A sample's null hypothesis indicates that no statistical relationship exists in a set of provided single observed variables. The null hypothesis is rejected for both parts A and B of the question.
A sample's null hypothesis indicates that no statistical relationship exists in a set of provided single observed variables.
Part A: The conclusion to be reached at the 5% level of significance-
The p value for the z score in the right tailed or upper test can be obtained from the z table, which is,
p-value=0.125=0.125
The p value of 0.125 is greater than the significance level of 0.06. As a result, the effect is not significant, and the null hypothesis cannot be rejected.
Part B: If the alternative hypothesis is Ha: p 0.6, the conclusion to reach at the 5% level is-
The unequal sign 2-test is carried out here. The p value for the 2-test is as follows:
The p-value is 0.25=0.25.
In this case, the p value of 0.25 is greater than the significance level of 0.05. As a result, the effect is not significant, and the null hypothesis cannot be rejected.
As a result, the null hypothesis fails to be rejected for both parts A and B of the question.
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A football club increases the number of seats in its stadium to 66,000 The manager of the club states they can now seat exactly 45% more people than previously. Explain why the manager cannot be correct
The manager's claim that the new stadium can seat exactly 45% more people than previously is not correct. Instead, the new stadium can accommodate 145% of the previous number of seats, which is an increase of 45% over the original capacity
If a football club increases the number of seats in its stadium by 45%, it means that the new stadium can accommodate 145% of the previous number of seats.
Suppose the previous capacity of the stadium was x seats. Then, the new capacity of the stadium can be calculated by multiplying the previous capacity by 1.45 (which represents the 45% increase in capacity):
New capacity = x × 1.45 = 1.45x
However, the manager of the club claims that the new capacity of the stadium is exactly 45% more than the previous capacity, which means that the new capacity should be:
New capacity = x + (0.45x) = 1.45x
This equation suggests that the increase in capacity is not 45%, but rather an increase of the original capacity by 45%. This is not the same as the increase of 45% over the original capacity.
Therefore, the manager's claim that the new stadium can seat exactly 45% more people than previously is not correct. Instead, the new stadium can accommodate 145% of the previous number of seats, which is an increase of 45% over the original capacity.
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Your checking account has a constant balance of $500. Let the function m represent the balance of your savings account after t years. The table shows the total balance of the accounts over time.
This equation can be used to predict the balance of the savings account after t years.
What is equation?Equation is a physical and mathematical statement that describing physical phenomena and the relationship between different physical quantity typically consist of variable simple presenting physical quantity and then it personal variable may be pointed such as your energy.
We can graph the data in a line graph to visualize the relationship between the two variables. The graph will have time (t) represented on the x-axis and the balance of the savings account (m) represented on the y-axis. The graph will demonstrate that the total balance of the two accounts increases over time.
The equation that models this data can be found by using the slope-intercept form, y = mx + b. Here, y represents the balance of the savings account (m), x represents the time (t), m is the slope of the line, and b is the y-intercept. To find the equation, we can first find the slope, which is the change in y divided by the change in x. This is (2,800 - 500) / (3 - 0) = 2,300 / 3 = 766.67. The y-intercept is the balance of the savings account when t = 0, which is 500. We can now write the equation as m = 766.67x + 500. This equation can be used to predict the balance of the savings account after t years.
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I need help on these two
The prepositional phrase would be B. of the bees.
The prepositional phrase would be C. he took shelter.
What is a prepositional phrase ?A prepositional phrase is a group of words that includes a preposition (such as "in," "on," "at," "by," "with," etc.) and a noun or pronoun object, along with any associated modifiers.
Prepositional phrases can be used in a variety of ways, including to describe location, time, possession, and other relationships between words in a sentence. In the given statements, the prepositional phrases would be "of the bees" and " he took shelter. "
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How to Calculate Z Scores?
The formula for calculating the z-score is z = (x-μ)/σ.
where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula suggests, the Z-score is simply the raw score minus the population mean divided by the population standard deviation.
A positive Z-score indicates that the raw score is higher than the average. For example, a z-score of +1 is one standard deviation above the mean.
A negative z-score indicates that the raw score is below average. For example, a Z-score of -2 is two standard deviations below the mean.
Another way to interpret z-scores is to create a standard normal distribution (also known as the z-score distribution or probability distribution).
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¿Cuáles de las siguientes opciones tienen el mismo valor que el
, de
81
?
The correct answer is option B and it is equal to 30% of 81 because it shows 0.30 x 81.
We have to find that how much is equal to the 30% of 81.
We can use the percentage formula here and that is,
%age = Obtained value/total value x 100
Here in this case, we have to find the obtained value and the total value is 81 and the percentage that we are going to get is equal to 30.
So, putting all the values, we get,
obtained value = 30 x 81/100
obtained value = 0.30 x 81
Hence, the value 0.30 x 81 is equal to 30 % of 81. hence option B is the correct answer.
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Complete question - Which of the following options have the same value as 30% of 81?
A. 0.03 x 81
B. 0.3 x 81
C. 0.03 x 81
D. 3 x 81
Use an algebraic method (substitution or elimination) to solve the system of equations. Show your work.
PLEASE HELP!!!!
4X − 2Y = 16
2X + 3Y = 32
Answer:
X=7
Y=6
Step-by-step explanation:
Divide the 1st equation by 2.
2x+3y=32
2x-y=8
When you subtract those 2 numbers, you get that 4y=24.
Y=6
32-6*3=14=2x
X=7
If parallelogram ABCD was reflected over the y-axis, reflected over the x-axis, and rotated 180°, where would point A' lie?
After all these transformations, point A' will lie on the left side of the y-axis with the same y-coordinate as A, and point A''' will lie on the right side of the y-axis with the same x-coordinate as A.
What is parallelogram ?
A parallelogram is a quadrilateral with two pairs of parallel sides.
If parallelogram ABCD is reflected over the y-axis, all the x-coordinates of the vertices will change sign while the y-coordinates remain the same. This means that A will be reflected to a point A' on the left side of the y-axis with the same y-coordinate as A.
If the reflected parallelogram is then reflected over the x-axis, all the y-coordinates of the vertices will change sign while the x-coordinates remain the same. This means that A' will be reflected to a point A'' below the x-axis with the same x-coordinate as A'.
Finally, if the reflected and flipped parallelogram is rotated 180 degrees, all points will be reflected over the origin. This means that A'' will be reflected to a point A''' on the right side of the y-axis with the same x-coordinate as A'' (which is the same as A).
Therefore, after all these transformations, point A' will lie on the left side of the y-axis with the same y-coordinate as A, and point A''' will lie on the right side of the y-axis with the same x-coordinate as A.
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In a recent year, Saudi Arabia produced 8.2 million
barrels of crude oil per day. The United States
produced 6.4 million barrels per day. Use a percent to
state how much more Saudi Arabia produced than the
United States did.
In a recent year, Saudi Arabia produced 8.2 million barrels of crude oil per day, compared to 6.4 million barrels per day for the United States. This was 28% more than what the United States produced.
Saudi Arabia now consumes 1.8 million more barrels of crude oil per day than the United States does. Crude Oil from Saudi Arabia: In January 2023, production was reported at 10,319,000 barrels per day, down from 10,475,000 barrels per day in the previous year.
In November 2022, crude oil production in Saudi Arabia was 10,468 thousand barrels per day, down from 10,957 thousand barrels per day the year before.
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which of the following methods can enid use to test her claim? choose the two correct answers. responses she could calculate the ratio change in calories burned to change in miles jogged for two data point and compare the results to the same ratio with two other data points. she could calculate the ratio change in calories burned to change in miles jogged for two data point and compare the results to the same ratio with two other data points. she could plot the data on a coordinate plane and see if a straight line starting at (0, 0) passes through all the data points. she could plot the data on a coordinate plane and see if a straight line starting at (0, 0) passes through all the data points. she could put the data in a table and check to see that the difference between the number of calories burned changes by the same amount for each row. she could put the data in a table and check to see that the difference between the number of calories burned changes by the same amount for each row. she could calculate the ratio between the number of calories burned for each pair of jogs and see if the ratio is always the same.
She should figure out how many calories she burned compared to how many miles she ran, which is the proper statement.
What is a ratio?When two amounts are quantitatively related, it is possible to see how many times one value contains or is contained by the other. for instance, "There are now two computers for every student."
Given here: The table containing data calories vs miles run.
Calculating the ratio between each pair of data in a table allows you to determine whether a proportional relationship is there. The table depicts a proportionate relationship if all of those ratios are the same.
Thus calculating the ratio for each pair of entries we get
1) ratio=118/1.2
=295:3a or unit rate 98.33 calories per mile
2) 190/2= 95:1 or unit rate 95 calories per mile
3) 226/2.4=565:6 or unit rate94.1
clearly, the ratios are different.
Hence, the table containing the data doesn't form a proportional relationship.
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A number increased by five percent result is then increased by five percent what is the percent of increase
Answer:
Step-by-step explanation:
A triangle has angles of −3,2+8, 3−17. Solve for g and find the measures of the
angles
The measures of the angles of the triangle are:
Angle 1: −3 degrees
Angle 2: 10 degrees
Angle 3: −14 degrees
To solve for g and find the measures of the angles of the triangle, we need to use the fact that the sum of the angles in a triangle is 180 degrees. This means that:
−3 + (2+8) + (3−17) = 180
Simplifying the equation, we get:
−3 + 10 + (−14) = 180
−7 = 180
To solve for g, we need to isolate it on one side of the equation. To do this, we can add 7 to both sides of the equation:
g = 180 + 7
g = 187
Now, we can plug g back into the original equation to find the measures of the angles:
−3 + (2+8) + (3−17) = 187
−3 + 10 + (−14) = 187
−7 = 187
Therefore, the measures of the angles of the triangle are:
Angle 1: −3 degrees
Angle 2: 10 degrees
Angle 3: −14 degrees
I hope this helps! Let me know if you have any further questions.
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Plsss help
Find the exact are of the shaded region. Point I marks the center of the circle.
The area of the shaded region that is the segment of the circle is equal to 4.6 ft² to the nearest tenth.
How to calculate the area of the circle segmentTo calculate the area of a segment of a circle, you need to know the radius of the circle, the central angle of the segment (in radians or degrees), and possibly the height of the segment (the distance from the chord of the segment to the center of the circle).
The formula for the area of a segment of a circle is:
A = θ/360° × πr² - 1/2 × r² × sinθ
where A is the area of the segment, r is the radius of the circle, and θ is the central angle of the segment in degrees.
From the question;
the central angle = 90°
radius r = 4ft
Area of the segment = 90°/360° × π4² - 1/2 × 4² × sin90°
Area of the segment = 1/4 × π × 4 × 4 - 1/2 × 4 × 4 × 1 {sin90° = 1}
Area of the segment = 4π × 8
Area of the segment = 4.5714
Therefore, the exact area of the shaded region that is the segment of the circle is equal to 4.6 ft² to the nearest tenth
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Carrie spent half of her allowance playing mini golf. She then earned $12 more babysitting. What is her weekly allowance if she has $24 left?
Answer: her allowance is $24
Step-by-step explanation:
1. A young man rents an apartment for $850 / month. His
sister buys a house with a $20,000 down payment and
monthly mortgage payments of $1,150. Determine who spends
more over the 30-year period that her mortgage lasts,
assuming that both living situations remain the same. Who is in
a better position as far as having a financial asset?
a) The sister spends more (making total payments of $414,000) over the 30-year period than the young man, who spends $306,000.
b) Assuming both living situations are the same, the sister is in a better position as far as having a financial asset is concerned.
How the numbers are determined:The numbers are determined using the mathematical operations of multiplication and subtraction.
The monthly rent for an apartment = $850
The total payments for 30 years = $306,000 (30 x 12 x $850)
Sister's Payment for a Mortgaged House:
Down payment = $20,000
Monthly mortgage payments = $1,150
The total payments for 30 years = $414,000 ($1,150 x 360)
The difference in total payments = $108,000 ($414,000 - $306,000)
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How likely is an event with a probability of 3/4?
1) unlikely
2) as likely as not
3) likely
4) certain
Answer:
3) Likely
Step-by-step explanation:
75% chance
special right triangles calculator
Online calculators can solve for sides, angles, area, and perimeter of special right triangles, such as 45-45-90 and 30-60-90.
There are numerous web-based number crunchers that can assist you with addressing exceptional right triangles, like the 45-45-90 and 30-60-90 triangles. A few instances of extraordinary right triangle mini-computers are:
Calculator.net: This site has a unique right triangle number cruncher that can settle for the sides, points, and area of 45-45-90 and 30-60-90 triangles.
Omni Number cruncher: This site has a unique right triangle mini-computer that can settle for the sides, points, region, and border of 45-45-90 and 30-60-90 triangles.
Mathsisfun.com: This site has an intelligent apparatus that can help you comprehend and address exceptional right triangles, including 45-45-90 and 30-60-90 triangles.
These adding machines can save you time and assist you with rapidly tackling extraordinary right triangles, particularly assuming that you are chipping away at calculation issues or developments.
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The complete question is:
Explain about special right triangles calculator.
When can a correlation coefficient based on an observational study be used to support a claim of cause and effect? A. Never B. When the correlation coefficient is close to -1 or +1. C. When the correlation coefficient is equal to -1 or +1. D. When the scatterplot of the data has little vertical variation.
Option A. Never. A correlation coefficient is a measure of the strength and direction of the linear relationship between two variables.
However, a strong correlation between two variables does not necessarily mean that one causes the other. This is because there may be other factors, known as confounding variables, that affect both variables simultaneously. Therefore, it is not appropriate to use a correlation coefficient to establish a causal relationship between two variables. In order to establish causality, more rigorous methods such as randomized controlled trials must be used, which attempt to isolate the effect of one variable on another while holding other factors constant.
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What is the slope?
Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Slope of the given graph is 2.
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
Let us take two points to find the slope of line.
The two points are (4, 1) and (6,5)
m=5-1/6-4
m=4/2
m=2
Hence, slope of the given graph is 2.
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Determine the type of distribution and the best measure of center and spread of the data set. 1, 7, 11, 14, 17, 17, 17, 21, 21, 23, 23, 26 The distribution is SO and (Type integers or decimals rounded to the nearest tenth as needed.) ... best represent the data set.
The mean of the data is 16.5 and the standard deviation is 7.28
What is the distribution of the dataThe given data set is a small sample of 12 observations.
To determine the type of distribution, we can first create a histogram or a boxplot of the data.
A histogram of the data shows that the distribution is unimodal and slightly right-skewed.
Alternatively, we can calculate the skewness of the data. If the skewness is close to zero, then the data is approximately symmetric. If the skewness is positive, then the data is right-skewed. If the skewness is negative, then the data is left-skewed.
Calculating the skewness of the data set, we get:
skewness = (n / ((n - 1) * (n - 2))) * Sum[(xi - x-bar)^3 / s^3]
where n is the sample size, x-bar is the sample mean, s is the sample standard deviation, and Sum is the sum of the values in the data set.
Using this formula, we get a skewness of approximately 0.456, which indicates that the distribution is slightly right-skewed.
Based on these findings, we can conclude that the distribution of the data set is approximately normal, but slightly right-skewed.
To find the best measure of center and spread, we can calculate the sample mean and sample standard deviation, respectively.
Sample mean:
mean = (1 + 7 + 11 + 14 + 17 + 17 + 17 + 21 + 21 + 23 + 23 + 26) / 12 = 16.5
Sample standard deviation:
s = sqrt((1/11) * [(1 - 16.5)^2 + (7 - 16.5)^2 + ... + (26 - 16.5)^2]) = 7.28
Therefore, the best measure of center for this data set is the sample mean of 16.5, and the best measure of spread is the sample standard deviation of 7.28.
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Use the trapezoid shown to mark each statement below as true or false. 10 If false, rewrite the statement correctly in the space below the statement. 4. The length of AB can be found using 32 + b² = 4². 5. The perimeter of the trapezoid shown is 22 units.
The statement is false, the perimeter of the trapezoid shown is 23 units
How to find the perimeter of a trapezoid?To find the perimeter of a trapezoid, you need to add the lengths of all four sides of the trapezoid.
A trapezoid is a four-sided figure with two parallel sides (the bases) of different lengths. The other two sides are called the legs.
The formula for the perimeter of a trapezoid is:
Perimeter = sum of all sides = base1 + base2 + leg1 + leg2
where base1 and base2 are the lengths of the parallel sides and leg1 and leg2 are the lengths of the non-parallel sides.
So, to find the perimeter of a trapezoid, you need to measure or know the length of all four sides and then add them up using the formula above.
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Use the trapezoid shown to mark each statement below as true or false. if false, please rewrite the statement correctly in the space below the statement!
Q: The perimeter of the trapezoid shown is 22 units.
True false?
What is antiderivative of sin?
The antiderivative of sin is -cos(x) + C.
The antiderivative of a function is the inverse operation of differentiation which is is also known as its indefinite integral.
So, the integral of sin
∫sin(x) = -cos(x) + C
where C = constant of integration.
This can be verified by differentiating -cos(x) + C with respect to x, which gives sin(x).
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A regular pentagon has an area of 118.25 square meters, and each side of the pentagon measures 4.3 meters.
What is the length of an apothem of the pentagon?
Answer: 11
Step-by-step explanation: Just took the test
The length of the apothem of the pentagon is approximately 11 meters.
What is a polygon?Polygon is defined as a geometric shape that is composed of 3 or more sides these sides are equal in length, and an equal measure of angle at the vertex,
Examples of polygons, equilateral triangles, squares, pentagons etc.
Here,
To solve this problem, we can use the formula for the area of a regular pentagon:
A = (5/2) × s × a
Where A is the area of the pentagon, s is the length of one side, and a is the apothem (the distance from the center of the pentagon to the midpoint of a side).
We know that the area of the pentagon is 118.25 square meters and the length of one side is 4.3 meters. We can substitute these values into the formula and solve for the apothem:
118.25 = (5/2) × 4.3 × a
Divide both sides by (5/2) x 4.3:
a = 118.25 / ((5/2) × 4.3)
= 118.25 / 10.75 ≈ 11 meters
Therefore, the length of the apothem of the pentagon is approximately 11 meters.
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Eva deposits $2,000 in a savings account with an interest rate of 5% compounded continuously. She decides not to deposit or withdraw any money after the initial deposit. She uses the formula A = Petto represent her account balance, A, after t years, where r is the rate of interest and P is the beginning balance, or principal. How long will it take for Eva's initial deposit to double?
It take 20 years for Eva's initial deposit to double?.
What is Simple interest?A quick and easy method of calculating the interest charge on a loan is called a Simple interest.
Given that;
Eva deposits $2,000 in a savings account with an interest rate of 5% compounded continuously.
Here, Principal amount = $2000
Amount = $4000
Interest rate = 5%
Now, We know that;
⇒ Interest = PRT
Substitute all the values, we get;
⇒ 2000 = 2000 × 0.05 × T
⇒ T = 1/0.05
⇒ T = 100/5
⇒ T = 20 years
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how many ounces to a milliliter
Answer:
1 fluid oz = 29.5735 milliliters
Step-by-step explanation:
What are units?A unit can be used for measurement, and is commonly found in mathematics to describe length, size, etc.
Converting these units:
1oz = 29.5735ml1ml = 0.033814ozTherefore, for every 1 fluid ounce, it is equivalent to 29.5735 milliliters.
Eliana's bedroom door is 7 feet tall and 4 feet wide. A new door would cost $5.09 per square foot. How much would a new bedroom door cost in total?
Therefore , the solution of the given problem of area comes out to be
a new bedroom door would cost $142.52 in total.
A Surface Area is what?Its surface area serves as a fair guide of its total size. The environment is considered when calculating the right triangle squared. Everything's overall size is determined by its surface area. The volume of water inside a cuboid is equal to the sum of the contents from each of its twelve vertical sides. To determine the box's dimensions, use the formula below: 2lh, 2lw, and 2hW symbolise the area (SA). The full surface area of the flexible shape serves as the area's representation.
Here,
The area of the door is:
Area = height x width
Area = 7 ft x 4 ft
Area = 28 sq ft
The cost per square foot is $5.09, so the total cost of the new door would be:
Cost = Area x cost per square foot
Cost = 28 sq ft x $5.09/sq ft
Cost = $142.52
Therefore, a new bedroom door would cost $142.52 in total.
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find the area of the surface generated when the given curve is revolved about the given axis.
The area of the surface generated when the given curve is revolved about the given axis 5x^1/3 is [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex].
The curve x = f(y)
The area of the surface around the y-axis from y = a → y = b is:
[tex]\int\limits^a_b {2\pi x\sqrt{1+(\frac{dx}{dy})^2 } } \, dx[/tex]
From the given curve:
[tex]y = 5x^{1/3}[/tex] ; assuming the region bounded by the curve is 0 ≤ y ≤ 1
So;
[tex]y = 5x^{1/3}[/tex]
⇒ 5x = y³
⇒ x = 1/5 y³
The differential of the above equation Is:
dx/dy = 1/5 (3y²)
⇒ dx/dy = 3/5 (y²)
Now, we have the area of the surface produced around the curve [tex]y = 5x^{1/3}[/tex] through the y axis from the region y = 0 to y = 1
∴ [tex]\int\limits^1_0 {2\pi \frac{1}{5}y^3\sqrt{1+(\frac{3}{5} y^2)^2} } \, dx[/tex]
= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{1+(\frac{9}{25} y^4)} } \, dy[/tex]
= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{\frac{25+9y^4}{25})} } \, dy[/tex]
= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{25+9} y^{4} )} } \, dy[/tex]
Let make u = [tex]\sqrt{25+9y^{4} }[/tex]
It implies that:
[tex]u^3 = (25+9y^4) \sqrt{25+9y^4}[/tex]
and, [tex]u = \sqrt{25+9y^{4} }[/tex]
⇒ [tex]du = \frac{1}{2\sqrt{25+9y^4} } (36y^3) dy[/tex]
⇒ [tex]y^3dy = \frac{1}{18} \sqrt{25+9y^{4} }du[/tex]
when y = 0 ;
[tex]u = \sqrt{25+9y^{4} }[/tex]
[tex]u = \sqrt{25+9{(0)^4} }[/tex]
[tex]u = \sqrt{25 }[/tex]
u = 5
when y = 1;
[tex]u = \sqrt{25+9{(1)^4} }[/tex]
[tex]u = \sqrt{34} }[/tex]
∴The equation [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{25+9} y^{4} )} } \, dy[/tex] can be written as:
= [tex]\frac{\pi }{225} [\frac{u^3}{3} ]\left \{ {{y=\sqrt{34} } \atop {x=5}} \right.[/tex]
= [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex]
Therefore,
The area of the surface generated when the given curve is revolved about the given axis 5x^1/3 is [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex].
Complete Question:
Find The area of the surface generated when the given curve is revolved about the given axis 5x^1/3.
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Which equation represents the relationship between Stephanie’s age and Chris’s age?
Answer:
y = x + 8 (second option)
Step-by-step explanation:
If Chris's age is x , then Stephanie who is 8 years older should be x + 8
y = x + 8
Determine the height of a picture when the total height of the picture with the frame is 38.5 cm.
A graph of the relationship modeled by the equation is shown in the image attached below.
If the graph continues, the point (28, 24) will not be on the graph.
The height of a picture when the total height of the picture with the frame is 38.5 cm would be 34.5 cm.
How to graph the relationship modeled by the equation?In this scenario, the total height of a picture in its frame (in centimeters) would be plotted on the y-axis of the graph while the height of the picture (in centimeters) would be plotted on the x-axis of the graph.
Based on the ordered pair (28, 24), we would determine if it lies on the graph of the equation:
y = x + 4
24 = 28 + 4
24 = 32 (False).
When the total height of the picture with the frame is 38.5 cm, the height of a picture can be calculated as follows;
y = x + 4
38.5 = x + 4
x = 38.5 - 4
x = 34.5 cm.
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Complete Question:
The equation y = x + 4 models the total height of a picture in its frame, y, in centimeters with respect to the height of the picture, x, in centimeters.
a. Graph the relationship modeled by the equation. Be sure to label the axes.
b. If the graph continues, will the point (28, 24) be on the graph?
c. Determine the height of a picture when the total height of the picture with the frame is 38.5 cm.
Drag and drop the labels to classify the pairs of angles formed by a transversal intersecting two parallel lines. Each label may be used once, more than once, or not at all.
Alternate interior angles - Congruent
Corresponding angles - Neither
Same side interior angles - Supplementary
What are alternate angles?Alternate angles are a type of angle formed when a straight line intersects two parallel lines. When this happens, alternate angles are formed on opposite sides of the transversal (the line that intersects the parallel lines).
Alternate angles are also known as "alternate interior angles" because they are located inside the parallel lines, but on opposite sides of the transversal. They are called "alternate" because they are located on opposite sides of the transversal and they are congruent, meaning they have the same angle measure.
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1. find the square roots of (a) 2i; (b) 1−√3i and express them in rectangular coordinates.
a) The square root of complex number, 2i is equals to the ±( 1 + i).
b) The square root of complex number, (1 - √3i) is equals to the ±1/√2(√3 - 1). The rectangular coordinates plane of both present in above figure.
Square root of a number is defined as a value which on multiplication by itself, results the original number. If p is the square root of q then it is denoted as p=√q. We have to determine the square root of complex numbers and express them in rectangular coordinates. For a nonzero complex number z, its square roots are √z =√r exp( i(2πk + θ)/2), k = 0,1
(a) The magnitude of 2i is r =√(2² + 0) = 2, and the principal argument is θ = π/2. So, (2i)½ = r½ exp(i (2πk + π/2)/2) , k = 0, 1
For first root, substuting, k = 0
=> (2i)½ = 2½ exp(i (2π×0 + π/2)/2)
=> (2i)½ = √2 exp(i (π/2)/2)
=> (2i)½ = √2 exp(i π/4) = √2(cos(π/4) + i sin(π/4)
=> (2i)½ = √2( 1/√2 + i(1 /√2)) = 1 + i
For second root substuting, k = 1
=> (2i)½ = 2½ exp(i (2π×1 + π/2)/2)
=> (2i)½ = √2 exp(i (5π/2)/2)
=> (2i)½ = √2 exp(i (5π/4) = √2( cos(5π/4) + i sin(5π/4)
=> (2i)½ = √2(- 1/√2 + i(-1 /√2)) = - (1 + i)
Hence, square root of 2i is ±( 1 + i).
b) 1 -√3i
The magnitude and principal argument of 1 −√3i are respectively, r = √(1² + (√3)²) = 2 and θ = tan⁻¹ ( -√3/1) = - π/3. Similar to part(a), square root of (1 - √3i) is written as (1 - √3i)½ = r½ exp(i (2πk - π/3)/2), k= 0, 1
For first root , k = 0
(1 - √3i)½ = √2 exp(i (2π×0 - π/3)/2)
=> (1 - √3i)½ = √2 exp(i (- π/6) )
=> (1 - √3i)½ = √2( cos(-π/6) - i sin(π/6))
=> (1 - √3i)½ = √2( cos(π/6) - i sin(π/6))
=> (1 - √3i)½ = √2( √3/2 - i (1/2)) = 1/√2(√3 - 1)
For second root, k = 1
(1 - √3i)½ = √2 exp(i (2π×1 - π/3)/2)
=> (1 - √3i)½ = √2 exp(i (5π/3)/2)
=> (1 - √3i)½ = √2 exp(i (5π/6))
=> (1 - √3i)½ = √2 (cos(5π/6) + i sin(5π/6))
=> (1 - √3i)½ = √2 (- √3/2 + i(1/2) ) = - 1/√2(√3 - i)
so, (√1- √3i ) is ± 1/√2(√3 - 1). Hence, we got all required square roots.
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Complete question:
1. Find the square roots of (a) 2i; (b) 1 - √3i and express them in rectangular coordinates.
