/15.

(c) A∩B = The experiment ends before the 6th toss and an even number of tosses are required.

To calculate the probability of this event, we need to consider only the outcomes that satisfy both conditions. These include outcomes that end after the second or fourth toss. Thus, we have:

P(A∩B) = P(outcome ends after 2 tosses) + P(outcome ends after 4 tosses)

= (1/2^2) + (1/2^4)

= 5/16

Therefore, the probability that the experiment ends before the 6th toss and an even number of tosses are required is 5/16.

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For Madison's investment of $8,000 at 7% for 3 years, she made $560, $1,120, and $1,680 in simple interest over each year respectively. Her balance at the end of Year 2 was $9,680. The compound interest earned over the 3-year period was $2,837.28.

The simple interest formula is

I = Prt

where I is the interest, P is the principal (the amount invested), r is the annual interest rate as a decimal, and t is the time in years.

For Madison's investment of $8,000 at 7% for 3 years, we have

P = $8,000

r = 7% = 0.07

t = 3 years

To find the simple interest for each year, we can use the formula above and multiply it by the number of years

I₁ = Prt = $8,000 x 0.07 x 1 = $560

I₂ = Prt = $8,000 x 0.07 x 2 = $1,120

I₃ = Prt = $8,000 x 0.07 x 3 = $1,680

To find the balance at the end of each year, we can add the interest to the principal

Year 1: $8,000 + $560 = $8,560

Year 2: $8,560 + $1,120 = $9,680

Year 3: $9,680 + $1,680 = $11,360

To find the compound interest, we can use the formula

[tex]A = P(1 + r/n)^{nt}[/tex]

where A is the amount of money at the end of the investment period, P is the principal, r is the annual interest rate as a decimal, n is the number of times interest is compounded per year, and t is the time in years.

Assuming the interest is compounded annually (once per year), we have

P = $8,000

r = 7% = 0.07

n = 1

t = 3 years

Using these values in the formula, we get

A = $8,000(1 + 0.07/1)¹ˣ³ = $10,837.28

To find the compound interest, we can subtract the principal from the amount

Compound interest = $10,837.28 - $8,000 = $2,837.28

Therefore, Madison made a total of $2,837.28 in interest over the 3-year period. At the end of Year 2, the balance of her investment was $9,680. The compound interest on the investment over the 3-year period was $2,837.28.

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The total surface area of the triangular prism is 204 square units.

To find the total surface area of a triangular prism, we need to find the areas of all its faces and add them up.

First, we need to find the area of each triangular face. We can use the formula:

Area of a triangle = 1/2 x base x height

For the triangle with base 4 and height 3, we have:

Area of triangle = 1/2 x 4 x 3 = 6

For the triangle with base 6 and height 8, we have:

Area of triangle = 1/2 x 6 x 8 = 24

Now, we need to find the area of each rectangular face. We can use the formula:

Area of a rectangle = length x width

For the rectangular face with length 6 and width 4, we have:

Area of rectangle = 6 x 4 = 24

For the rectangular face with length 8 and width 4, we have:

Area of rectangle = 8 x 4 = 32

Finally, we add up all the areas to get the total surface area:

Total surface area = 2 x (area of triangle) + 3 x (area of rectangle)

Total surface area = 2 x (6 + 24) + 3 x (24 + 32)

Total surface area = 60 + 144

Total surface area = 204

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Find The Total Surface Area Of This Triangular Prism:

We are given two sets A and B, and we are asked to describe some sets that can be formed using these sets. We will use set operations such as union and intersection to form new sets and provide descriptions of these sets in words.

a. A ∩ B: This set includes all current students who are attending both ABC and Harvard at the same time.

b. A ∪ B: This set includes all current students who are attending either ABC, Harvard, or both.

c. A ∩ B: (same as 'a') This set includes all current students who are attending both ABC and Harvard at the same time.

d. (A ∩ B): (same as 'a') This set includes all current students who are attending both ABC and Harvard at the same time.

e. (A ∪ B): (same as 'b') This set includes all current students who are attending either ABC, Harvard, or both.

3. Let A = {2 ∈ Z | x = 6a for some integer a} and B = {y ∈ Z | y = 36 for some integer b}. To prove that A ⊂ B, we need to show that every element of A is also an element of B.

Let x be an arbitrary element of A.

Since x = 6a for some integer a, we can write x as 6a = 2 * 3a.

Because 3a is also an integer (since a is an integer), we can say x = 2 * (3 * a), which implies x = 36 * a for some integer a. Thus, x ∈ B.

Since every element of A is also an element of B, we have proven that A ⊂ B.

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1.39 is the median of the distribution

Creating a table:

X :___0 __ 1 __ 2 ___ 3 ____ 4 ___ 5 ___ 6

P(x):0.02_0.03_0.07, 0.12, 0.30_ 0.28_0.18.

The standard deviation = √(Var(x))

Var(x) = Σx²*p(x) - E(x)²

E(x) = ΣX*p(x)

E(x) = (0*0.02) + (1*0.03) + (2*0.07) + (3*0.12) + (4*0.30) + (5*0.28) + (6*0.18) = 4.21

Var(X) :

[tex]((0^2*0.02) + (1^2*0.03) + (2^2*0.07) + (3^2*0.12) + (4^2*0.30) + (5^2*0.28) + (6^2*0.18)) - 4.21^2[/tex]

19.67 - 17.7241

= 1.9459

Standard deviation = √(Var(X))

Standard deviation = √(1.9459)

Standard deviation = 1.3949551

= 1.39

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Full Question ;

Hannah has a chicken coop with 6 hens. Let X be the total number of eggs the hens lay on a randomly chosen day. The distribution for X is given in the table.

A 2-column table with 7 rows. Column 1 is labeled number of eggs with entries 0, 1, 2, 3, 4, 5, 6. Column 2 is labeled probability with entries 0.02, 0.03, 0.07, 0.12, 0.30, 0.28, 0.18.

What is the standard deviation of the distribution?

1.39

1.95

2.16

4.67

The correct answer is option b.

To find the median, we need to first put the observations in order:

5, 7, 10, 10, 10, 14, 14, 15, 18, x

Since there are 10 observations, the median is the average of the 5th and 6th observations, which are both 10. Therefore, the median is 10.

To find the mode, we need to find the observation that appears most frequently. Here, both 10 and 14 appear three times each, so the data has two modes: 10 and 14.

To find the standard deviation, we need to first find the mean of the data. We know that the sum of the observations is:

5 + 7 + 10 + 10 + 10 + 14 + 14 + 15 + 18 + x

= 103 + x

Since we know that X = 11.6, we can substitute to get:

Sum of observations = 103 + 11.6 = 114.6

The mean is then:

Mean = (Sum of observations) / (Number of observations)

Mean = 114.6 / 10 = 11.46

To find the standard deviation, we need to calculate the deviation of each observation from the mean, square each deviation, find the average of the squared deviations, and then take the square root.

Deviation of 5 = 11.46 - 5 = 6.46

Deviation of 7 = 11.46 - 7 = 4.46

Deviation of 10 = 11.46 - 10 = 1.46

Deviation of 10 = 11.46 - 10 = 1.46

Deviation of 10 = 11.46 - 10 = 1.46

Deviation of 14 = 11.46 - 14 = -2.54

Deviation of 14 = 11.46 - 14 = -2.54

Deviation of 15 = 11.46 - 15 = -3.54

Deviation of 18 = 11.46 - 18 = -6.54

Deviation of x = 11.46 - x

To find the standard deviation, we need to find the average of the squared deviations.

Average of squared deviations = [(6.46)^2 + (4.46)^2 + (1.46)^2 + (1.46)^2 + (1.46)^2 + (-2.54)^2 + (-2.54)^2 + (-3.54)^2 + (-6.54)^2 + (11.46 - x)^2] / 10

= (41.7316 + 19.8916 + 2.1316 + 2.1316 + 2.1316 + 6.4516 + 6.4516 + 12.5316 + 42.8916 + (11.46 - x)^2) / 10

= (136.786) / 10

= 13.6786

Finally, we take the square root of the average of the squared deviations to find the standard deviation:

Standard deviation = sqrt(13.6786) = 3.8356

Therefore, the correct answer is option b.

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Answer:

c

Step-by-step explanation:

c is correct

To calculate Melanie's score, we first need to find out how many total points she earned and how many points were deducted for incorrect answers.

Melanie earned 6 points for each of the 45 questions she answered correctly, which gives her a total of 6 x 45 = 270 points.

For the 33 questions she answered incorrectly, Melanie received a penalty of 5 points for each one. So, the total points deducted for incorrect answers is 5 x 33 = 165 points.

To find Melanie's score, we need to subtract the points deducted for incorrect answers from the total points earned:

Score = Total points earned - Points deducted for incorrect answers
Score = 270 - 165
Score = 105

Therefore, Melanie's score on the college test is 105.

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Answer: 9x=45

Step-by-step explanation:

“product” means the result from multiplying two numbers, so if the product of the two numbers is 45, then the two numbers provided (9 and x) are being multiplied.  

If we were to solve for x, we could divide both sides by 9 to get 5.

There are three types of arcs in a circle: minor arcs, major arcs, and semicircles. The method for finding the measure of each arc depends on its type.

1. Minor arcs are arcs that measure less than 180 degrees. To find the measure of a minor arc, simply measure the angle that it subtends at the center of the circle. This angle is equal to the arc's measure.

2. Major arcs are arcs that measure greater than 180 degrees but less than 360 degrees. To find the measure of a major arc, subtract the measure of its corresponding minor arc from 360 degrees. For example, if the minor arc measures 60 degrees, the major arc measures 360 - 60 = 300 degrees.

3. Semicircles are arcs that measure exactly 180 degrees. To find the measure of a semicircle, simply divide the measure of the full circle (360 degrees) by 2. Therefore, a semicircle always measures 180 degrees.

Remember, when finding the measure of an arc, it is important to identify the type of arc and use the appropriate method.

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62.8 cubic inches is the volume of a can of peanuts with a height of 5 in. and a lid that is 4 in. wide

We have to find the  volume of a can of peanuts with a height of 5 in. and

a lid that is 4 in. wide

Volume of cylinder =πr²h

h is height which is 5 in

r is radius of can which is 2 in

Plug in values of h and r

Volume = 3.14×4×5

=62.8 cubic inches

Hence, 62.8 cubic inches is the volume of a can of peanuts with a height of 5 in. and a lid that is 4 in. wide

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The optimal value for x in the stochastic model is a range of values from x=6 to x=8. The stochastic model causes problems due to the uncertainty in the optimal solution and the assumptions.

With a=5, the optimal solution is x=8. However, with the addition of stochasticity and possible values for a of 3, 4, and 6, the optimal value for x becomes a range of values.

The expected value of a is 4.5, which means that there is a higher probability of a lower value for a, resulting in a lower optimal value for x. Therefore, the optimal value for x becomes x=6 when a=3 or a=4, x=7 when a=5, and x=8 when a=6.

The stochastic model causes problems because the optimal solution is no longer a fixed value but rather a range of values that are dependent on the probability distribution of a.

Additionally, this model assumes that the production time is the only constraint on production, which may not always be the case in real-world production scenarios. Therefore, the stochastic model may not accurately reflect the actual production process and could lead to suboptimal production decisions.

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Answer:

Step-by-step explanation:

To find the difference in temperature between the day's high and low, we need to subtract the low temperature from the high temperature.

The high temperature is 18 degrees, and the low temperature is -6 degrees.

So, the difference in temperature between the day's high and low is:

18 degrees - (-6 degrees)

= 18 degrees + 6 degrees

= 24 degrees

Therefore, the difference in temperature between the day's high and low is 24 degrees.

Sample = 102 nests
Failed nests = 64

Proportion of failed nests = p = 64/102 = 0.6275

95% interval is given as:

p ± z x √ ( p( 1-p /n))

Note that

z = z-score related to 95% = 1.96

so

0.6275 ± 1.96 x (√(0.6275 (1-0.6275) /102) )

0.6275  ±  0.09382660216

95% Confidence interval  = (0.721, 0.031)


b) H⁰ : P = 0.29
Ha : p > 0.29

z = (0.6275 - 0.29) / √(0.29(1-0.29)/102)

= 7.51182894275

= 7.51

Since the test is greater than the critical value, we must reject the null hypothesis.

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Full Question:

One difficulty in measuring the nesting success of birds is that the researchers must count the number of eggs in the nest, which is disturbing to the parents. Even though the researcher does not harm the birds, the flight of the bird might alert predators to the presence of a nest. To see if researcher activity might degrade nesting success, the nest survival of 102 nests that had their eggs counted, was recorded. Sixty-four of the nests failed (i.e. the parent abandoned the nest.)

a) Construct and interpret a 95% confidence interval for the proportion of nest failures in the population I

b) The usual nest failure rate of these birds is 29%. Based on the confidence interval from part (a), is this consistent with the theory that the researcher's activity affects nesting success? Justify your answer with an appropriate statistical

The number of tables that have 2 chairs each, if there are 50 tables at the restaurant and 40% have 2 chairs each, based on the percentage, therefore is 20 tables

A percentage is a representation of a part of a quantity, expressed as a fraction of 100.

The number of tables in the restaurant = 50 tables

The percentage of the table that have 2 chairs = 40%

The percentage of the table that have 4 chairs = 60%

The percentage of the tables that have 2 chairs each indicates;

The number of tables that have 2 chairs = (40/100) × 50 = 20

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Therefore, the five one-foot rulers laid end-to-end would be equal to 60 inches.

One foot or 12 inches is equivalent to one ruler. Three feet make up a yard. Three rulers make up a yardstick. To measure shorter distances, use rulers.  A foot is made up of 12 inches. Typically, a ruler is 12 inches long. Yardsticks are longer rulers with a length of 3 feet (or 36 inches, which is equivalent to one yard).

Larger things like this teacher's desk are measured in length using a ruler, which is commonly used to represent one foot. The length of the teacher's desk is equal to the edge of five rulers, or around five feet.

There are 12 inches in one foot, so five one-foot rulers laid end-to-end would be equal to:

5 feet × 12 inches/foot = 60 inches

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In the land of Paclandia, the Ok and Tok tribes have been rivals for centuries, fighting for control of border farms. The outcome of their latest skirmish can be analyzed using a game tree, where the Ok act as maximizers and the Tok as minimizers, assuming the Tok will react optimally.

The Talok tribe, after observing the conflicts, decides to get involved. They have unique powers of suggestion and can coerce the Ok into misinterpreting terminal utilities. If the Talok tricks the Ok into thinking that a terminal utility z is now valued as y = 22 + 22 + 6, this could affect the actions taken by the Ok.

However, the final outcome depends on how the game tree is structured and the terminal utilities assigned to each node. If the new perceived value of y leads the Ok to choose different actions than they would have without the Talok's intervention, it will indeed affect the actions taken by the Ok.

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The probability of selecting blue marble and green marble is 1/13.

A probability is a number that reflects the chance or likelihood that a particular event will occur. The certainty of an event is 1 which is equivalent to 100%.

Probability = sample space/total outcome

total outcome = 13

The probability of picking blue in the first pick = 6/13

since there is no replacement, the total outcome for the second pick = 12

The probability of picking green in the second pick = 2/12 = 1/6

Therefore the probability of selecting blue and green marble = 6/13 × 1/6

= 1/13

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Thus, the third-order Taylor polynomial for f(x) = cos(x) at a = 0 is: [tex]P_3(x) = 1 - x^2 / 2! + x^4 / 4![/tex].

Taylor Polynomials:

We have f(x) = cos(-5x) = cos(0 - 5x), so we can use the Taylor series for cos(x) centered at a = 0:

cos(x) = Σ[tex](-1)^n * x^(2n) / (2n)![/tex]

Thus, we have:

[tex]P_1(x) = cos(0) + (-5x) * (-sin(0)) = 1\\P_2(x) = 1 + 0 + (-5x)^2 / 2! = 1 + 12.5x^2\\P_3(x) = 1 + 0 + (-5x)^2 / 2! + 0 + (-5x)^4 / 4! = 1 + 12.5x^2 + 52.0833x^4\\P_4(x) = 1 + 0 + (-5x)^2 / 2! + 0 + (-5x)^4 / 4! + 0 + (-5x)^6 / 6! = 1 + 12.5x^2 + 52.0833x^4 + 136.7188x^6[/tex]

Interval of Convergence:

The power series given is:

Σ[tex](2k+1)*(x-2)^k[/tex]

Using the ratio test, we have: limit:

[tex]|(2k+3)(x-2)^(k+1) / ((2k+1)(x-2)^k)| = |x-2| lim |2k+3| / |2k+1| = |x-2|[/tex]

So, the series converges for |x - 2| < 1, or 1 < x < 3. Thus, the interval of convergence is (1, 3).

Polar Coordinates:

Using the Pythagorean theorem, we have:

r = [tex]\sqrt{(x^2 + y^2)\\\\\sqrt{(5^2 + 73.5^2) }\\[/tex]

r= 73.790

Using trigonometry, we have:

θ = arctan(y/x) = arctan(73.5/5) = 1.493 rad = 85.758°

In the range 0 ≤ θ < 2π, this point can be expressed in polar coordinates as (73.790, 85.758°) or (73.790, 445.242°).

In the range -π < θ ≤ π, this point can be expressed in polar coordinates as (73.790, -94.242°).

Third-Order Taylor Polynomial:

The Taylor series for cos(x) centered at a = 0 is:

cos(x) = [tex]1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...[/tex]

Taking the first four terms, we have:

[tex]P_3(x) = 1 - x^2 / 2! + x^4 / 4! = cos(x) + x^6 / 6![/tex]

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Mathematical induction can be used to prove that for every integer k ≥ 5, k^2 - 3k ≥ 10.

Base Case: Let k = 5,

Then,  k^2 - 3k = 5^2 - 3(5) = 10

Since 10 >= 10 is true, the base case holds.

Inductive Step: Assume that for some integer n >= 5, n^2 - 3n >= 10 is true.

We want to prove that (n + 1)^2 - 3(n + 1) >= 10 is also true.

Expanding the left-hand side of the inequality, we get:
(n + 1)^2 - 3(n + 1) = n^2 + 2n + 1 - 3n - 3

On simplifying ,we get:
n^2 - n - 2 >= 0
On factoring,we get:

(n - 2)(n + 1) >= 0

Since n >= 5, n - 2 >= 3, and n + 1 >= 6, so both factors are positive. Therefore, the inequality is true for all n >= 5.

By mathematical induction, we have proved that for every integer k >= 5, k^2 - 3k >= 10.

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Chaneece did not find the correct solution

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

x and y are the integers

So, we have

x + y ≤ 40

x - y ≥ 20

Add the equations

So, we have

2x = 60

Divide

x = 30

Next, we have

30 + y ≤ 40

So, we have

y ≤ 10

This means that

x = 30 or between 20 and 30

y = 10 or between 0 and 10

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Brenda can use 1000 bookshelves.

Given that, Brenda has 40 math books and 25 science books we need to find that what is the greatest number of bookshelves Breanda can use,

So, the greatest number of books = 40 x 25 = 1000

Hence, Brenda can use 1000 bookshelves.

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Answer:5+3x/4

Step-by-step explanation:

Answer:2x+1

1

​

Step-by-step explanation:

The roots of the quadratic equation x² + 2x - 7 are x = -1 + 2√2 and x = -1 - 2√2

To find the roots of the quadratic equation x² + 2x - 7 using the quadratic formula, we need to first identify the values of a, b, and c in the equation.

In this case, a = 1, b = 2, and c = -7.

The quadratic formula is:

x = (-b ± √(b² - 4ac)) / 2a

We can substitute the values of a, b, and c into the formula and simplify:

x = (-2 ± √(2² - 4(1)(-7))) / 2(1)

x = (-2 ± √(4 + 28)) / 2

x = (-2 ± √(32)) / 2

x = (-2 ± 4√2) / 2

We can simplify this expression further by dividing both the numerator and denominator by 2:

x = -1 ± 2√2

The roots of a quadratic equation represent the values of x that make the equation equal to zero. The quadratic formula provides a method for finding these roots for any quadratic equation, regardless of the values of a, b, and c.

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The length of the line segments m and n which are halves of the diagonals AC and BD in the parallelogram ABCD are 6 and 11 respectively.

A parallelogram is a quadrilateral, and the diagonals always bisect each other. However, diagonals only form right angles if the parallelogram is a rhombus or a square.

For the parallelogram ABCD; the lines AC and BD are its diagonals, and they both bisect each other, that is they cut each other to form two equal parts.

So AP and PC are equal halves of the line AC, while BP and PD are equal halves of the line BD

Therefore, since PC = 6 then m = 6, and for PD = 11, then n = 11 because they form diagonals of the parallelogram ABCD.

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(a) The is the solution to the integral equation is:

f(t) = (t-2)/2 + (1/2) e^(7-t) f(t-7) - (1/2) S e^(7-t) f'(t-7) dt

(b) The is the solution to the integral equation is:

f(t) = L^-1[F(s)] = (1/2) sin(t) + (1/2) cos(t-T) u(t-T)
where u(t-T) is the unit step function.

To solve integral equations, we need to use techniques such as integration by substitution or integration by parts. Let's start with the given equations:

(a) t - 2f(2)= S e---)f(t – 7)dt

To solve this integral equation, we need to integrate the function on the right-hand side with respect to t. Let u = t - 7, then du = dt. The integral becomes:

S e---)f(t – 7)dt = S e---)f(u)du

We can then apply integration by parts, using u = f(u) and dv = e^-u du, which gives us:

S e^-u f(u) du = -e^-u f(u) + S e^-u f'(u) du

Substituting back in for u, we get:

S e---)f(t – 7)dt = -e^(7-t) f(t-7) + S e^(7-t) f'(t-7) dt

Now we can substitute this into the original equation:

t - 2f(2) = -e^(7-t) f(t-7) + S e^(7-t) f'(t-7) dt

To solve for f(t), we need to isolate it on one side of the equation. Rearranging, we get:

f(t) = (t-2)/2 + (1/2) e^(7-t) f(t-7) - (1/2) S e^(7-t) f'(t-7) dt

This is the solution to the integral equation (a).

(b) f(t) = cost + Stef(t – T)dt = е

To solve this integral equation, we can take the derivative of both sides with respect to t. Using the chain rule, we get:

f'(t) = -sinf(t) + s e^(-T) f(t-T)

Now we can substitute this back into the original equation:

f(t) = cost + S e^(-T) f(t-T)dt

To solve for f(t), we need to isolate it on one side of the equation. Rearranging, we get:

f(t) - S e^(-T) f(t-T) = cost

Now we can take the Laplace transform of both sides of the equation:

L[f(t) - S e^(-T) f(t-T)] = L[cos(t)]

Using the properties of the Laplace transform, we get:

F(s) - e^(-Ts) F(s) e^(-Ts) = s/(s^2 + 1)

Simplifying, we get:

F(s) = s/(s^2 + 1) / (1 - e^(-Ts))

Now we can take the inverse Laplace transform to get the solution to the integral equation:

f(t) = L^-1[F(s)] = (1/2) sin(t) + (1/2) cos(t-T) u(t-T)

where u(t-T) is the unit step function. This is the solution to the integral equation (b).

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The probability that the randomly selected point is in the bullseye is: 1/3

We are told that there are two concentric circles.

Now, the circles that have a common centre are referred to as concentric circles and have different radii. In other words, it is defined as two or more circles that have the same centre point. The region between two concentric circles are of different radii is known as an annulus.

Now, the bulls eye diameter of 4 cm since the radius is 2 cm

Meanwhile, the outer part forms a diameter of 8 cm.

Thus:

Probability of hitting the bulls eye = 4/12 = 1/3

Probability of hitting the outer part = 8/12 = 2/3

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The formula for the surface area of a sphere is derived from the formula for its volume by taking its derivative with respect to the radius.

Deriving the formula for the surface area of a sphere depends on knowing the formula for its volume because it involves taking the derivative of the volume formula with respect to the radius.

The volume formula for a sphere is  [tex]V = (4/3)πr^3[/tex], where r is the radius, and π is a constant. If we differentiate this formula with respect to r, we get dV/dr = [tex]4πr^2[/tex], which gives us the formula for the surface area of the sphere, A = [tex]4πr^2.[/tex]

Therefore, the formula for the surface area of a sphere is derived from the formula for its volume by taking its derivative with respect to the radius.

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