Evaluate the double integral. 2y2 dA, D is the triangular region with vertices (0, 1), (1, 2), (4, 1) D
Answers
Answer:
[tex]\mathbf{\iint _D y^2 dA= \dfrac{22}{3}}[/tex]
Step-by-step explanation:
From the image attached below;
We need to calculate the limits of x and y to find the double integral
We will notice that y varies from 1 to 2
The line equation for (0,1),(1,2) is:
[tex]y-1 = \dfrac{2-1}{1-0}(x-0)[/tex]
y - 1 = x
The line equtaion for (1,2),(4,1) is:
[tex]y-2 = \dfrac{1-2}{4-1}(x-1) \\ \\ y-2 = -\dfrac{1}{3}(x-1)[/tex]
-3(y-2) = (x -1)
-3y + 6 = x - 1
-x = 3y - 6 - 1
-x = 3y - 7
x = -3y + 7
This implies that x varies from y - 1 to -3y + 7
Now, the region D = {(x,y) | 1 ≤ y ≤ 2, y - 1 ≤ x ≤ -3y + 7}
The double integral can now be calculated as:
[tex]\iint _D y^2 dA= \int ^2_1 \int ^{-3y +7}_{y-1} \ 2y ^2 \ dx \ dy[/tex]
[tex]\iint _D y^2 dA= \int ^2_1 \bigg[ 2xy ^2 \bigg]^{-3y+7}_{y-1} \ dy[/tex]
[tex]\iint _D y^2 dA= \int ^2_1 \bigg[2(-3y+7)y^2-2(y-1)y^2 \bigg ] \ dy[/tex]
[tex]\iint _D y^2 dA= \int ^2_1 \bigg[-6y^3 +14y^2 -2y^3 +2y^2 \bigg ] \ dy[/tex]
[tex]\iint _D y^2 dA= \int ^2_1 \bigg[-8y^3 +16y^2 \bigg ] \ dy[/tex]
[tex]\iint _D y^2 dA= \bigg[-8(\dfrac{y^4}{4}) +16(\dfrac{y^3}{3})\bigg ] ^2_1[/tex]
[tex]\iint _D y^2 dA= \bigg[-8(\dfrac{16}{4}-\dfrac{1}{4}) +16(\dfrac{8}{3}-\dfrac{1}{3})\bigg ][/tex]
[tex]\iint _D y^2 dA= \bigg[-8(\dfrac{15}{4}) +16(\dfrac{7}{3})\bigg ][/tex]
[tex]\iint _D y^2 dA= -30 + \dfrac{112}{3}[/tex]
[tex]\iint _D y^2 dA= \dfrac{-90+112}{3}[/tex]
[tex]\mathbf{\iint _D y^2 dA= \dfrac{22}{3}}[/tex]
To answer this question, we need, first get the equations for the lines that enclosed the surface, and integrate according to the limits these equations give.
The solution is:
A = 22/3 square units
Let´s call points:
P ( 0 , 1 ) Q ( 1 , 2 ) and R ( 4 , 1 )
The equation for the line between P and R is:
y = 1
The equation for the line between P and Q is:
Slope-intercept equation is y = m×x + b
The slope m₁ = ( 2 - 1 ) / ( 1 - 0 ) m₁ = 1
and the line passes over the point x = 0 y = 1 ; then
1 = 0 +b b = 1
y = x + 1 ⇒ x = y - 1
The equation for the line between Q and R is:
m₂ = ( 1 - 2 ) / ( 4 - 1) m₂ = - 1/3
y = ( -1/3)× x + b
when x = 1 y = 2
2 = ( - 1/3)×(1) + b
2 + 1/3 = b
b = 7/3
y = - (x/3) + 7/3 ⇒ x = 7 - 3×y
The double integral becomes:
A = 2×∫∫ y² dx dy ⇒ A = 2 ×∫₁² y²dy ∫dx | (y - 1 ) y ( 7 - 3y)
A = 2 ×∫₁² y²dy × x | ( y - 1 ) y ( 7 - 3y)
A = 2 ×∫₁² y²dy × [ 7 - 3×y - ( y - 1 )]
A = 2 ×∫₁² y²dy × (8 - 4×y ) ⇒ A = 2 ×∫₁² (8×y² - 4×y³ ) dy
A = 2 × [ (8/3)×y³ - y⁴ | ₁²
A = 2 × [ 64/3 - 16 - (8/3) + 1 ]
A = 2 × ( 56/3 - 15 )
A = 2 × ( 56 - 45 /3)
A = 2 × 11/3
A = 22/3 square units
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Step-by-step explanation:
Find the length of side x in simplest radical form with a rational denominator.
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Step-by-step explanation:
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Log_4 (x-5)=3
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4x-20=3
+20 +20
——————-
4x=23
— —
4. 4
X= 5 3/4 WHICH ALSO EQUALS 5.75
Joe’s garden is the shape of a hexagon. The measures of 5 of the angles are: 160°, 90°, 60°, 160°, and 80°. What is the measure of the remaining angle? *
Answers
Answer:
1800000000000000000
The solution is 170°, is the measure of the remaining angle.
What is addition?Addition is a way of combining things and counting them together as one large group. ... Addition in math is a process of combining two or more numbers.
here, we have,
given that,
The measures of 5 of the interior angles of a hexagon are: 160°, 90°, 60°, 160°, and 80°.
so, we have,
Sum of all the six interior angles of a hexagon is 720°
now, we get,
The sum of 5 interior angles is = 160°+ 90°+ 60°+ 160°+ 80° degrees
= 550
The missing 6th angle is = 720° - 550degrees
=170
So the remaining angle of hexagon is 170 degrees.
Therefore, the answer is 170°.
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The average height of a male ROTC cadet is 70 in. Suppose that 10 percent of the male cadets are taller than 72 in. Assuming that the heights of the cadets are normally distributed, what is the standard deviation
Answers
Answer:
1.56
Step-by-step explanation:
Given that :
Average height (μ) = 70 in
10% are taller than 72 in ; the standard deviation Given a normal distribution :
P(x > Z) = 0.10
Using a Z probability calculator, the Z value corresponding to P(x > Z) = 0.10 is 1.282
Z = 1.282
Using the Z relation:
Z = (x - μ) / σ
σ = standard deviation x = 72
1.282 = (72 - 70) / σ
1.282σ = 2
σ = 2 / 1.282
σ = 1.5600624
σ = 1.56
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Answer:
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Step-by-step explanation:
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Which coordinate grid shows Point A at (0.25, −1.75)?
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Answer:
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Step-by-step explanation:
you go out .25 on the x axis and down 1.75
Answer:
im pretty sure it's a
Step-by-step explanation:
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Mrs. Milavec had to take out a loan, to pay for Oreos. She borrowed $2,000 at 8% for 2 years. What is the interest Mrs. Milavec will have to pay?
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Answer:
$320.00
Step-by-step explanation:
(7-b)+(3b+2)+(7-b)+(3b+2)
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Answer:
4b+18
Step-by-step explanation:
Answer:
4b+18
Step-by-step explanation:
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Li can walk 2 miles in 45 minutes. At this rate how far can she walk in 135 minutes?
Answers
Answer:
6 miles
Step-by-step explanation:
SO first divide 135 by 45
You’ll get 3
Now using 3 we multiply that by 2
youll get 6 miles
135 minutes/ 45 minutes = 3
2 miles x 3 = 6
They can walk 6 miles.
Find the value of the variable in the parallelogram
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Select the correct answer.
The probability of event Als x, and the probability of event Bis y. If the two events are independent, which of these conditions must be true?
O A P(BA) = y
OB. P(
AB) = y
OC. P(BA) = x
OD. P(A and B) = x + y
OE
P(A and B)
P(A)
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The condition that must be true about the probabilities is P(A and B) = xy
The probabilities of the events A and B are given as:
P(A) = x
P(B) = y
Given that events A and B are independent events, then we have the following equation
P(A and B) = P(A) * P(B)
The equation becomes
P(A and B) = x * y
P(A and B) = xy
Hence, the condition that must be true is P(A and B) = xy
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I need to find the identity of this equation for pre calc
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Expand the left side using the angle sum identity for sine:
sin(x + π/2) = sin(x) cos(π/2) + cos(x) sin(π/2)
cos(π/2) = 0 and sin(π/2) = 1, so the right side reduces to
sin(x + π/2) = cos(x)
as required.
Somebody please help me asap I need help noww
Answers
Answer:
x = -7
Step-by-step explanation:
sum of interior angles in any triangle 180°
84 + 59 + 51 = 194
180 - 194 = -14
14 ÷ 2 = -7 x = -7
check
-7 + 59 = 52
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THIS IS URGENT
Is 6 less than or equal to 6
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Graph the system of equations given below on the provided graph.
2x- 3y = -18
3x+ y = -5
Then, use the Mark Feature tool to plot the solution to the system.
Answers
1st equation (blue line) is 2/3x + 6
2nd equation (red line) is -5 - 3x
they intersect at (-3,4)
The intersection of both the graphs gives the solution to the system of equations , The point of intersection is (- 3 , 4).
What is a System of Equation ?The set of equations that have common solution is called System of Equations.
The equations given in the question is
2x -3y = -18
3x +y = -5
The graph has been plotted and attached with the answer ,
The intersection of both the graphs gives the solution to the system of equations.
The point of intersection is (- 3 , 4) , x = -3 and y = 4.
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chloe has three dogs. Corky weighs twice as much as muffin.Rufus weighs twice as much as corky. the total weight of the dog is 77 pounds. how much does each dog weigh?
Answers
Answer:
Corky = c = 22 pounds
Muffin = m = 11 pounds
Rufus = r = 44 pounds
Step-by-step explanation:
Let the weight of each dog be represented as
Corky = c
Muffin = m
Rufus = r
The total weight of the dog is 77 pounds.
c + m + r = 77
Corky weighs twice as much as muffin.
c = 2m
m = c/2
Rufus weighs twice as much as corky.
r = 2c
Hence
c + c/2 + 2c = 77
3.5c = 77
c = 77/3.5
c = 22 pounds
m = c/2
m = 22/2
m = 11 pounds
r = 2c
r = 2 × 22
r = 44 pounds
Each dog weighs:
Corky = c = 22 pounds
Muffin = m = 11 pounds
Rufus = r = 44 pounds
Use the transformations on the function f(x) = x| to select the correct graph of
Answers
Answer: A
Step-by-step explanation:
The correct graph to the parent function after the transformation is shown in the picture.
What is a function?
It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
We have a parent function:
y = |x|
Applying first transformation:
Multiply by 1/3 the graph of the function will stretch
[tex]\rm y = \dfrac{1}{3} |x|[/tex]
Second transformation:
Add 1 to the function, the graph will shift upside by 1 units.
[tex]\rm y = \dfrac{1}{3} |x|+1[/tex]
Thus, the correct graph to the parent function after the transformation is shown in the picture.
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You are downsizing. Your new residence will have much less closet space than does your current home. You have 12 pair of long pants, 6 pair of short pants and 17 t-shirts. You plan to give some away.
a. In how many ways can a random selection of 9 pair of long pants, 4 pair of short pants and 13 t-shirts be arranged?
b. You randomly select 25 of the items of clothing to give away. What is the probability you select 10 pair of long pants, 3 pair of short pants and 12 t-shirts?
Answers
Answer:
a) there are 7,854,000 number of ways the random selections can be arranged
b) the required probability is 0.0445
Step-by-step explanation:
Given that;
you have 12 pair of long pants, 6 pair of short pant and 17 t-shirt which in total;
Total = 12 + 6 + 17 = 35
a) number of ways a random selection of 9 pair of long pants, 4 pair of short pants and 13 t-shirts can be arranged;
⇒ [ 12 × [ 6 × [ 17
9 ] 4 ] 13 ]
= (12! / (9! 3!)) (6! / (4! 2!)) (17! / (13! 4!))
= (220) (15) (2380)
= 7,854,000
Therefore there are 7,854,000 number of ways the random selections can be arranged
b)
Number of ways of selecting 25 item from 35 items;
⇒ [ 35
25 ]
=
(35! / (25! 10!) = 183.579,396
now number of ways to select 10 pair of long pants, 3 pairs of short pantts and 12 t-shirt;
⇒ [ 12 × [ 6 × [ 17
10 ] 3 ] 12 ]
= (12! / (10! 2!)) (6! / (3! 3!)) (17! / (12! 5!))
= (66) (20) (6188)
= 8,168,160
So, Required probability = 8,168,160 / 183,579,396
= 0.0445
So the required probability is 0.0445
Question Attached! ^w^
Answers
Answer
thats right
Step-by-step explanation:
Answer:
A. p+18=41; p = 23
Step-by-step explanation:
Number of hardcover books 18, + number of paperback book add up to make a total of 41, which makes the equation, p+18=41, to solve this, you want to have p by itself, simply subtract 18 from both sides, and you get p = 23.
Hope this helped :)
solve -36 4/9-(-10 2/9)-(18 2/9)
Answers
Answer:
negative 400 over 9−
400
9
=negative 44 and 4 over 9− 44
4
9
≈negative 44.444−44.444
Step-by-step explanation:
19 more than 9 times a number is less than or equal to 163. Which of the following represents the solution set of this problem?
Answers
Answer:
x ≤ 6
Step-by-step explanation:
Let the unknown number be x;
9 times the number is expressed as 9x
9 more than 9 times a number is expressed as 9x + 9
9 more than 9 times a number is less than or equal to 163 is then expressed as;
9x + 9≤ 63
≤ is the less than or equal to sign
Find the solution;
9x ≤ 63 - 9
9x≤54
x ≤ 54/9
x≤ 6
Hence the required solution is x ≤ 6
The equation 2(x + 32) = 96 models this situation. Which choice shows the sequence of operations needed to solve this problem, using both arithmetic and algebra?
Answers
2x +64 =96
2x = 96 - 64
2x= 32
x= 16
Which of the numbers below is greater than zero? Select all that apply
A) 0.5
B) 1.25
C) - 5/2
D) 2
E) 1/4
Answers
Answer:
0.5, 1.25, 2 and 1/4
Step-by-step explanation:
Answer:
a,b,d
Step-by-step explanation:
all are more then zero
Select all of the quadratic expressions in vertex form.
(x-1)2+10
(x-5)(x-4)
(x-5)2
O x2-4x+4
Ox(x - 4)
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x^2-4x+4
Solve the system using any method.
y=3.50-2
y=-1.9x+4.48
The system has no solution, ().
Answers
Answer:
Did you mean 3.50x? If so, It would be (1.2, 2.2). If you meant just 3.50, it would be (1.56, 1.5)
Hope this helps!
Dominic has $600 in a savings account at the beginning of the summer. He wants to have at least $100 in the account by the end of the summer. He withdraws $25 each week for food, clothes, and movie tickets. Write an inequality that represents Dominic's situation. How many weeks can Keith withdraw money from his account
Answers
at what x coordinate would a line whose equation is y=2x-3 intersect a perpendicular line whose y intercept is 17
Answers
Answer:
Step-by-step explanation:hj
Using linear function concepts, it is found that these lines intersect at x = 8.
The equation of a line is given by:
[tex]y = mx + b[/tex]
In which:
m is the slope, which is the rate of change.b is the y-intercept, which is the value of y when x = 0.If two lines are perpendicular, the multiplication of their slopes is -1.Line perpendicular to y = 2x - 3.
Multiplication of the slopes is -1, thus:
[tex]2m = -1[/tex]
[tex]m = -\frac{1}{2}[/tex]
y-intercept of 17 means that b = 17, thus, the equation of the line is:
[tex]y_p = -\frac{1}{2}x + 17[/tex]
They intersect at the value of x for which:
[tex]y = y_p[/tex]
Then
[tex]2x - 3 = -\frac{1}{2}x + 17[/tex]
[tex]2x + \frac{1}{2}x = 20[/tex]
[tex]\frac{5x}{2} = 20[/tex]
[tex]5x = 40[/tex]
[tex]x = \frac{40}{5}[/tex]
[tex]x = 8[/tex]
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