The supplied statement states that the iterated integral after being converted to interpolation is 32/3.
What is the definition of an integral?In mathematics, an integral is either a number representing the region beneath a function's graph over a certain interval or just a new function, the reverse of which contains the original mechanism (indefinite integral). To complete the whole, an essential component is required. The term "essential" is almost a synonym in this context. Trigonometric functions of functional and equations are a concept in mathematics. Integral is a derivative of Middle English, Latin integer, and Medieval Latin integralis, each of which mean "making up a whole."
Therefore, the area is,
[tex]R=\left\{(x, y) \mid 0 \leq x \leq 2,0 \leq y \leq \sqrt{2 x-x^2}\right\}[/tex]
So,
[tex]y=\sqrt{2 x-x^2}[/tex]
y² = 2x - x²
x² - 2x + y² = 0
x² - 2x + 1 + y² = 1
(x-1)² + y² = 1
In other words, the area is the middle section of the disk with a radius of 1 and a center at (0,1).
Region R's polar coordinates are
x = r cos∅, y = r sin∅
So,
[tex]y=\sqrt{2 x-x^2}[/tex]
y² = 2x - x²
x² + y² = 2x
r² = 2r cos∅
r = 2 cos∅
The polar coordinates of the sphere of integration as,
[tex]R=\left\{(r, \theta) \mid 0 \leq \theta \leq \frac{\pi}{2}, 0 \leq r \leq 2 \cos \theta\right\}[/tex]
The provided integral then becomes,
[tex]\begin{aligned}& \int_0^2 \int_0^{\sqrt{2 x-x^2}} 6 \sqrt{x^2+y^2} d y d x \\= & \int_0^{\frac{\pi}{2}} \int_0^{2 \cos \theta} 6 r \cdot r d r d \theta \\= & \int_0^{\frac{\pi}{2}} \int_0^{2 \cos \theta} 6 r^2 d r d \theta \\= & \int_0^{\frac{\pi}{2}} 6\left(\frac{r^3}{3}\right)_0^{2 \cos \theta} d \theta\end{aligned}[/tex]
[tex]\begin{aligned}& =\int_0^{\frac{\pi}{2}} 2\left(8 \cos ^3 \theta-0\right) d \theta \\& =16 \int_0^{\frac{\pi}{2}} \cos ^2 \theta \cdot \cos \theta d \theta \\& =16 \int_0^{\frac{\pi}{2}}\left(1-\sin ^2 \theta\right) \cdot \cos \theta d \theta\end{aligned}[/tex]
Let u = sin∅ then du = cos ∅ d∅
If ∅ = 0 then u = 0 and if ∅ = π/2 then u = 1.
The above integral thus becomes,
[tex]\begin{aligned}& \int_0^2 \int_0^{\sqrt{2 x-x^2}} 6 \sqrt{x^2+y^2} d y d x \\= & 16 \int_0^1\left(1-u^2\right) d u \\= & 16\left(u-\frac{u^3}{3}\right)_0^1 \\= & 16\left(1-\frac{1}{3}\right)\end{aligned}[/tex]
= 32/3
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I need help solving this math for Algebra: The problem is in the picture, THANKS!
Answer:
Step-by-step explanation:
The first step to determine the implied domain of the function is to identify any restrictions on the variables x and y. In this case, the function 2y + x^(1/3) = 3x + x^2 does not have any restrictions on x and y, therefore the domain is all real numbers.
We can express this as an interval notation:
(-∞, ∞)
This means that the function is defined for any real value of x and y, which means the domain of the function is all real numbers.
1 7/9 x 2 IXL Y.6
Please help :)
Answer: 32/9, or 3 5/9
Step-by-step explanation:
1. convert the mixed number to an improper fraction. It would turn into 16/9 x 2
2. Calculate the product : 32/9
3. Use division to get a mixed number: 3 5/9.
According to state census information, the population of lavergne, tennessee in 2012 was 3.3777 * 104. the population of murfreesboro, tennessee in 2012 was 1.14038 * 105. what was the combined population of both cities?
(a) The Population of LaVergne, Tennessee in Standard Form is 33,777.
(b) The combined population of both cities is 147,815
In the given question, according to state census information, the population of LaVergne, Tennessee in 2012 was 33.777 x [tex]10^3[/tex].
The population of Murfreesboro, Tennessee in 2012 was 1.14038 x [tex]10^5[/tex].
(a) We have to write the Population of LaVergne, Tennessee in Standard Form.
The population of LaVergne, Tennessee in 2012 = 33.777 x [tex]10^3[/tex]
The population of LaVergne, Tennessee in 2012 = 33.777 x 1000
Now multiplying
The population of LaVergne, Tennessee in 2012 = 33,777
(b) We have to find the combined population of both cities.
The population of LaVergne, Tennessee in 2012 = 33,777
The population of Murfreesboro, Tennessee in 2012 = 1.14038 x 100,000
The population of Murfreesboro, Tennessee in 2012 = 114038
The combined population of both cities = The population of LaVergne, Tennessee + The population of Murfreesboro, Tennessee
The combined population of both cities = 33,777 + 114038
The combined population of both cities = 147,815
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If during her first three months at
college a student's long distance
phone bills are $119. 68, $84. 36,
and $68. 49, and her local phone
bill is $21. 00 per month, what is her
total average monthly phone bill?
Please help
Her entire monthly phone bill is $153.37, which includes long distance calls of $119.68, $84.36, and $68.49, as well as local calls of $21.00.
How to solve this equation?To figure out the student's total average monthly phone expense, combine her long distance phone bills ($119.68 + $84.36 + $68.49) and her local phone bill ($21.00), then divide the total by three (the number of months). The sum of the three long distance phone bills is $272.53, which is added to the $21.00 local phone charge for a total of $293.53. By dividing this sum by three, we get a monthly phone charge of $153.37. In summary, the student's total average monthly phone bill is $153.37, which includes long distance calls of $119.68, $84.36, and $68.49, as well as local calls of $21.00.
Here,
Long distance phone bills total:
$119.68 + $84.36 + $68.49
= $272.53
Local phone bill: $21.00
Total: $272.53 + $21.00
= $293.53
Average monthly phone bill:
$293.53 / 3
= $153.37
Her entire monthly phone cost is $153.37, which includes $119.68, $84.36, and $68.49 for long distance calls and $21.00 for local calls.
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What is an equation of the line that passes through the point (1, 7) and is parallel to the line 32 -y= 1?
The general form of a linear Equation in two variables is ax + by + c = 0.
32-y=1
0.x+y+32 = 0
Hence Proved.
what is 8 divided by 6.48 long division
Answer:
0.81
Step-by-step explanation:
Terrignonium-220 has a half life of 6 hours. Write an equation to model how much Terrignonium-220 would remain from a 50 g sample after x hours.
y = 50 (0.5)* y = 50 (0.5) 6x y = 50 (0.5)x/6 y = 50 (0.5) 6/x
Answer:
sorry i got to
Step-by-step explanation:
15 of Austen's dominoes are standing and the other 34 dominoes have fallen over. What is the ratio of the number of fallen dominoes to the number of dominoes still standing?
The ratio of the number of fallen dominoes to the number of dominoes still standing is 34 : 15
How to find ratios?Ratio refers to a number representing a comparison between two or more quantities.
Number of Austen's dominoes that are standing = 15
Number of Austen's dominoes that have fallen over= 34
The ratio of the number of fallen dominoes to the number of dominoes still standing = 34 : 15
Therefore, 34 : 15 is the ratio of fallen dominoes to standing dominoes.
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Multiply. x^2-1/5xy * x^2y/1+x
The required expression is x(x - 1)/5 which is determined by multiplication.
The expression is given in the question, as follows:
(x² - 1)/5xy × x²y/(1 + x)
We have to determine the solution to the given expression by multiplication.
As per the given expression, we have
⇒ (x² - 1)/5xy × x²y/(1 + x)
Cancel out the equivalent terms in the above expression,
⇒ (x - 1)(x + 1)/5 × x/(1 + x)
Reduce the same in the above expression,
⇒ x(x - 1)/5
The required expression is x(x - 1)/5.
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Solve the equation. 4x - 6 = 2x + 8 x = _____
Answer:
x=-1
Step-by-step explanation:
4x-6=2x+8x; 4x-6=10x; -6=6x; x=-1.
Answer:
x=-1
Step-by-step explanation:
4x-6=2x+8x=
4x-6=10x
-6=10x-4x
-6=6x
x=6/-6
x=-1
If two sides of a triangle are 6cm and 10cm then the lenght of the third side can be
Answer:
6cm
Step-by-step explanation:
Given, the length of two sides of a triangle are 6 cm and 10 cm.
We have to find the length of the third side.
We know that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
Sum of two sides = 6 + 10
= 16 cm
We know that the third side must be greater than the difference between two sides and less than the sum of two sides.
Difference of two sides = 10 - 6
= 4 cm
So, the third side must be greater than 4 cm and less than 16 cm.
Therefore, the third side is 6 cm.
nine times the sum of a number and 4 is equal to 7
Use the variable w for the unknown number. Write it into an equation
Answer:
9(w+4)7
Step-by-step explanation:
I got this answer too, just like the person before me
Find the Exponents!
(2 points each)
For numbers 5-7, use the properties of exponents to determine what numbers should
replace each variable written as an exponent below that will make the equations true.
LOOK AT THE PHOTO
DO NOT DO NUMBER 8
I ONLY NEES 5,6,7
Using the property of exponents the numbers are determined as follows
5. x = 9
6. b = -4
7. n = 0
How to determine the numbersAn exponential equation can be solved using the same bases because to the property of equality of exponential functions. It states that if all sides of an exponential equation have identical bases, then the exponents must likewise be equal.
The property is applied in the following equations
5. 8² * 8ˣ = 8¹¹
equating the powers
2 + x = 11
x = 11 - 2
x = 9
6. 5⁷ * 5ᵇ = 5³
equating the powers
7 + b = 3
b = 3 - 7
b = -4
7. x¹² * xⁿ = x¹²
equating the powers
12 + n = 12
n = 12 - 12
n = 0
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Your team scores 10 points in the first game, 3 points in the second game, 2 points in the third game, 5 points in the fourth game, and 0 points in the fifth game. What was the mean number of points your team scored for all 5 games?
The mean number of points for all 5 games is 4 points.
What is arithmetic mean?The arithmetic mean is defined as the ratio of the sum of observations to the total number of observations. It can be referred to as the average of a specific set of data or the arithmetic mean.
The formula for the mean of a given set of data is as follows:
Mean = Sum of Observations/Total number of observations
As per the question,
Here, n = 5
⇒ ∑x = Sum of Observations
⇒ ∑x = 10 + 3 + 2 + 5 + 0 = 20
Mean = ∑x/n
Mean = 20/5
Mean = 4
Hence, the mean of the given data is 4.
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the half-life of strontium-90 is 28 years, meaning that after 28 years a given amount of the substance will have decayed to half the original amount. find the exponential decay midel of strontium-90
Answer:i had to
Step-by-step explanation: mb
Use the ALEKS Ruler to find the length of the stick.
Write your answer to the nearest millimeter
we have 4 women and 5 men and want to create a committee with 2 women and 2 men. in how many ways we can do this?
126 number of ways we can arrange the men, women in order to committee with 2 women and 2 men ,the combination is explained below as
We can use the combination formula to find out the number of ways to choose 2 women and 2 men from a group of 4 women and 5 men.
as we know the formula for combination as :
C(n ,k) = n! / (k!(n-k)!)
where n is the total number of people, k is the number of people we want to choose, and ! denotes factorial to find the number of combinations (i.e. the product of all integers up to that number, e.g. 4! = 432*1 = 24).
In this case, to choose the number of individuals as = 4 + 5 = 9 and k = 2 + 2 = 4. So, the number of ways to choose the committee is:
C(9,4) = 9! / (4!(9-4)!) = 126 ways
Hence the number of ways to choose committee is 126 ways.
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which function could represent the value of an antique vase that decreases over time? type of problem
If the value of an antique vase decreases at a rate of 7% a year , and the price of vase a years ago was $1200 , then the current value of vase is $1116 .
The value of the antique vase a years ago was = $1200 ;
the percent decrease in the value of the antique vase is = 7% ,
So , let the present value of the vase be = $x ;
the present value of the vase ca be calculated as = [tex]1200-7\% \; of 1200[/tex] ;
= [tex]1200-0.07\times 1200[/tex] ;
Simplifying further ,
we get ;
= [tex]1200-84[/tex] ;
= 1116 .
Therefore , the current value of the antique vase is $1116 .
The given question is incomplete , the complete question is
The value of an antique vase decreases at a rate of 7% a year. If the vase was valued at $1200 years ago, what would its current value be?
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Find the area of the following figure. Pictures are not drawn to scale. Round answers to the nearest tenth.
The area of the trapezium is 36.3 unit squared.
How to find area of a figure?The figure above is a trapezium. Therefore, the area of a trapezium can be represented as follows:
Hence,
area of a trapezium = 1 / 2 (a + b)h
where
h = heighta and b are the top and base of the trapeziumTherefore,
area of a trapezium = 1 / 2 (a + b)h
where
a = 4b = 7 + 1.5 = 8.5Using Pythagoras's theorem,
c² = a² + b²
where
a and b are the legsc is the hypotenuseh² = 6² - 1.5²
h²= 36 - 2.25
h = √33.75
h = 5.80947501931
h = 5.81
Therefore,
area of the trapezium = 1 / 2 (4 + 8.5)5.8
area of the trapezium = 1 / 2 (12.5)5.8
area of the trapezium = 72.5 / 2
area of the trapezium = 36.3 units²
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Find the z-score for the value 93, when the mean is 100 and the standard deviation is 3.
The z-score for the value 93, when the mean is 100 and the standard deviation is 3 is solved to be
2.3How to solve for z scoreZ scores is used to determine the amount of standard deviations a sample, X is from the mean
The z score is given by the formula
z = (X - μ) / σ
Definition of the parameters
mean, μ = 100
standard deviation, σ = 3
sample score, X = 93
z score for the score of 93
z = (X - μ) / σ
substituting into the formula
z = (93 - 100) / 3
= -7 / 3
= -2.333
Hence we can say that the z score is 2.333
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-2x+5y=14 3x-3y=-3 by elimination
Answer:
x = -5
y = -4
Step-by-step explanation:
-2x + 5y = 14
3x - 3y = -3 --divide by 3 --> x - y = -1 --multiply by 2 --> 2x - 2y = -2
-2x + 5y = 14
2x - 2y = -2
Add the two equations
-3y = 12
y = -4
PLUG INTO x - y = -1
x + 4 = -1
x = -5
Hope this helps :)
Have a great day!
Answer:x=3, y=4
sorry if im wrong
Step-by-step explanation:
12. Vreni took part in a charity walk.
She walked a distance of 20 kilometres.
a) She raised money at a rate of $12.50 for each kilometre.
i. How much money did she raise by walking the 20 kilometres?
(1 mark)
ii. The money she raised in part (a)(i) was 5
52
of the total money raised.
Work out the total money raised.
(2 marks)
iii. In the previous year the total money raised was $2450.
Calculate the percentage increase on the previous year’s total.
(2 marks)
11b) Part of the 20 kilometres was on a road and the rest was on a footpath.
The ratio road distance : footpath distance was 3 : 2.
i. Work out the road distance.
(2 marks)
ii. Vreni walked along the road at 3km/h and along the footpath at 2.5km/h.
How long, in hours and minutes, did Vreni take to walk the 20 kilometres?
(2 marks)
iii. Work out Vreni’s average speed.
(1 mark)
iv. Vreni started at 08 55. At what time did she finish?
(1 mark)
12
c) On a map, the distance of 20 kilometres was represented by a length of 80 centimetres.
The scale of the map was 1 ∶ .
Calculate the value of .
12. i) By walking the 20 kilometers in the charity walk, Vreni raised $250 at a unit rate of $12.50 per kilometer.
12. ii) The total money raised by all participants in the charity walk was $3,100, using proportions.
12. iii) The percentage increase on the previous year's total is 26.53%.
Part B:
11. i) Based on a ratio of 3:2 of road and footpath distance, the road distance covered 12 kilometers.
11. ii) Using the distance, time, and speed formula, Vreni took 7 hours and 12 minutes to walk the 20 kilometers.
11. iii) Vreni's average speed is 2.78 km/h.
11. iv) Vreni finished the charity walk at 16:07.
Part C: The scale of the map to the total distance covered by Vreni was 1: 4 or a scale factor of 25%.
What is the ratio?The ratio is the quotient of a value compared to another.
Ratios are proportionate values, which are equated to each other.
The average speed and scale factor show the ratios of one value and the whole.
12) Charity Walk and Vreni:
The total distance walked = 20 kilometers
The unit rate of raising money per kilometer = $12.50
i) The total amount raised = $250 ($12.50 x 20)
Money raised in 12. 1 = 5/52 of the total money raised
ii) The total money raised in the charity walk = $3,100 ($250 ÷ 5/52)
iii) The total amount raised in the previous year = $2,450
The total amount raised in the current year = $3,100
Difference between current and previous year = $650 ($3,100 - $2,450)
Percentage increase on the previous year's total = 26.53% ($650/$2,450 x 100)
11b) Ratio of road and footpath walk = 3:2
The sum of ratios = 5
Road distance = 3/5 x 20 = 12 kilometers
Footpath distance = 2/5 x 20 = 8 kilometers
Vreni's walking speed along the road = 3 km/h
Road distance = 12 kilometers
Time to walk the road = 4 hours (12/3)
Vreni's walking speed along the footpath = 2.5 km/h
Footpath distance = 8 kilometers
Time to walk the footpath = 3.2 hours or 3 hours and 12 minutes (8/2.5) or 192 minutes (8/2.5 x 60)
Total time spent by Vreni = 7 hours and 12 minutes (4 hr + 3 hr + 12 min)
Vreni's average speed = Distance/Time
= 20 km/7.2 hrs
= 2.78 km/h
Starting time = 08:55
Total hours = 7 hours and 12 minutes
Finishing time = 16:07 (08:55 + 7 hrs + 12 min)
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How do you know if there are no solutions?
If you attempt to solve an equation and the result is false, the problem has no solutions.
If both sides of a linear equation have unique constant values but the same variable term, then the problem has no solutions.
For instance, in the equation [tex]3x+8=1+3x[/tex],
the variable term is the same, [tex]3x[/tex], but the constant terms, 8 and 1, are on opposite sides of the equation. It, therefore, lacks solutions. Try the equation to determine why.
[tex]3x+8=1+3x\\8=1[/tex]
following mathematics, the term is not possible. so, here equation has no solutions. so how we can know that equation has no solution if mathematical inequality is wrong.
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The equation y=-4.9x^2+9.8x+2 models the trajectory of a ball thrown in the air by a man, where y = the distance the ball is from the ground in meters and x = the elapsed time in seconds. what is the greatest distance the ball is from the ground?
On solving the provided question, we can say that here, in equation the greatest distance the ball is from the ground is -y = -4.9 + 9.8 +2 = 6.9
What is equation?An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to explain the connection between two sentences on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.
here,
the equation y=-4.9x^2+9.8x+2
where y = the distance the ball is from the ground in meters
x = the elapsed time in seconds
the greatest distance the ball is from the ground is -
[tex]y=-4.9x^2+9.8x+2[/tex]
x = 1
y = -4.9 + 9.8 +2 = 6.9
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Last month the average price of gas was $2.20 per gallon. This month, the average price has increased by 75%. What is the average price of gas per gallon this month?
The average price of gas per gallon this month is
(don't forget $)
The average price of gas per gallon this month: $3.85 per gallon.
Define the term percentage increase?The degree, intensity, extent, or value of a variable can change to a greater or lesser extent as measured by percent rise and percent decrease. By comparing the original (or before) as well as final (or after) amounts in accordance with a predetermined formula, the figures are derived.For the stated question-
Average price of gas last month = $2.20 per gallon.
Percentage increase = 75%.
75% of 2.20 = 75*2.20 / 100
= 1.65
Average price of gas this month = $2.20 + $1.65 per gallon
= $3.85 per gallon
Thus, the average price of gas per gallon this month: $3.85 per gallon.
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you measure the distance between the finges of a diffraction pattern as follows: distance (mm): 3.27, 3.23, 3.15 you measure the distance seven additional times to obtain the following ten values: distance (mm): 3.27, 3.23, 3.15, 3.16, 3.28, 3.14, 3.16, 3.12, 3.32, 1.46 what values for the distance and uncertainty would you report using the first three measurements and the entire set of ten measurements? group of answer choices
The distance is 3.21 +/- 0.04 mm using the first three measurements, and 3.16 +/- 0.08 mm using the entire set of ten measurements.
What is deviation ?
Deviation is a measure of how much a set of values differs from a certain value, typically the mean (average) of the set. There are two types of deviation ,
The absolute deviation is the difference between each value and the mean.
The standard deviation is the square root of the average of the squared absolute deviations.
The mean distance = (3.27 + 3.23 + 3.15) / 3 = 3.21 mm
The standard deviation = sqrt(((3.27 - 3.21)^2 + (3.23 - 3.21)^2 + (3.15 - 3.21)^2) / 2) = 0.04 mm
So, using the first three measurements, we can report the distance as 3.21 +/- 0.04 mm
To report the distance and uncertainty using the entire set of ten measurements, we can calculate the mean and standard deviation of the ten measurements.
The mean distance = (3.27 + 3.23 + 3.15 + 3.16 + 3.28 + 3.14 + 3.16 + 3.12 + 3.32 + 1.46) / 10 = 3.16 mm
The standard deviation = sqrt(((3.27 - 3.16)^2 + (3.23 - 3.16)^2 + (3.15 - 3.16)^2 + (3.16 - 3.16)^2 + (3.28 - 3.16)^2 + (3.14 - 3.16)^2 + (3.16 - 3.16)^2 + (3.12 - 3.16)^2 + (3.32 - 3.16)^2 + (1.46 - 3.16)^2) / 9) = 0.08 mm.
The distance is 3.21 +/- 0.04 mm using the first three measurements, and 3.16 +/- 0.08 mm using the entire set of ten measurements.
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pls help
giving 100 points
The value of the expression [tex]\frac{999999999\times 999999999}{1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1}[/tex] , which involves the division of a large number is obtained by making use of a scientific calculator is; 12345678987654321
How calculators can be used to divide large numbers?Depending on the processor of the calculator, and the number of digits the calculator can handle, the exact value following the operations with very large numbers can be obtained. Graphing calculators can also be made to increase the number of digits of precession, up to 9999 digits.
The expression in the question can be presented as follows;
999999999 × 999999999 ÷ (1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1)
The value can be computed using a calculator;
999999999 × 999999999 = 999999998000000001
(1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1) = 81
999999999 × 999999999 ÷ (1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1) = 999999998000000001 ÷ 81 = 12345678987654321
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The sum of a number and negative six greater than 9
The expression becomes x-6 > 9 from the given condition The sum of a number and negative six greater than 9.
What is Algebraic expression ?
Algebraic expression can be defined as the combination of variables and constants.
Given,
To find the sum of a number and negative six , greater than 9 .
let the given number be x .
so,
the expression becomes ,
x - 6 > 9
LHS = x-6
as in the question its given the sum of number and negative of six
So whereas in RHS = 9
As it is given greater than 9 .
Hence, the expression becomes x-6 > 9 from the given condition.
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y+4=3(x-3) rewrite in Ax+By=C
Answer:
0=3x-y=-13
Step-by-step explanation:
Answer:
In the equation y+4=3(x-3), we can see that x and y are the variables and 4 and -3 are the constants.
To convert the equation to the form Ax + By = C, we'll first distribute the 3 on the right side:
y + 4 = 3x - 9
Next, we'll move all the terms with x to one side of the equation and all the terms with y to the other side:
-3x + y = - 13
Now we have the equation in the form Ax + By = C, where A = -3, B = 1, and C = -13.
a student is randomly eelected from mr.willets class and that studenr is also in ap satts what is the probability that they are as senior?
