What is the distance, in units, between the points (−3, 1) and (2, −1)?
\sqrt{x} 3\\

\\
\sqrt{x} 5\\

\sqrt{x} 21\\
\\
\sqrt{x} 29

The equation for the circle with a center at (-2,5) and a radius of 3 is (x + 2)² + (y - 5)² = 9.

Given that:

Center, (h, k) = (-2, 5)

Radius, r = 3

Let r be the radius of the circle and the location of the center of the circle be (h, k). Then the equation of the circle is given as,

(x - h)² + (y - k)² = r²

The equation for the circle with a center at (-2,5) and a radius of 3 is given as,

(x + 2)² + (y - 5)² = 3²

(x + 2)² + (y - 5)² = 9

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Answer:

3.16 x 10⁵ or 316,000

Step-by-step explanation:

We can start by dividing the mass of the sun by the combined mass of the moon, Earth, and Pluto:

1.898 x 10³⁰ / 5.986 x 10²⁴ ≈ 3.16 x 10⁵

This means that approximately 3.16 x 10⁵ (or 316,000) combined masses of the moon, Earth, and Pluto are needed to equal the mass of the sun.

The p-value of 0.6901, we did not find enough evidence to say a significant difference exists between the true average fuel economy today and 22.74 MPG (option 5).

Based on the given information, the null hypothesis is that the true average fuel economy today is equal to 22.74 MPG, while the alternative hypothesis is that it is not equal to 22.74 MPG. The p-value of 0.6901 indicates that there is a 69.01% chance of obtaining the observed sample mean or one more extreme, assuming the null hypothesis is true.

Since the p-value is greater than the significance level of 5%, we fail to reject the null hypothesis. This means that we do not have enough evidence to say that the true average fuel economy today is significantly different from 22.74 MPG.

Therefore, the appropriate conclusion would be option 5: We did not find enough evidence to say a significant difference exists between the true average fuel economy today and 22.74 MPG.

In this scenario, we are conducting a one-sample mean hypothesis test to determine whether the average fuel economy today is different from 22.74 MPG. The null hypothesis (u = 22.74) states that there is no significant difference, while the alternative hypothesis (u ≠ 22.74) states that there is a significant difference.

We are using a 5% level of significance to make our decision. The p-value observed in this test is 0.6901, which is greater than the significance level of 0.05. Therefore, we do not have enough evidence to reject the null hypothesis.

In conclusion, based on the given information and the p-value of 0.6901, we did not find enough evidence to say a significant difference exists between the true average fuel economy today and 22.74 MPG (option 5).

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A valid claim for the population is given as follows:

The estimated number is between 115 and 120.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

Out of 50 chocolates, 17 are milk chocolates, hence the probability that a chocolate piece is of milk is given as follows:

p = 17/50.

Out of 350 pieces, the expected amount of chocolate pieces is given as follows:

E(X) = 350 x 17/50

E(X) = 7 x 17

E(X) = 119. -> between 115 and 120.

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The calculated volume of the sphere is 50939.2 cubic foot

Given that the radius of the sphere is represented as

Radius, r = 23 ft

The formula of the volume of the sphere is represented as

V = 4/3πr³

Where

By substitution, we have

V = 4/3 * 3.14 * 23^3

Evaluate the products in the above equation

V = 50939.2 cubic foot

Hence, the calculated volume of the sphere is 50939.2 cubic foot

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Answer:

Perimeter = 48 inches

Area = 144 inches²

Step-by-step explanation:

Perimeter = 4 * side

Perimeter = 4*12 in

Perimeter = 48 inches

Area = side²

Area = (12in)²

Area = 144 in²

Your answer: A  −14 × {23 + (−47)} = [−14 × 23] + [−14 × (−47)]  This example demonstrates the distributive property of multiplication over addition for rational numbers, which states that for any rational numbers a, b, and c: a × (b + c) = (a × b) + (a × c).

The correct answer is A: −14 × {23 + (−47)} = [−14 × 23] + [−14 × (−47)]. This is an example of the distributive property of multiplication over addition for rational numbers because we are multiplying −14 by the sum of 23 and −47, and we can distribute the multiplication to each term inside the parentheses by multiplying −14 by 23 and −14 by −47 separately, and then add the two results together. This is the basic definition of the distributive property of multiplication over addition. Option B shows the distributive property of multiplication over subtraction, option C shows the product of multiplication and addition, and option D is not a valid equation.

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If in 2007, the population of the city was about 166,025, then K is approximately 0.0454.

The term "population" refers to the total number of humans, animals, or other living things in a certain area or region. When referring to human populations, it means the whole population of a certain city, state, nation, or even the entire planet. Births, deaths, immigration, and emigration are a few examples of things that might have an impact on a region's population.

We are given that the population of the city in 2007 was about 166,025. Since t = 0 corresponds to the year 2000, this means that in 2007, t = 7.

So we have P = 166.025 and t = 7, and the equation is:

P = 120.8[tex]e^{kt}[/tex]

Substituting these values, we get:

166.025 = 120.8[tex]e^{k*7}[/tex]

Dividing both sides by 120.8, we get:

166.025/120.8 =[tex]e^{k * 7}[/tex]

Taking the natural logarithm of both sides, we get:

ln(166.025/120.8) = k× 7

Simplifying the left-hand side, we get:

ln(1.372) = k× 7

Using a calculator to evaluate ln(1.372), we get:

0.3188 ≈ k × 7

Dividing both sides by 7, we get:

k ≈ 0.0454

Hence, if in 2007, the population of the city was about 166,025, then K is approximately 0.0454.

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The expression c⁸(d⁶ / c²)³ is equivalent to c⁴d¹⁸. Then the correct option is B.

Given that:

Expression, c⁸(d⁶ / c²)³

The equivalent is the expression that is in different forms but is equal to the same value.

Simplify the expression, then we have

⇒ c⁸(d⁶ / c²)³

⇒ c⁴d¹⁸

Thus, the correct option is B.

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The prompt on arcs and chords is to test your ability to recognize congruent patterns.

G) Note that ∡AB = 82°

H) UV = 6; and

I) Secant QR = 15

G) Note that ∡ED = 82° and is given to be congruent to ∡AB hence, ∡AB = 82°. This is because the degree measure of a minor arc is equal to the measure of the central angle that intercepts it. Since the central angle here is 82°, hence, the arc AB is also 82°

H) UV is a chord. So is UT. They both are congruent, that is =67°.

Since the we know that the Chord UT = 6, then Line segment UV must also be 6.

I) In this case we are given two sets of diagrams.

The first indicates that the Secant is of length 15 and creates an arc of 120°.  If that is the case, and QR is also a secant creating an arc ∡QR of 120° then Secant QR must also be 15.

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Since the sample size is less than 30, the appropriate type of confidence interval to use in this situation is the t-interval.

This is because the t-distribution takes into account the added uncertainty due to the estimation of the population standard deviation from the sample standard deviation. As the sample size increases, the t-distribution approaches the standard normal distribution, and the z-interval becomes appropriate. When estimating a population mean from a sample mean and standard deviation, there is inherent uncertainty in the estimate due to sampling variation. The t-distribution takes into account this added uncertainty by adjusting for the fact that the population standard deviation is estimated from the sample standard deviation, rather than known exactly. The t-distribution has fatter tails than the standard normal distribution, which allows for the increased uncertainty in the estimate.

As the sample size increases, the sample standard deviation becomes a more accurate estimate of the population standard deviation, and the added uncertainty due to estimation decreases. This means that the t-distribution approaches the standard normal distribution in shape, and the z-interval becomes appropriate to use. In general, if the sample size is large (say, greater than 30), it is common to use the z-interval instead of the t-interval, as the difference between the two becomes negligible.

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The Volume of the solid which is made of small-cubes is 15 yd³.

The "Volume" of a cube is the measure of the amount of space that the cube occupies in three-dimensional space. It is calculated by multiplying the cube's three side lengths.

In the figure, we can see that, the solid consists of 5×3 = 15 cubes,

The side-length of each small cube is = 1 yd,

So, the volume of each small-cube is = (side)×(side)×(side),

⇒ Volume = 1 yd³.

To find the volume of the entire solid, we multiply the volume of single-cube, by the "total-number" of cubes,

So, Volume of Solid is = 15×1 = 15 yd³.

Therefore, the volume of the solid is 15 yd³.

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Therefore, the probability that exactly 5 of the students in a study group of 10 have studied in the last week is approximately 0.2461.

To calculate the probability that exactly 5 of the students in a study group of 10 have studied in the last week, we need to use the binomial distribution formula:

[tex]P(X=k) = C(n,k) * p^k * (1-p)^{(n-k)}[/tex]

where:

P(X=k) is the probability that k students have studied in the last week

n is the total number of students in the group (n = 10)

k is the number of students who have studied in the last week (k = 5)

p is the probability that a student has studied in the last week

Since we don't have information about p, let's assume that it's 0.5, which means that each student has an equal chance of having studied in the last week or not.

Using this assumption and plugging in the values, we get:

[tex]P(X=5) = C(10,5) * 0.5^5 * 0.5^{(10-5)[/tex]

= 252 * 0.03125 * 0.03125

= 0.2461

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The value of the diameter of the cone part of the funnel to the nearest tenth is,

⇒ d = 7.2 cm

We have to given that;

A funnel can hold 159π cm³ of fluid.

And, Its height (without the stem) is 12 cm.

Now, We know that;

Volume of cone = πr²h/3

Where, r is radius of cone.

Hence we get;

⇒ 159 = 3.14 × r² × 12 / 3

⇒ 159 = 12.56 r²

⇒ r² = 12.65

⇒ r = √12.65

⇒ r = 3.556

⇒ r = 3.6 cm

Thus, The value of the diameter of the cone part of the funnel to the nearest tenth is,

⇒ d = 2r

⇒ d = 2 x 3.6

⇒ d = 7.2 cm

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The average price of milk in 2018 was 6.45 dollars per gallon.

The average price of milk in 2021 was 189.95 dollars per gallon.

The given function is:

Price = 3.55 + 2.90(1 + x)³

where x is the number of years after 2018.

In order to find the average price of milk in 2018, we need to set x = 0:

Price in 2018 = 3.55 + 2.90(1 + 0)³ = 3.55 + 2.90(1) = 6.45 dollars per gallon

Price in 2021 = 3.55 + 2.90(1 + 3)³ = 3.55 + 2.90(64) = 189.95 dollars per gallon

So, the average price of milk in 2021 was 189.95 dollars per gallon.

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The probability of the spinner first not landing on green, spun again, and then landing on green is P ( A ) = 0.2475

Given data ,

Let the probability of the spinner first not landing on green, spun again, and then landing on green is P ( A )

Now , Since the likelihood of the spinner landing on green on each spin is independent and constant, the chance that it will do so on the second spin is 0.45.

We add the probabilities together to determine the likelihood that the spinner won't land on green at first, then will spin again and land on green.

On the first spin, there is a 0.55 percent chance of not landing on green.

The second spin's probability of landing on green is 0.45.

Probability of landing on green after not landing on green is equal to 0.55 times 0.45.

P ( A ) = 0.2475

Hence , the probability that the spinner will first miss landing on green, spin again, and finally land on green is 0.2475, or 24.75%

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The area of the given polygon is calculated as: 847 square inches

The given composite figure can be broken down into a trapezium and a square.

Formula for area of a square is:

Area = side * side

Formula for area of a trapezium is:

Area = ¹/₂(sum of parallel sides) * height

Thus, total area of composite figure is:

Area = (¹/₂(22 + 11) * 22) + (22 * 22)

= 363 + 484

= 847 in²

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The absolute maximum of f(x) is e^(2pi), which occurs at x = 2pi, and the absolute minimum of f(x) is approximately -1.30, which occurs at x = 5*pi/4

a. To find the absolute maximum and minimum values of f(x), we can use the first derivative test and the endpoints of the given interval.

First, we find the first derivative of f(x):

f'(x) = e^xcos(x) - e^xsin(x)

Then, we find the critical points of f(x) by setting f'(x) = 0:

e^xcos(x) - e^xsin(x) = 0

e^x(cos(x) - sin(x)) = 0

cos(x) = sin(x)

x = pi/4 or x = 5*pi/4

Note that these critical points are in the domain [0, 2*pi].

Next, we find the second derivative of f(x):

f''(x) = -2e^xsin(x)

We can see that f''(x) is negative for x in [0, pi/2) and (3pi/2, 2pi], and f''(x) is positive for x in (pi/2, 3*pi/2).

Therefore, x = pi/4 is a relative maximum of f(x), and x = 5*pi/4 is a relative minimum of f(x). To find the absolute maximum and minimum of f(x), we compare the values of f(x) at the critical points and the endpoints of the domain:

f(0) = e^0cos(0) = 1

f(2pi) = e^(2pi)cos(2pi) = e^(2pi)

f(pi/4) = e^(pi/4)cos(pi/4) ≈ 1.30

f(5pi/4) = e^(5*pi/4)cos(5pi/4) ≈ -1.30

Therefore, the absolute maximum of f(x) is e^(2pi), which occurs at x = 2pi, and the absolute minimum of f(x) is approximately -1.30, which occurs at x = 5*pi/4.

b. To find the intervals on which f(x) is increasing, we look at the sign of f'(x) on the domain [0, 2pi]. We know that f'(x) = 0 at x = pi/4 and x = 5pi/4, so we can use a sign chart for f'(x) to determine the intervals of increase:

x 0 pi/4 5*pi/4 2*pi

f'(x) -e^0 0 0 e^(2*pi)

f(x) increasing relative max relative min decreasing

Therefore, f(x) is increasing on the interval [0, pi/4) and decreasing on the interval (pi/4, 2*pi].

c. To find the x-coordinate of each point of inflection of the graph of f, we need to find where the concavity of f changes. We know that the second derivative of f(x) is f''(x) = -2e^xsin(x), which changes sign at x = pi/2 and x = 3*pi/2.

Therefore, the point (pi/2, f(pi/2)) and the point (3pi/2, f(3pi/2)) are the points of inflection of the graph of f.

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Assume while in college you are not required to pay off your loan yet, and the interest will NOT compound, the amount you owe right after graduation without interest is $48,000.

The loan amount (without interest) at the end of the college years is a product of the multiplication of the number of semesters and the loan per semester without compounded interest.

Multiplication is one of the four basic mathematical operations, including addition, subtraction, and division.

Multiplication operation involves the multiplicand, the multiplier, and the product.

The typical loan per semester of an average student = $6,000

The number of college years = 4 years

The total number of semesters for a 4-year college degree = 8 (4 x 2)

The total amount of student loan incurred = $48,000 ($6,000 x 8)

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Answer:

Olivia has 7.75 meters of string left

Step-by-step explanation:

15.1-7.35=7.75

We are given that f(x) = √x, and we want to find the equation of the tangent line to this function at x = 81. We can use the formula for the equation of the tangent line:

y - f(a) = f'(a)(x - a),

where f'(a) is the derivative of f(x) evaluated at x = a.

First, we calculate the derivative of f(x) as:

f'(x) = 1/(2√x)

Evaluated at x = 81, we get:

f'(81) = 1/(2√81) = 1/18

So the equation of the tangent line to f(x) at x = 81 is:

y - √81 = (1/18)(x - 81)

Simplifying:

y - 9 = (1/18)x - (1/2)

y = (1/18)x + 8.5

Now we can use this equation to approximate √81.1:

√81.1 ≈ (1/18)(81.1) + 8.5

≈ 9.0139

Therefore, using linear approximation with the tangent line, we can approximate √81.1 as 9.0139.

The steady-state pmf of the Markov chain: π₀ ≈ 0.48, π₁ ≈ 0.32, π₂ ≈ 0.20

To solve for the steady-state pmf of this Markov chain, we need to find the probabilities of being in each state in the long run, assuming that the chain has stabilized. We can do this by solving the system of equations:

π = πP

where π is the row vector of state probabilities and P is the transition matrix of the Markov chain. In this case, the transition matrix is:

P =
0.6 0.7 0.4
0.1 0.1 0.8
0   0.2 0.3

and the initial state probabilities are:

π = (1 0 0)

Substituting these values into the equation, we get:

π = πP
(1 0 0) = (1 0 0)P
1 = 0.6π1 + 0.1π2
0 = 0.7π1 + 0.1π2 + 0.2π3
0 = 0.4π1 + 0.8π2 + 0.3π3

Simplifying these equations, we get:

π1 = 0.4π2
π3 = 2π2

Substituting these values back into the second equation, we get:

0 = 0.7π1 + 0.1π2 + 0.4π2
0 = 0.7π1 + 0.5π2
π2 = 1.4π1

Substituting these values into the first equation, we get:

1 = 0.6π1 + 0.1(1.4π1)
1 = 0.76π1
π1 = 1/0.76 ≈ 1.3158

Substituting this value back into the other equations, we get:

π2 ≈ 1.7368
π3 ≈ 3.4737

Therefore, the steady-state pmf of this Markov chain is:

π ≈ (0.4103 0.5789 0.0108)

This means that in the long run, the probability of the computer disk drive being in state 0 (idle) is about 0.41, the probability of being in state 1 (read) is about 0.58, and the probability of being in state 2 (write) is very low at about 0.01.

The steady-state pmf of the Markov chain for the computer disk drive in states 0 (idle), 1 (read), and 2 (write) can be found by solving a system of linear equations. Given the Markov chain transition probabilities:

0.6 0.7 0.4
0.1 0.1 0.6
0.3 0.2 0.0

Let π = [π₀, π₁, π₂] be the steady-state probabilities.

We have the following system of linear equations:

π₀ = 0.6π₀ + 0.1π₁ + 0.3π₂
π₁ = 0.7π₀ + 0.1π₁ + 0.2π₂
π₂ = 0.4π₀ + 0.6π₁ + 0.0π₂
π₀ + π₁ + π₂ = 1

Solving the system, we find the steady-state pmf of the Markov chain:

π₀ ≈ 0.48
π₁ ≈ 0.32
π₂ ≈ 0.20

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Answer:

2.9

Step-by-step explanation:

cos 70° = 0.342020

1/cos 70° = 1/0.342020 = 2.9238

Answer: 2.9

In a right triangle where tan(θ) = -1, the other trigonometric functions of θ can be evaluated as sin(θ) = -1/√2, cos(θ) = 1/√2, cosec(θ) = -√2, sec(θ) = √2, and cot(θ) = -1.

If tan(θ) = -1, we can determine that the opposite side of the angle is equal to the adjacent side, or in other words, they are both equal to a negative square root of 2 (-√2). Using the Pythagorean theorem, we can find the hypotenuse of the right triangle:

a² + b² = c²

(-√2)² + (-√2)² = c²

2 + 2 = c²

4 = c²

c = 2

Now, we can use the definitions of sine, cosine, tangent, cotangent, secant, and cosecant to determine their values:

sin(θ) = opposite/hypotenuse = (-√2)/2

cos(θ) = adjacent/hypotenuse = (-√2)/2

tan(θ) = opposite/adjacent = -1

cot(θ) = adjacent/opposite = -1

sec(θ) = hypotenuse/adjacent = -√2

csc(θ) = hypotenuse/opposite = -√2

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Answer:

missing angle measure = 100°

Step-by-step explanation:

the sum of the interior angles of a polygon is

sum = 180° (n - 2 ) ← n is the number of sides

a pentagon has n = 5 , then

sum = 180° × (5 - 2) = 180° × 3 = 540°

given sum of 4 angles = 440° , then

missing angle = 540° - 440° = 100°

Answer:

D

Step-by-step explanation:

Dustin's mean number of situps is 42.5714285714, while Jacob's is 41.4285714286.

You can get the mean of Dustin's data by adding all of the data of Dustin's situps together and dividing by 7, and the same thing with Jacob's.

Null Hypothesis is the claim that no difference or relationship exists between two sets of data or variables being analyzed.

Type I error: A. Reject the null hypothesis that the percentage of high school students who graduate is equal to 55% when that percentage is actually equal to 55%.

Type II error: D. Fail to reject the null hypothesis that the percentage of high school students who graduate is equal to 55% when the percentage is actually different from 55%.

In statistics, the null hypothesis is a statement that there is no significant difference between two populations or sets of data. It is a hypothesis that is assumed to be true until proven otherwise. The null hypothesis is often denoted as H0 and is used to test the validity of an alternative hypothesis (Ha). The goal of hypothesis testing is to determine whether the evidence supports rejecting the null hypothesis in favor of the alternative hypothesis.

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The proper equations in Pattern A with the appropriate values on the right side are:

3.675 • 10 = 36.750

3.675 • 100 = 367.500

3.675 • 1,000 = 3,675.000

The correct equations in Pattern B with the precise values on the right side are:

3.675 • 0.1 = 0.3675

3.675 • 0.01 = 0.03675

3.675 • 0.001 = 0.003675

Pattern A:

The equations in Pattern A are incorrect due to not possessing the adequate number of decimal places in the products. In each equation, the end result should have one additional decimal place as compared to the initial number being multiplied.

Pattern B:

The equations in Pattern B are flawed because they inaccurately reflect the number of decimal places in the outcomes. In each equation, the results should display one fewer decimal place when compared to the original number being multiplied.

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For the given geometric sequence we have:

a) R = 3/4

b) 243/256

c) Aₙ = 3*(3/4)⁽ⁿ⁻¹⁾

The common reason is given by the quotient between two consecutive terms, so using the first two, we will get:

R = (9/4)/3

R = 3/4

That is the common reason.

b) To find the fifth therm, we need to take the fourth one and multiply it by the common reason, so we will get:

81/64*(3/4) = 243/256

c) The n-th term of the sequence is the first term of the sequence multiplied by the common reason (n - 1) times, it gives:

Aₙ = 3*(3/4)⁽ⁿ⁻¹⁾

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The required sample size is 907.

We have,

To determine the required sample size for estimating the fraction of new car buyers who prefer foreign cars over domestic with a 95% confidence level and an error of at most 0.03, we'll use the following formula:
n = (Z² x p  (1 - p)) / E²
Where:
- n is the required sample size
- Z is the Z-score for the desired confidence level (1.96 for a 95% confidence level)
- p is the estimated population proportion (0.31)
- E is the margin of error (0.03)

Step-by-step calculation:
1. Calculate Z²: 1.96² = 3.8416
2. Calculate p (1 - p): 0.31 x (1 - 0.31) = 0.2139
3. Calculate E²: 0.03² = 0.0009
4. Substitute these values into the formula: n = (3.8416 x 0.2139) / 0.0009 = 906.92

Since we need to round up to the next integer, the required sample size is 907.

Thus,

The required sample size is 907.

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