We can conclude that for all integers a and b, if a|b, then a ≤ b.
To prove that for all m and n, if m ≠ n, then m+n ≠ Q, we will use proof by contradiction.
Assume that for some m and n, m ≠ n, and m+n = Q, where Q is a rational number. By the definition of a rational number, Q can be expressed as the ratio of two integers, p and q, where q ≠ 0.
Thus, we have:
m + n = p/q
Multiplying both sides by q, we get:
mq + nq = p
Rearranging, we get:
mq = p - nq
Since p, n, and q are integers, p - nq is also an integer. Therefore, mq is an integer.
But we know that m and n are integers and m ≠ n, which implies that m and n have different prime factorizations. Therefore, mq cannot be an integer, as it would require m and q to have a common factor, which is not possible.
This contradicts our assumption that m+n = Q, and hence, we can conclude that for all m and n, if m ≠ n, then m+n ≠ Q.
To prove that for all integers a and b, if a|b, then a ≤ b, we will use direct proof.
Assume that a and b are integers such that a|b, i.e., there exists an integer k such that b = ak.
To prove that a ≤ b, we need to show that a is less than or equal to k times a, i.e., a ≤ ka.
Dividing both sides of the equation b = ak by a (which is possible as a ≠ 0 since it is a divisor of b), we get:
b/a = k
Since k is an integer, we know that b/a is also an integer. Therefore, a must be less than or equal to b/a.
Multiplying both sides of the inequality a ≤ b/a by a (which is a positive number since a > 0), we get:
[tex]a^2 ≤ ab[/tex]
Since a and b are both positive integers, we know that [tex]a^2 ≤[/tex] ab implies that [tex]a ≤ b[/tex].
Therefore, we can conclude that for all integers a and b, if a|b, then a ≤ b.
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Three tennis balls are stored in a cylindrical container with a height of 8.2 inches and a radius of 1.32 inches. The circumference of a tennis ball is 8 inches. Find the amount of space within the cylinder not taken up by the tennis balls. Round your answer to the nearest hundredth.
The amount of space within the cylinder not taken up by the tennis balls is 18.9 [tex]inches^3[/tex]
The volume of a tennis ball:
The circumference of the tennis ball is 8 inches.
The tennis ball is the form of sphere whose circumference is given by formula [tex]2\pi r[/tex], where r is the radius.
Thus, if r is the radius then according to condition,
[tex]2\pi r[/tex] = 8 or
r = 8/2[tex]\pi[/tex] inches.
Now, the volume of the sphere of radius r is [tex]\frac{4}{3}\pi r^3[/tex] hence, find the volume of the given tennis ball by substituting r = 8/2[tex]\pi[/tex] inches in [tex]\frac{4}{3}\pi r^3[/tex] and simplify:
Volume = [tex]\frac{4}{3}\pi[/tex] × [tex](\frac{8}{2\pi } )^3[/tex]
Volume = 8.65[tex]inches^3[/tex]
Hence the required volume of the tennis ball is 8.65[tex]inches^3[/tex]
The volume of three tennis balls is (3 × 8.65) [tex]inches^3[/tex] = 25.96 [tex]inches^3[/tex]
Find the volume of the cylinder:
The volume of the cylinder with radius r units and height h units is given by [tex]\pi r^2h[/tex] Hence the volume of the given cylinder with radius 1.32 inches , 8.2 height inches is:
[tex]\pi (1.32)^2[/tex] × 8.2
= 3.14 × [tex](1.32)^2[/tex] × 8.2
= 44.86 [tex]inches^3[/tex]
Hence the volume of the cylinder is 44.86 [tex]inches^3[/tex]
Find the amount of space within the cylinder not taken up by the tennis balls.
The required volume can be obtained by subtracting the volume three tennis balls from the volume of the cylinder as follows:
Volume of cylinder - volume of three tennis balls = (44.86 - 25.96) = 18.9 [tex]inches^3[/tex]
Hence, the amount of space within the cylinder not taken up by the tennis balls is 18.9 [tex]inches^3[/tex]
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About 75% of MCC students believe they can achieve the American dream and about 65% of Ferris State Universtiy students believe they can achieve the American dream. Construct a 99% confidence interval for the difference in the proportions of Montcalm Community College students and Ferris State University students who believe they can achieve the American dream. There were 100 MCC students surveyed and 100 FSU students surveyed. a. With 99% confidence the difference in the proportions of MCC and FSU students who believe they can achieve the American dream is (round to 3 decimal places) and (round to 3 decimal places). b. If many groups of 100 randomly selected MCC students and 100 randomly selected FSU students were surveyed, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population proportion of the difference in the proportions of MCC students and FSU students who believe they can achieve the American dream about percent will not contain the true population difference in proportions.
a. With 99% confidence, the difference in proportions is between -0.023 and 0.223. b. 99% of the confidence intervals will contain the true population proportion, and about 1% will not contain the true population difference in proportions.
a. To construct a confidence interval for the difference in proportions, we can use the following formula:
CI = (p1 - p2) ± zα/2 * √((p1*(1-p1)/n1) + (p2*(1-p2)/n2))
where:
p1, p2 = proportion of MCC, FSU students who believe they can achieve the American dream
n1, n2 = sample size of MCC, FSU students
zα/2 = critical value from the standard normal distribution for a 99% confidence level, which is 2.576
So,
CI = (0.75 - 0.65) ± 2.576 * √((0.75*(1-0.75)/100) + (0.65*(1-0.65)/100))
CI = 0.10 ± 0.123
CI = (−0.023, 0.223)
Therefore, with 99% confidence, the difference in proportions of MCC and FSU students who believe they can achieve the American dream is between -0.023 and 0.223.
b. Approximately 99% of these confidence intervals will contain the true population proportion of the difference in proportions of MCC students, and FSU students who believe they can achieve the American dream.
And about 1% will not contain the true population difference in proportions. This is because we constructed a 99% confidence interval.
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Due to a power outage, the sales clerk manually prepares a sale receipt to her customer. Which one of the following diagrams represents this activity?
a. trapezoid to curvy side rectangle to curved rectangle
b. curved side rectangle to circle to curved rectangle
c. circle to curved side rectangle to curved rectangle
d. trapezoid to curved side rectangle to circle
Based on the given options, the diagram that best represents the activity of a sales clerk manually preparing a sale receipt during a power outage would be option D.
A trapezoid could represent the shape of a receipt, a curved side rectangle could represent the shape of the clerk's desk or the paper she is using, and a circle could represent the shape of a calculator or cash register. Therefore, a trapezoid to a curved side rectangle to a circle could represent the process of the clerk manually calculating and recording the sale amount and inputting it into a calculator or cash register to produce a receipt.
It is important to note that during a power outage, technology-dependent activities such as electronic sales and transactions may be disrupted, and manual methods may have to be used as a backup. This highlights the power of technology in our daily lives and the impact that power outages can have on businesses and individuals.
The question is about selecting the correct diagram that represents the sales clerk manually preparing a sale receipt due to a power outage. Given the options:
a. trapezoid to curvy side rectangle to curved rectangle
b. curved side rectangle to circle to curved rectangle
c. circle to curved side rectangle to curved rectangle
d. trapezoid to curved side rectangle to circle
The appropriate answer for this question cannot be determined based on the provided information. Diagrams typically require visual representation, and the description of the shapes alone is insufficient to convey the activity of preparing a receipt manually. Moreover, the terms "power," "outage," "clerk," "receipt," "trapezoid," "curved," and "curved" don't necessarily correspond to the shapes given in the options.
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Find the area of a triangle with a base length of 4 units and a height of 5 units.
Answer:
Step-by-step explanation:
1/2*b*h
1/2*4*5
4/2*5
2*5
10
What is the
midpoint of AD?
y↑
10
9
8
7
6
5
4
3
2
1
1
A
O (3,6)
O (7,6)
O (6,9)
(4,3)
coordinate for the
2 3
D
4
B
C
5 6 7 8
9
10
+x
X
Answer:
6 is the middle point of AD
(a) Determine the mean and standard deviation of the sampling distribution of X. The mean is Hy = 176.5. (Type an integer or a decimal. Do not round.) The standard deviation is on = 1.28 . (Type an integer or a decimal. Do not round.) (b) Determine the expected number of sample means that fall between 174.2 and 177.2 centimeters inclusive. sample means (Round to the nearest whole number as needed.)
The expected number of sample means falling between 174.2 and 177.2 can be estimated as:
Expected number = A * Total number of sample means.
(a) The mean of the sampling distribution of X is given as 176.5 and the standard deviation is 1.28.
(b) To determine the expected number of sample means that fall between 174.2 and 177.2 centimeters inclusive, we need to calculate the z-scores corresponding to these values and find the area under the normal curve between these z-scores.
The z-score for 174.2 can be calculated as:
z1 = (174.2 - 176.5) / 1.28
Similarly, the z-score for 177.2 can be calculated as:
z2 = (177.2 - 176.5) / 1.28
Using a standard normal distribution table or a calculator, we can find the area between these two z-scores.
Let's assume the area between z1 and z2 is A. The expected number of sample means falling between 174.2 and 177.2 can be estimated as:
Expected number = A * Total number of sample means.
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how do you do this i dont understand ir
Answer: The answer is 2 and 3 or B and C which ever way you want it.
The reason its 2and3 is because you can see its 60 degree angle.
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The reason its 3 also is because they are all congruent and its the only other right answer that fits.
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Hence, The answer is 2 and 3 or B and C.
Step-by-step explanation: Please give Brainliest.
Hope this helps!!!!
I can answer more questions if you want.
what are the exact values of the cosecant, secant, and cotangent ratios of -pi/4 radians?
The exact values of cosecant, secant, and cotangent ratios of -pi/4 radians is:
[tex]csc=\frac{\pi }{4}=\frac{hypontenuse}{opposite}=\frac{\sqrt{2} }{1}=\sqrt{2}[/tex]
[tex]sec=\frac{\pi }{4}=\frac{hypotenuse}{adjacent}= \frac{\sqrt{2} }{1}=\sqrt{2}[/tex]
[tex]cot=\frac{\pi }{4}=\frac{adjacent}{opposite}=\frac{1}{1}=1[/tex]
[tex]\frac{\pi }{4}[/tex] radians is the same as 90 degrees. So, first draw a right triangle with an angle of [tex]\frac{\pi }{4}[/tex]:
This creates a 45-45-90 triangle, also known as a right isosceles triangle. This is a very special triangle, and we know that both of its legs will be the same length, and the hypotenuse will be the length of one of the legs times √2.
The three functions are just the inverses of the first three. Cosecant is the inverse of sine, secant is the inverse of cosine, and cotangent is the inverse of tangent.
[tex]csc=\frac{\pi }{4}=\frac{hypontenuse}{opposite}=\frac{\sqrt{2} }{1}=\sqrt{2}[/tex]
[tex]sec=\frac{\pi }{4}=\frac{hypotenuse}{adjacent}= \frac{\sqrt{2} }{1}=\sqrt{2}[/tex]
[tex]cot=\frac{\pi }{4}=\frac{adjacent}{opposite}=\frac{1}{1}=1[/tex]
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The perimeter of a rectangle is 52 cm. If its width is 2 cm more than one-third of its length, find the dimensions of rectangle.
The area of a circle is 4π in². What is the circumference, in inches? Express your answer in terms of pi
The circumference of the area of a circle is 4π in² using the formula A = πr², in inches is 4π inches.
The formula for the area of a circle is A = πr², where A is the area and r is the radius. Given that the area is 4π in², we can solve for the radius by taking the square root of both sides:
√(A/π) = √(4π/π) = 2 in
The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. Substituting the value of r, we get:
C = 2π(2 in) = 4π in
Therefore, the circumference of the circle is 4π inches.
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Answer!!!!! Tysm!!!!
The angle measurement of the triangle would be 9. 59 degrees
How to determine the valueThe different trigonometric identities are given as;
sinetangentcotangentcosinesecantcosecantThe ratios of the trigonometric identities are represented as;
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
From the information given, we have;
Opposite side = 1
Hypotenuse side = 6
Using the sine identity, we have;
sin θ = 1/6
Divide the values
sin θ = 0. 1666
Find the inverse of the sine
θ = 9. 59 degrees
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Determine whether each statement is True or False. Select the correct cell in each row. Statement True False T h e s u m o f − 9 a n d 18 2 i s e q u a l t o 0. The sum of −9 and 2 18 is equal to 0. T h e s u m o f − 14 2 a n d 7 i s g r e a t e r t h a n 0. The sum of − 2 14 and 7 is greater than 0. T h e s u m o f 6 , − 4 , a n d − 2 i s e q u a l t o 0. The sum of 6, −4, and −2 is equal to 0. T h e s u m o f 7 , − 9 , a n d 2 i s l e s s t h a n 0. The sum of 7, −9, and 2 is less than 0.
Each of the statements should be marked correctly as follows;
The sum of −9 and 18/2 is equal to 0: True.
The sum of −14/2 and 7 is greater than 0: False.
The sum of 6, −4, and −2 is equal to 0: True.
The sum of 7, −9, and 2 is less than 0: False.
What is an inequality?In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;
Greater than (>).Less than (<).Greater than or equal to (≥).Less than or equal to (≤).Next, we would evaluate each of the statements as follows;
-9 + 18/2 = -9 + 9 = 0
Therefore, the sum of −9 and 18/2 is truly equal to 0.
-14/2 + 7 = -7 + 7 = 0.
Therefore, the sum of −14/2 and 7 is not greater than 0.
6 - 4 - 2 = 0
Therefore, the sum of 6, −4, and −2 is truly equal to 0.
7 - 9 + 2 = 0
Therefore, the sum of 7, −9, and 2 is not less than 0.
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find the range of f(x) = 2x -3 when the domain is { -2 , 0, 1/2 , 5}
The range of f(x) = 2x -3 include the following: {-7, -3, -2, 7}.
What is a domain?In Mathematics and Geometry, a domain is the set of all real numbers for which a particular function is defined.
When the domain is -2, the range of this function can be calculated as follows;
f(x) = 2x - 3
f(-2) = 2(-2) - 3
f(-2) = -7.
When the domain is 0, the range of this function can be calculated as follows;
f(x) = 2x - 3
f(0) = 2(0) - 3
f(0) = -3.
When the domain is 1/2, the range of this function can be calculated as follows;
f(x) = 2x - 3
f(1/2) = 2(1/2) - 3
f(1/2) = -2.
When the domain is 5, the range of this function can be calculated as follows;
f(x) = 2x - 3
f(5) = 2(5) - 3
f(5) = 7.
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f(x)= - 3(x - m)2 + pParabola vertical point T(2,5), how much m + p equal
If f(x)= - 3(x - m)2 + p Parabola vertical point T(2,5), then m + p is equal to 27.
Since the given parabola is vertical and has a vertex at T(2,5), we know that the equation is of the form f(x) = a(x-2)^2 + 5, where a is a constant.
We also know that f(x) = -3(x-m)^2 + p, which is in the same form as the first equation.
So, we can equate the two equations and get:
a(x-2)^2 + 5 = -3(x-m)^2 + p
Expanding the squares, we get:
a(x^2 - 4x + 4) + 5 = -3(x^2 - 2mx + m^2) + p
Simplifying and collecting like terms, we get:
ax^2 + (-4a + 6m)x + (4a - 3m^2 + p - 5) = 0
Since this equation must hold for all values of x, the coefficients of x^2 and x must be equal to zero.
Therefore, we have:
a = -3 (from the given equation f(x) = -3(x-m)^2 + p)
-4a + 6m = 0 (from the equation above)
-4(-3) + 6m = 0
12 + 6m = 0
m = -2
Substituting m = -2 and a = -3 into the equation above, we get:
4a - 3m^2 + p - 5 = 0
4(-3) - 3(-2)^2 + p - 5 = 0
-12 - 12 + p - 5 = 0
p = 29
Therefore, m + p = -2 + 29 = 27.
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what is the quartile of 84,75,90,87,99,91,85,88,76,92,94
Answer: 84
Step-by-step explanation:
Sorry if this is wrong.
A slot machine consists of 4 reels, and each real consists of 16 stops. To win the jackpot of $10,000, a player must get a wild symbol on the center line of each reel. If each reel has two wild symbols, find the probability of winning the jackpot. A. 1/4,096 B. 1/8,192 C. 1/8 D. 4/65,536
Answer: The probability of winning the jackpot is 1/4096, which corresponds to option A
Step-by-step explanation:
To find the probability of winning the jackpot, we need to find the probability of getting a wild symbol on the center line of each of the 4 reels.
Since there are 16 stops on each reel and 2 wild symbols on each reel, the probability of getting a wild symbol on the center line of a single reel is 2/16 or 1/8.
The probability of getting a wild symbol on the center line of all 4 reels is the product of the probabilities for each reel. Thus:
(1/8) * (1/8) * (1/8) * (1/8) = 1/4096
Find the value of c using the given chord and secant lengths in the diagram shown to the right.
The value of each variable in the circle is:
b = 90°
c = 47°
a = 43°
How to find the value of each variable in the circle?A circle is a round-shaped figure that has no corners or edges. It can be defined as a closed shape, two-dimensional shape, curved shape.
An angle inscribed in a semicircle is a right angle. Thus,
b = 90°
c = 360 - 133 - 180 (sum of angle in a circle)
c = 47°
a = 180 - 90 - 47 (sum of angle in a triangle)
a = 43°
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We have to make choices every day. Some choices may affect our lives for years, like the colleges we attend.
Other decisions have short-term effects, like where we should eat lunch.
Read the options below. Which option would you choose?
A. Option 1: Receive $1,000,000 today.
B. Option 2: Receive $25,000 every day for a month (30 days).
C. Option 3: Start with 1 penny, then double it every day for a month (30 days).
Answer:
C
Step-by-step explanation:
The reason I would choose C is that the penny doubling each day might seem small but the amount would continue to grow exponentially giving you a might higher payoff than the rest of the options. I don't know the exact amount you would get by it is around 3 mill.
Option B gives you linear growth, which means that by the end of 30 days, you would only have 750,000.
Option A is the worst potion only leaving you with 1 million.
Write an expression that represents the quotient of 24 and 3 plus x
(24 ÷ 3) + x expresses the expression that represents the quotient of 24 and 3 plus x.
The expression refers to a mathematical phrase with two or more numbers or variables with mathematical operations such as addition, subtraction, division, multiplication, exponential, and so on. Examples of expression include 2a + 3p, and 9p.
To convert the given phrase into the expression, we have to start with the first operation which is a division that is represented by the word quotient. Thus we get 24 ÷ 3
Then we add the operation of addition which is represented by the word plus to the existing equation and we get (24 ÷ 3) + x
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Mr. Barth is painting an arrow on the school parking lot. He draws the edges between the following points on the coordinate plane: (–2, 2), (5, 2), (5, 6), (12, 0), (5, –6), (5, –2), (–2, –2).
please quickly, it's for today
The area of the arrow of the painting is A = 70 units²
Given data ,
Mr. Barth is painting an arrow on the school parking lot.
The coordinates are (-2, 2), (5, 2), (5, 6), (12, 0), (5, -6), (5, -2), (-2, -2)
The area of the arrow would be:
Area of Arrow = Area of Triangle + Area of Rectangle
Let the base of the triangle be = 12 units
Let the height of the triangle is = 7 units
So , area of triangle = 42 units²
Area of rectangle = 7 x 4 = 28 units
Hence , the area of arrow A = 70 units²
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The shape of a logo is made up of a triangle, a rectangle, and a parallelogram.
What is its area in square centimeters? Round your answer to the nearest tenth (1 decimal place).
The value of area of logo is,
A = 139.5 cm²
We have to given that;
The shape of a logo is made up of a triangle, a rectangle, and a parallelogram.
Here, Base of triangle = 9 cm
Height of triangle = 7 cm
Length of rectangle = 9 cm
Width of rectangle = 9 cm
Hence, We get;
The value of area of logo is,
A = (1/2 × 9 × 7) + (3 × 9) + (9 × 9)
A = 31.5 + 27 + 81
A = 139.5 cm²
Thus, The value of area of logo is,
A = 139.5 cm²
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An artist is painting a mural at the Hattiesburg Zoo. She draws
a diagram of the mural before painting,
2 ft.
10 ft.
5 ft.1
7
14 ft
8 ft.
104 square feet is the total area of the mural.
From the figure, we can say that the total area of the mural can be found by using basic algebra,
So,
The total area of the mural = area of the rectangle +area of the triangle- an area of a smaller triangle.
So, First for a bigger triangle,
the height of the triangle = 5 +2 = 7 ft.
the base of the triangle = 14 ft.
the area = 1/2*base*height
Thus the area = 49 square feet.
For the rectangle,
Height of the rectangle = 8 ft.
Width of the rectangle = 10 ft.
Thus the area of the rectangle = width * height
The area of the rectangle = 80 square feet
For the smaller triangle,
the height of the triangle = 5 ft.
the base of the triangle = 10 ft.
the area = 1/2*base*height
Thus the area = 25 square feet.
Hence,
Total area of the mural = 80 + 49 -25
Therefore, The total area of the mural is 104 square feet
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According to a national survey of asthma: On May 1, 2010, the number of residents of Oklahoma who had been diagnosed with asthma at any time during their life was 230,147. The population on June 30, 2010, was 3,325,128. During the same year, the number of new cases of asthma was 15,124. The incidence rate of asthma (per 100,000) is: O 6921 O 6571 O 454 O None of the above
Answer:
we find that the incidence rate of asthma in Oklahoma during that year was 454 cases per 100,000 population.
Step-by-step explanation:
The incidence rate of asthma is a measure of the number of new cases of asthma in a given population over a specific period. It is usually expressed per 100,000 population to allow for easier comparison between populations of different sizes. In this case, we are given the number of new cases of asthma and the total population of Oklahoma during the same year.
To calculate the incidence rate, we divide the number of new cases of asthma by the total population, and then multiply by 100,000. Applying this formula, we find that the incidence rate of asthma in Oklahoma during that year was 454 cases per 100,000 population.
This means that for every 100,000 residents of Oklahoma, there were 454 new cases of asthma during that year.
This information can be useful for public health officials and policymakers in identifying areas where more resources may be needed to prevent and manage asthma. It can also help in the evaluation of the effectiveness of interventions aimed at reducing the incidence of asthma.
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The probability distribution of a 3-coin toss is shown in the table. Find the expected number of heads.
The expected number of heads = 1.5
The correct answer is an option (B)
We know that the formula for the expected value is:
E (x) = ∑ x P ( x )
where P(x) represents the probability of outcome X
and E(x) is the expected value of x
We need to find the expected number of heads.
From the probability distribution table of a 3-coin toss, the expected number of heads would be,
E(H) = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8)
E(H) = 0 + 3/8 + 6/8 + 3/8
E(H) = 12/8
E(H) = 1.5
The correct answer is an option (B)
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Answer:
Step-by-step explanation:
Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x^2+y^2=144 and x^2-12x+y^2=0
The area between the two circles in the first quadrant is 18π square units.
To evaluate this integral, we first need to find the polar equations of the two circles.
For the circle [tex]x^2 + y^2 = 144,[/tex] we can convert to polar coordinates using the substitutions x = r cos θ and y = r sin θ, which gives:
[tex]r^2 = x^2 + y^2 = 144[/tex]
r = 12 (since r must be non-negative in polar coordinates)
For the circle [tex]x^2 - 12x + y^2 = 0[/tex], we can complete the square to get:
[tex]x^2 - 12x + 36 + y^2 = 36[/tex]
[tex](x - 6)^2 + y^2 = 6^2[/tex]
Again using the substitutions x = r cos θ and y = r sin θ, we get:
(r cos θ - 6[tex])^2[/tex] + (r sin θ[tex])^2[/tex] = [tex]6^2[/tex]
[tex]r^2 cos^2[/tex] θ - 12r cos θ + 36 +[tex]r^2 sin^2[/tex] θ = 36
r^2 - 12r cos θ = 0
r = 12 cos θ
Now we can set up the integral to find the area between the two circles in the first quadrant. Since we are in the first quadrant, θ ranges from 0 to π/2. We can integrate over r from 0 to 12 cos θ (the radius of the inner circle at the given θ), since the area between the two circles is bounded by these two radii.
Thus, the integral to evaluate is:
∫[θ=0 to π/2] ∫[r=0 to 12 cos θ] r dr dθ
Integrating with respect to r gives:
∫[θ=0 to π/2] [(1/2) r^2] from r = 0 to r = 12 cos θ dθ
= ∫[θ=0 to π/2] (1/2) (12 cos θ)^2 dθ
= ∫[θ=0 to π/2] 72 cos^2 θ dθ
Using the trigonometric identity cos^2 θ = (1 + cos 2θ)/2, we can simplify this to:
∫[θ=0 to π/2] 36 + 36 cos 2θ dθ
= [36θ + (18 sin 2θ)] from θ = 0 to θ = π/2
= 36(π/2) + 18(sin π - sin 0)
= 18π
Therefore, the area between the two circles in the first quadrant is 18π square units.
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f(x)= [tex]f(x)=\frac{x^{2} +7}{x^{2} +4x-21}[/tex]
The value of the function f(x) = ( x² + 7 ) / ( x² + 4x - 21 ) for x = 5 is equal to 4/3.
The function is equal to,
f(x) = ( x² + 7 ) / ( x² + 4x - 21 )
find the value of f(x) at x=5 by substituting x=5 into the given function we have,
⇒ f(5) = ( 5² + 7 ) / ( 5² + 4(5) - 21 )
⇒ f(5) = ( 25 + 7 ) / ( 25 + 20 - 21 )
⇒ f(5) = 32 / 24
Now reduce the fraction by taking out the common factor of the numerator and the denominator we get,
⇒ f(5) = ( 8 × 4 ) / ( 8 × 3 )
⇒ f(5) = 4/3
Therefore, the value of the function f(x) at x=5 is 4/3.
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The given question is incomplete, I answer the question in general according to my knowledge:
Find the value of the function f(x) at x = 5.
f(x) = ( x² + 7 ) / ( x² + 4x - 21 )
the domain of function f is (-oo, oo). the value of the function what function could be f
The domain of rational function f(x) = (x² - 36) / (x - 6) is x ∈ (- ∞, + ∞).
How to find a function associated with a given domain
In this question we must determine what function has a domain, that is, the set of all x-values, that comprises all real numbers. According to algebra, polynomic functions have a domain that comprises all real numbers.
Herein we need to determine what rational function is equivalent to a polynomic function. Polynomic functions are expression of the form:
[tex]y = \sum\limits_{i = 0}^{n} c_{i}\cdot x^{i}[/tex]
Where:
[tex]c_{i}[/tex] - i-th Coefficient of the polynomial.[tex]x^{i}[/tex] - Power of the i-th term of the polynomial.y - Dependent variable.Now we check the following expression by algebra properties:
f(x) = (x² - 36) / (x - 6)
f(x) = [(x - 6) · (x + 6)] / (x - 6)
f(x) = x + 6
The rational function f(x) = (x² - 36) / (x - 6) is equivalent to polynomial of grade 1 and, thus, its domain comprises all real numbers.
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Do not answer with another chegg expert solution, i will dislike the answer, It is NOT (C)Question 1
Please see the Page 27 in the PowerPoint slides of Chapter 8. If the first boundary condition
becomes Y'(0)=1, what is the correct SOR formula for this boundary condition?
OY'1 = 1
OY₁ =1/6∆ (4Y₂ - Y3)
O Y₁ = y0+y2/2-0.05∆z(Y₂-Yo)
O Y₁ = 1
O Y₁ = (4Y₂ - Y₁ - 2∆x)
OY₁ = 2∆z + Y3
The correct SOR formula for the boundary condition Y'(0) = 1 is:
OY₁ = (1 - ω/4)(Y₁ - Y₂) / 6 + (1 - ω/4)(Y₁ + Y₂) / 3 + (ω/4)(∆²f₁ + Yᵢ₊ₙ + Yᵢ₋ₙ - ∆)
To derive the correct SOR formula for the boundary condition Y'(0) = 1, we start with the standard SOR formula:
OYᵢ = (1 - ω)Yᵢ + (ω/4)(Yᵢ₊₁ + Yᵢ₋₁ + Yᵢ₊ₙ + Yᵢ₋ₙ - ∆²fᵢ)
where i and j are indices corresponding to the discrete coordinates in the x and y directions, ω is the relaxation parameter, and ∆ is the grid spacing in both directions.
To incorporate the boundary condition Y'(0) = 1, we use a forward difference approximation for the derivative:
Y'(0) ≈ (Y₁ - Y₀) / ∆
Substituting this into the original equation gives:
(Y₁ - Y₀) / ∆ = 1
Solving for Y₀ gives:
Y₀ = Y₁ - ∆
Now we can use this expression for Y₀ to modify the SOR formula at i = 1:
OY₁ = (1 - ω)Y₁ + (ω/4)(Y₂ + Y₀ + Yᵢ₊ₙ + Yᵢ₋ₙ - ∆²f₁)
Substituting the expression for Y₀, we get:
OY₁ = (1 - ω)Y₁ + (ω/4)(Y₂ + Y₁ - ∆ + Yᵢ₊ₙ + Yᵢ₋ₙ - ∆²f₁)
Simplifying:
OY₁ = (1 - ω/4)(Y₁ - Y₂) / 6 + (1 - ω/4)(Y₁ + Y₂) / 3 + (ω/4)(∆²f₁ + Yᵢ₊ₙ + Yᵢ₋ₙ - ∆)
So the correct SOR formula for the boundary condition Y'(0) = 1 is:
OY₁ = (1 - ω/4)(Y₁ - Y₂) / 6 + (1 - ω/4)(Y₁ + Y₂) / 3 + (ω/4)(∆²f₁ + Yᵢ₊ₙ + Yᵢ₋ₙ - ∆)
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(1 point) find the limit. use l'hospital's rule if appropriate. use inf to represent positive infinity, ninf for negative infinity, and d for the limit does not exist. \lim\limits {x\rightarrow \infty} \dfrac{8 x}{2 \ln (1 2 e^x)}
The limit of the function as x approaches infinity is infinity.
To evaluate this limit, we can use L'Hospital's rule, which says that if we have an indeterminate form of the type 0/0 or infinity/infinity, we can differentiate the numerator and denominator separately with respect to the variable of interest, and then take the limit again.
In this case, we have infinity/infinity, so we can apply L'Hospital's rule:
\begin{aligned}
\lim_{x\rightarrow\infty} \frac{8x}{2\ln(12e^x)} &= \lim_{x\rightarrow\infty} \frac{8}{\frac{2}{12e^x}}\\
&= \lim_{x\rightarrow\infty} \frac{8}{\frac{1}{6e^x}}\\
&= \lim_{x\rightarrow\infty} 48e^x\\
&= \infty
\end{aligned}
Therefore, the limit of the function as x approaches infinity is infinity.
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During a lab experiment, the temperature of a liquid changes from 625°F to 1034°F.
What is the percent of increase in the temperature of the liquid?
Enter your answer in the box as a percent rounded to the nearest hundredth
Therefore, the percent increase in temperature is approximately 65.44%.
The percent increase in temperature, we need to find the difference between the initial and final temperatures, divide that by the initial temperature, and then multiply by 100 to get a percentage:
Calculate the variation between the initial and end values. Subtract the beginning value from its absolute value. Add 100 to the result.
Even in a low-emission scenario, the earth is predicted to rise by two degrees Celsius by 2050, suggesting that we might not be able to keep the Paris Agreement. Compared to the average temperature between 1850 and 1900, the global temperature has increased by 1.1°C.
percent increase = ((final temperature - initial temperature) / initial temperature) x 100
In this case:
percent increase = ((1034 - 625) / 625) x 100
percent increase = (409 / 625) x 100
percent increase = 65.44%
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