Find the mass and center of mass of the lamina that occupies the region D and has the given density function p. D is bounded by the parabolas y = x2 and x = y2; p(x, Y) = 19 x
Answers
The mass and center of mass of the lamina that occupies the region D and has the given density function p is 57/14 and (14/27, 7/18) respectively.
The center of mass (x―,y―) of a lamina with density function ρ(x,y) is given by
x = M(y)/m, y = M(x)/m
Where, m=∫∫[tex]_{D}[/tex]ρ(x,y)dA
Mx=∫∫[tex]_{D}[/tex] yρ(x,y)dA
My=∫∫[tex]_{D}[/tex] xρ(x,y)dA
Given that, D is bounded by y=x^2 and x=y^2
And ρ(x,y)=19√x
Now, for the point of intersection of y=x^2,x=y^2
we have,
x = (x^2)^2
x = x^4
Subtract x^4 on both side
x - x^4 = 0
x(x^3−1) = 0
x = 0, 1
Now, x=0⇒y=0 and x=1⇒y=1
The points of intersection are (0,0),(1,1)
So, the region D can be written as
D={(x,y): 0≤x≤1, x^2≤y≤x}
So,
m = [tex]\int_{0}^{1}\int_{x^2}^{\sqrt x}19 \sqrt xdydx[/tex]
m = [tex]19\int_{0}^{1}\sqrt{x}[y]^{x^2}_{x}dx[/tex]
m = [tex]19\int_{0}^{1} \sqrt x(\sqrt{x}-x^2)dx[/tex]
m = [tex]19\int^{1}_{0}(x-x^{5/2})dx[/tex]
m = [tex]19[\frac{x^2}{2}-\frac{x^{7/2}}{7/2}]^1_{0}[/tex]
m = [tex]19[\frac{1}{2}(1^2-0)-\frac{2}{7}(1^{7/2}-0)][/tex]
m = 19(1/2−2/7)
m = 57/14
m = 5714
Now,
Mx = [tex]\int_{0}^{1}\int_{x^2}^{\sqrt x}(19xy)dydx[/tex]
Mx = 19[tex]\int^{1}_{0}x[\frac{y^2}{2}]_{x^2}^{\sqrt x}dx[/tex]
Mx = [tex]\frac{19}{2}\int^{1}_{0}x[(\sqrt{x})^2-(x^2)^2}dx[/tex]
Mx = [tex]\frac{19}{2}\int^{1}_{0}(x^2-x^5)dx[/tex]
Mx = [tex]\frac{19}{2}[\frac{x^3}{3}-\frac{x^6}{6}]_{0}^{1}[/tex]
Mx = [tex]\frac{19}{2}[\frac{1}{3}(1^3-0)-\frac{1}{6}(1^6-0)][/tex]
Mx = 19/2 (1/3−1/6)
Mx = 19/12
And
My = [tex]\int^{1}_{0}\int_{x^2}^{\sqrt x}x(19\sqrt{x})dydx[/tex]
My = [tex]19\int_{0}^{1}x^{3/2}[y]^{\sqrt{x}}_{x^2}dx[/tex]
My = [tex]19\int_{0}^{1}x^{3/2}(\sqrt{x}−x^2)dx[/tex]
My = [tex]19\int_{0}^{1}(x^2-x^{7/2})dx[/tex]
My = [tex]19[\frac{x^3}{3}-\frac{x^{9/2}}{9/2}]_{0}^{1}[/tex]
My = 19[1/3(1^3−0)−2/9(1^{9/2}−0)]
My = 19(1/3−2/9)
My = 19/9
So, x = My/m
x = (19/9)/(57/14)
x = (19/9)×(14/57)
x = 14/27
y = Mx/m
y = (19/12)/(57/14)
y = (19/12)×(14/57)
y = 7/18
Therefore, the solutions are (14/27, 7/18).
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How do you find the mean 10?
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The complete question is-
How do you find the mean of first 10 natural numbers ?
What are the three types of parameters in Python?
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We come to the conclusion that there are three different kinds of parameters that can be used with functions in Python. Default, keyword, and arbitrary arguments are these.
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PLEASE HELP I WILL GIVE BRAINLIEST FOR THE CORRECT ANSWER!!
Use mathematical induction to prove that 1 + 2 + 3 + … + n = (1/2) n (a1 + an).
Use the result to find the sum of 1 + 2 + 3 + ... + 500.
Answers
Answer:
See below for proof.
[tex]S_{500}=125250[/tex]
Step-by-step explanation:
Given arithmetic series:
1 + 2 + 3 + … + nTherefore:
[tex]S_n=1+2+3+...+(n-2)+(n-1)+n[/tex]
[tex]S_n=1+(1+1)+(1+2)+...+(1+n-3)+(1+n-2)+(1+n-1)[/tex]
[tex]S_n=1+(1+1)+(1+2(1))+...+(1+(n-3)(1))+(1+(n-2)(1))+(1+(n-1)(1))[/tex]
Let:
a = first term = 1d = common difference = 1n = nth termTherefore:
[tex]S_n=a+(a+d)+(a+2d)+...+(a+(n-3)d)+(a+(n-2)d)+(a+(n-1)d)[/tex]
Reverse the order:
[tex]S_n=(a+(n-1)d)+(a+(n-2)d)+(a+(n-3)d)+...+(a+2d)+(a+d)+a[/tex]
Add the two expressions for Sₙ:
[tex]2S_n=(2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d)[/tex]
Therefore, the term (2a + (n – 1)d) has been repeated n times:
[tex]2S_n=n(2a+(n-1)d)[/tex]
Divide both sides by 2:
[tex]S_n=\dfrac{1}{2}n(2a+(n-1)d)[/tex]
[tex]S_n=\dfrac{1}{2}n(a+a+(n-1)d)[/tex]
Replace a with a₁ (first term) and a + (n – 1)d with aₙ (last term):
[tex]S_n=\dfrac{1}{2}n(a_1+a_n)[/tex]
To find the sum of the series 1 + 2 + 3 + ... + 500, substitute the following values into the formula:
a₁ = 1aₙ = 500n = 500Therefore:
[tex]\implies S_{500}=\dfrac{1}{2}(500)(1+500)[/tex]
[tex]\implies S_{500}=250(501)[/tex]
[tex]\implies S_{500}=125250[/tex]
What is meant by rigid transformation?
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Which is not a test for similarity 1⃣ SSS 2⃣ SAS 3⃣ AAA 4⃣ ASA?
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Step-by-step explanation:
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Grading may be done using letters, numbers, percentages, points, or descriptions. The most prevalent grades are in the form of letters, and they usually correspond to a variety of percentages or points. Many people consider "A" and "B" grades to be "good." D and F grades are referred to as "poor grades." Commonly regarded as average is a "C."
We can say that
In Korea,
7th grades = 13 or 14 year old
10th grades = 16 or 17 year old ,
11th grade is = 17 or 18 and 12th grade is = 18 or 19.
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Work out the value of
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The following three shapes are based only on squares, semicircles,
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Assuming a 10% rate of orders that are incorrect. The accuracy rate seems to be satisfactory.
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Given: A study of the accuracy of fast food drive-through orders, one restaurant had 37 orders that were not accurate among 333 orders observed. Significance level-0.05
Explanation: Assuming a 10% rate of orders that are incorrect. The accuracy rate seems to be satisfactory.
Let a be the population proportion of the orders that were not accurate.
Then according to the claim we have,
[tex]H_{o} :[/tex] [tex]p[/tex] = [tex]0.10[/tex]
[tex]H{a} :p[/tex] ≠ [tex]0.10[/tex]
Since the alternative hypothesis is two-tailed so the hypothesis test is a two-tailed test.
For example,
n = 333
The proportion of the orders that were not accurate = [tex]p{'}[/tex][tex]=\frac{37}{333}[/tex] ≈[tex]0.112[/tex]
Test statistics for population proportion:-
Assuming a 10% rate of orders that are incorrect. The accuracy rate seems to be satisfactory.
Let a be the population proportion of the orders that were not accurate.
Then according to the claim we have,
[tex]H_{o} :p=0.10[/tex]
[tex]H_{a} :p[/tex] ≠ [tex]0.10[/tex]
Since the alternative hypothesis is two-tailed so the hypothesis test is a two-tailed test.
For example,
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The proportion of the orders that were not accurate = [tex]p^{'} =\frac{37}{333}[/tex] ≈ [tex]0.112[/tex]
Test statistics for population proportion:-
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can someone pls explain this, I'm stuck
Answers
Answer:
A = 6
B = 4
C = 8
D = 10
surface area ❀
leftmost rectangle : 8 x 4 = 32
middle rectangle : 10 x 4 = 40
rightmost rectangle : 6 x 4 = 24
top and bottom triangles : (8 x 6) = 48
total = 32 + 40 + 24 + 48 = 144cm²
Step-by-step explanation:
not sure if its correct. but i tried.
The solution is
The measure of A = 6 cm
The measure of B = 4 cm
The measure of C = 8 cm
The measure of D = 10 cm
b) The net surface area of the triangular prism = 144 cm²
What is the surface area of the prism?
The surface area of the prism is calculated as
The formula for the surface area of a prism is SA=2B+ph, where B, is the area of the base, p represents the perimeter of the base, and h stands for the height of the prism
Surface Area of the prism = 2B + ph
The area of the triangular prism is A = ph + ( 1/2 ) bh
Given data ,
Let the measure of the side A = 6 cm
The measure of side B = 4 cm
The measure of side C = 8 cm
The measure of side D = 10 cm
The surface area of the triangular prism is = 2B + ph
The surface area A = ( 6 x 8 ) + ( 6 x 4 ) + ( 10 x 4 ) + ( 8 x 4 )
The surface area A = 48 + 24 + 32 + 40
The surface area A = 144 cm²
Therefore , the value of A is 144 cm²
Hence , the area of the triangular prism is 144 cm²
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The graph of a slope of 3 is a straight line.
What is a slope?It is possible to determine a line's direction and steepness by looking at its slope. Finding the slope of lines in a coordinate plane can aid in anticipating whether the lines are parallel, perpendicular, or none at all without actually using a compass. Any two distinct places along the line can be used to determine the slope of any line. When two different places on a line are considered, the slope of a line formula compares the "vertical change" to the "horizontal change" in each case. The difference between the line's y and x coordinate changes is known as the slope of the line.Here, let us graph a line with slope = 3Let us assume the co-ordinates
[tex](x _1, y_1) = (2,1)\\(x _2, y_2) = (5,10)\\\\\text{We know} ,\\\\m =\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\\\m = \frac{10-1}{5-2} \\[/tex]
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m = 3 (Refer image for graph)
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For what values of k the roots of the quadratic equation 9x 2 )+ 3kx 4 0 are equal?
Answers
The value of k is ±4 so that the roots of the quadratic equation 9x² + 3kx +4 are equal
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Quadratic equation, ax2 + bx + c = 0 has equal roots if (b2 - 4ac) = 0
Hence in quadratic equation, 9x2 + 3kx +4 = 0 has equal roots if
(3k)² - 4×9×4 = 0
Hence 9k² = 144 or k² = 16 or k = ±4
In order for the roots of the quadratic equation 9x2 + 3kx + 4 to be equal, the value of k must be 4.
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7 divided by three-eighths
Answers
Answer:
18 2/3
Step-by-step explanation:
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The correct solution of the given function is g'(t)=20 for some t in the interval. The correct option is (B).
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Given that g(10)= 220, g(0)=20
Explanation: A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.
For a given function, y = F(x), if the value of y is increasing on increasing the value of x, then the function is known as an increasing function and if the value of y is decreasing on increasing the value of x, then the function is known as a decreasing function.
So putting values in condition of differentiation for increasing function we get:
=[tex]\frac{g(10)-g(0)}{10-0}[/tex]=[tex]\frac{220-20}{10}[/tex]=[tex]\frac{200}{10}[/tex]=20
The correct solution for given function is g'(t) = 20 for some t in the interval.
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Help help meeeeee help meee please asap please !!??
Answers
Step-by-step explanation:
12 boxes * 8 candies = 96 total
divided among 30 kids:
each kid gets 3 pieces with (96 - (30 *3)) = 6 remaining.
What is the nature of the roots of the quadratic equation √ 3x² 10x 7 √ 3 0?
Answers
The nature of the roots of a given quadratic equation √3x²+10x+7√3=0 is real and distinct because of the value of b²-4ac>0.
where a, b, and c are the coefficients in the standard form of the quadratic equation ax²+bx+c=0
Given quadratic equation is √3x²+10x+7√3=0 on comparing with the standard form of a quadratic equation ax²+bx+c=0. Hence a=√3,b=10,
and c=7√3
Now b²-4ac=10²-4×√3×7√3=100-28×3=16>0.
here the value of b²-4ac is greater than 0.
We know if the value of b²-4ac then the roots of the given equations are real and distinct.
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select all the true: ninety-six golf balls were picked up at the driving range and placed into two buckets. one bucket has twenty-eight more balls then the other bucket. how many gold balls are in each bucket?
Answers
The number of golf balls that are in each bucket would be = bucket X = 62, bucket y = 34
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The number of golf balls that are picked up at driving range = 96
The number of buckets present= 2( X and y)
Bucket X = 28 + a
Bucket y = a
To calculate for each bucket first find a.
But 96 = 28+ a+ a
96 = 28 + 2a
2a = 96- 28
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Answer:
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find the number of students who study
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2. Spanish only
3. none of the languages
4. at least one language
Answers
Answer:
1. 38 = 88 - (10 + x + 25)
38 = 88 - 35 - x
x = 53 - 38
x = 15
There are 15 students who study 3 languages2. Spanish only = 65 - (5 + 15 + 10)
= 65 - 30
= 35
There are 35 students who study Spanish only3. x = 180 - (65 + 25 + 38 + 15 + 5 + 10 + 35)
x = 180 - 193
x = -13
There are 13 students who studied none4. 65 + 38 + 35
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There are 138 students who studied one language onlyStep-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-Purchasing 5 bananas and a $3.50 pineapple costs the same as purchasing 7 bananas and a pear that costs 90 cents. What is the cost of each banana?
Answers
Answer:
$0.52
Step-by-step explanation:
the pear is 0.90 and the total for the 5 bananas and one pineapple is 3.50
3.50-0.90=2.6 2.6/5=$0.52
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Answers
After rotating 90 degrees anticlockwise, B's new coordinates are (y, -x).
By switching the x and y coordinates and negating the new x coordinate, we can rotate a point (x, y) 90 degrees clockwise around the origin.
After rotating the triangle 90 degrees anticlockwise around the origin, note the new coordinates of B.
Let (x, y) be point B's coordinates.
Fllowing a 90° anticlockwise turn:
B'x, the modified x coordinate, will now be y.
B'y, the modified y coordinate, will now be -x.
Therefore, B's new coordinates are (y, -x).
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At a real estate agency, an agent sold a house for $375,000. The commission rate is 6.5% for the real estate agency and the commission rate for the agent is 25% of the amount the real estate agency gets. How much did the agency make on the house? How much did the agent earn in commission?
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The real estate agency made $24,375 on the house. The agent earned $6,093.75 in commission.
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In this case, the ordered list of numbers is: 2, 3, 3, 4, 5, 6, 7.
The median of this list is 4.
The mathematical median is what?Sorting all the data points and selecting the central one are required to determine the middle number (or if there are two middle numbers, taking the mean of those two numbers). Example: The number 4 is in the middle when the numbers are grouped (1, 4, 7), making it the median of the numbers (4, 1 and 7).
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What is set in math 6?
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Set is a collection of unique objects. The objects are unique, that is there is no repetition in set, making it distinct from a group. A set is well defined.
In mathematics, the objects belonging to a set are called the elements of that particular set.
These elements will be listed in a pair of curly brackets. For example, A is a set of first ten natural numbers, that is its elements are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. According to the notation
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What is the solution to the system of linear equations?
Answers
Answer:
(0,2)
Step-by-step explanation:
The equations must be derived from the graph:
First, find the slope of f(x):
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex] let [tex](x_1,y_1)=(-3,3)[/tex] and [tex](x_2,y_2)=(3,1)[/tex]
[tex]m=\frac{(1)-(3)}{(3)-(-3)}\\m=\frac{-2}{6}\\m=-\frac{1}{3}[/tex]
Then use [tex]y-y_1=m(x-x_1)[/tex]
[tex]y-3=-\frac{1}{3}[x-(-3)]\\y-3=-\frac{1}{3}x-1\\y=-\frac{1}{3}x+2\\f(x)=-\frac{1}{3}x+2[/tex]
For g(x), let:
[tex](x_1,y_1)=(-3,0)\\(x_2,y_2)=(0,2)[/tex]
[tex]m=\frac{(2)-(0)}{(0)-(-3)}\\m=\frac{2}{3}[/tex]
[tex]y-0=\frac{2}{3}[x-(-3)]\\y=\frac{2}{3}x+2\\g(x)=\frac{2}{3}x+2[/tex]
Now, to solve for x, allow the two expressions written in terms of x equal each other.
[tex]-\frac{1}{3}x+2=\frac{2}{3}x+2\\(-\frac{1}{3}x+2)+\frac{1}{3}x=(\frac{2}{3}x+2)+\frac{1}{3}x\\2=x+2\\(2)-2=(x+2)-2\\x=0[/tex]
To solve for y, substitute 0 for x in f(x):
[tex]y=-\frac{1}{3}x+2\\y=-\frac{1}{3}(0)+2\\y=0+2\\y=2[/tex]
So, the solution to this system of equations would be the ordered pair (0,2). This solution is seen on the graph as the intersection of the two lines.
1. A study of the accuracy of fast food drive-through orders, one restaurant had 37 orders that were not accurate among 333 orders observed. Use a 0.05 significance level to test the claim that the rate of inaccurate orders is equal to 10%.
Answers
Assuming a 10% rate of orders that are incorrect. The accuracy rate seems to be satisfactory.
Let a be the population proportion of the orders that were not accurate.
Then according to the claim we have,
[tex]H_{o} : p=0.10\\H_{a} : p\neq 0.10[/tex]
Since the alternative hypothesis is two-tailed so the hypothesis test is a two-tailed test.
For example,
n = 333
The proportion of the orders that were not accurate = [tex]p'=\frac{37}{333}[/tex] ≈ [tex]0.112[/tex]
Test statistics for population proportion:-
[tex]z=\frac{p'-p}{\sqrt{\frac{p(1-p)}{n} } }\\= \frac{0.112-0.10}{\sqrt{\frac{0.10(0.90)}{333} } }[/tex] ≈[tex]0.75[/tex]
By using the standard normal distribution table,
The p-value : [tex]2(z > 0.75)= 2.55[/tex]
Since the p-value is greater than the significance level (0.05), so we do not reject the null hypothesis.
Hence, we conclude that the accuracy rate appears to be acceptable.
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What is the nature of the roots of the quadratic equation 9 x2 6x =- 9?
Answers
The nature of the roots of the quadratic equation x² - 6x + 9 = 0 are real and equal
Given that :
The quadratic equation x² - 6x + 9 = 0
To find :
The nature of the roots of the quadratic equation x² - 6x + 9 = 0
x² - 6x + 9 = 0
Find discriminant of the equation
Comparing the given Quadratic equation x² - 6x + 9 = 0 with the general equation ax² + bx + c = 0 we get
a = 1 , b = - 6 , c = 9
The discriminant of the quadratic equation
= b² - 4ac
= (-6)² - 4* 1*9
=36 - 36
=0
Find nature of the roots of the quadratic equation
Since the discriminant of the quadratic equation = D = 0
Hence the roots of the quadratic equation x² - 6x + 9 = 0 are real and equal
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Given the quantity of 3 x squared plus x minus 4, all over x minus 1, what are the domain and range? domain is x is all real numbers where x does not equal negative 1 comma range is y is all real numbers where y does not equal 1 domain is x is all real numbers where x does not equal 1 comma range is y is all real numbers where y does not equal negative 1 domain is x is all real numbers where x does not equal negative 1 comma range is y is all real numbers where y does not equal negative 7 domain is x is all real numbers where x does not equal 1 comma range is y is all real numbers where y does not equal 7
Answers
The domain and range are equal to: D. domain is x is all real numbers where x does not equal 1 comma range is y is all real numbers where y does not equal 7.
What is a domain?In Mathematics, a domain can be defined as the set of all real numbers for which a particular function is defined. This ultimately implies that, a domain represents the input values (x-values) to a function.
What is a range?In Mathematics, a range can be defined as the set of all real numbers that connects with the elements of a domain. This ultimately implies that, a range simply refers to the set of all possible output numerical values (real numbers), which are shown on the y-coordinate (y-axis) of a graph.
Translating the word sentence into an algebraic expression (function), we have;
Function = (3x² + x - 4)/(x - 1)
Function = (3x² + 4x - 3x - 4)/(x - 1)
Function = [(3x + 4)(x - 1)]/(x - 1)
Function = 3x + 4
Note: The domain is all real numbers except one (1) because one (1) would make the function to become undefined.
When x = 1, the range of this function are all real numbers except;
Range = 3x + 4
Range = 3(1) + 4
Range = 7
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