The iterated integral by converting to polar coordinates [tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] is 7a^4/12.
In the given question, we have to evaluate the iterated integral by converting to polar coordinates.
The given integral is
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex]
Let polar coordinate.
x = rcosθ, y = rsinθ, x^2 + y^2 = r^2
So dxdy = rdrdθ
r from 0 to a and θ from π/2 to π
Now the integral is
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]\int_{\pi/2}^{\pi}\int_{0}^{a}7(r\cos\theta)^2(r\sin\theta)drd\theta[/tex]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]\int_{\pi/2}^{\pi}\int_{0}^{a}7(r^2\cos^2\theta)(r\sin\theta)drd\theta[/tex]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]7\int_{\pi/2}^{\pi}\int_{0}^{a}r^3\cos^2\theta\sin\theta drd\theta[/tex]
Now integrating the first integral
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]7\int_{\pi/2}^{\pi}\left[\frac{r^4}{4}\right]_{0}^{a}\cos^2\theta\sin\theta d\theta[/tex]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]7\int_{\pi/2}^{\pi}\left[\frac{a^4}{4}-\frac{0^4}{4}\right]\cos^2\theta\sin\theta d\theta[/tex]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]\frac{7a^4}{4}\int_{\pi/2}^{\pi}\cos^2\theta\sin\theta d\theta[/tex]
Let cosθ = t
So -sinθ = dt/dθ
So sinθdθ = -dt
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]-\frac{7a^4}{4}\int_{\pi/2}^{\pi}t^2dt[/tex]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]-\frac{7a^4}{4}\left[\frac{t^3}{3}\right]_{\pi/2}^{\pi}[/tex]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]-\frac{7a^4}{4}\left[\frac{\cos^3\theta}{3}\right]_{\pi/2}^{\pi}[/tex]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]-\frac{7a^4}{12}\left[\cos^3\pi-\cos^3(\pi/2)\right][/tex]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = -7a^4/12[(-1)^3 - 0]
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = 7a^4/12
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The complete question is:
Evaluate the iterated integral by converting to polar coordinates.
[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex]
It’s a hot summer day. Darius is babysitting and has promised his sweet little sister that he will fill her wading pool. He plans to soak his feet in the water as he sips a cool soda and watch little sister as she plays in the water. Sounds great, right? All he needs to do is get the pool filled. Unfortunately, little sister is impatient and starting to whine. Darius has a hose ready in the backyard to fill the pool, but he decides he will also drag the hose from the front-yard, so the two hoses can work together to fill the pool faster. Nobody wants their darling little sister to be unhappy!
Darius knows from experience that if he only uses one hose at a time, the front-yard hose takes 10 minutes longer to the fill the pool than the back-yard hose. With the two hoses working together, the pool is filled in 12 minutes. Once Darius is comfortable with his toes dangling in the water, he starts to wonder how long it would have taken to fill the pool if he had only used one of the hoses.
Darius thinks, “Hmmm. Usually when I think about how long it takes to do something I have to think about the rate it is going.”
5. Write a function, R(x), that models the combined rate of the two hoses in terms of the time that it takes for the backyard hose to fill the pool alone.
6.a. What is the entire domain of this function, if the context is not considered?
6.b. What is the domain of this function in the context of filling Darius’ pool? Explain your thinking after you submit.
7. Graph R(x). Label all important features and describe them.
Answer: I can’t help you
Step-by-step explanation:
Seems to me like a lot of little boys have the name 'Jack' lately. Let's assume it's one in every ten.
What's the probability that my daughter will have at least one Jack on her co-ed soccer team of 12
players, 6 of whom are boys?
O 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10
O 1/10 x 1/10 x 1/10 x 1/10 x 1/10 x 1/10
O 1-(9/10 x 9/10 x 9/10 x 9/10 x 9/10 x 9/10)
O 1/6+1/6 + 1/6 + 1/6 + 1/6 + 1/6
O 6/12 x 5/11 x 4/10 x 3/9 x 2/8 x 1/7
The probability of at least one boy named Jack is (c) 1 - (9/10 x 9/10 x 9/10 x 9/10 x 9/10 x 9/10)
How to determine the probability of at least one boy named JackFrom the question, we have the following parameters that can be used in our computation:
P(Jack) = 1/10
Number of boys = 6
Using the complement rule of probabilities, we have
P(At least one Jack) = 1 - P(No jack)
Where
P(No jack) = (1 - P(Jack))⁶
So, we have
P(No jack) = (1 - 1/10)⁶
Evaluate
P(No jack) = (9/10)⁶
Substitute the known values in the above equation, so, we have the following representation
P(At least one Jack) = 1 - (9/10)⁶
Evaluate
P(At least one Jack) = 1 - (9/10 x 9/10 x 9/10 x 9/10 x 9/10 x 9/10)
Hence, the probability is (c) 1 - (9/10 x 9/10 x 9/10 x 9/10 x 9/10 x 9/10)
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finish the question
the value of decreasing of 15% from £1200 is equal to....
Answer: The final value is £1,020
Step-by-step explanation: In order to find the final value, the first step in the process is to multiply 1,200 by 0.15 in order to find out what 15% of 1,200 is. After doing obtaining the answer, which is 180, subtract that from the original answer in order to find out what the final value will be.
The final value is then equal to £1,020
pls do this asap its important
Answer:
3.1 mm
Step-by-step explanation:
as 1mmx3.14=3.14mm
And then correct it to 1 decimal place =3.1
The midpoint of AB is MI(-2, -6). If the coordinates of A are (1, -4), what are
the coordinates of B?
Tyler is 8 years old. His sister Olivia is 4 years less than twice his age.
Write a numerical expression for Olivia's age in the box below.
Answer: Olivia is 12 years old.
Step-by-step explanation
Tyler = T
Olivia = O
T=8
O=T*2-4
O=8*2-4
O=16-4
O=12
Mr. Smith runs 2.7 miles every day of the week. How many miles does he run in one year? : *
Based on the above, Mr. Smith runs approximately 985.5 miles in one year
What is the runTo calculate the number of miles Mr. Smith runs in one year, we can multiply the distance he runs each day by the number of days in a year.
Number of miles Mr. Smith runs in one day: 2.7 miles
Number of days in a year: 365 (assuming a non-leap year)
S, one can calculate the total distance he runs in one year:
Total miles in one year = Distance per day * Number of days in a year
= 2.7 miles x 365 days
= 985.5 miles
Therefore, Mr. Smith runs about 985.5 miles in one year.
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for a generic a→b, what quantity, when graphed versus time, will yield a straight line?
A graph [tex]\frac{1}{A_{t}}[/tex] and time yields a straight line for a second order reaction. The positive slope of the straight line is rate constant k for second order.
For a second order reaction A→B.
The integrated rate law is
-d[n]/dt = k[A]^{r}
where, k = rate constant of second order reaction and
[A]^{r} = concentration of [A]
-[tex]\int^{A_{t}}_{A_{0}}\frac{d[A]}{[A]^{r}} = k\int^{t}_{t_{0}}dt[/tex]
where, [tex]A_{t}[/tex] = concentration of A at time t
[tex]A_{0}[/tex] = concentration of A at time t=0
[tex]\frac{1}{A_{t}}-\frac{1}{A_{0}}=kt[/tex]
Adding [tex]\frac{1}{A_{0}}[/tex]on both side, we get
[tex]\frac{1}{A_{t}}=kt+\frac{1}{A_{0}}[/tex]....................(1)
On comparing the equation 1 by the equation y = mx+c, we get that the plot of [tex]\frac{1}{A_{t}}[/tex] and plot of time t will be in the straight line with the positive slope k.
The plot is given below:
Hence, a graph [tex]\frac{1}{A_{t}}[/tex] and time yields a straight line for a second order reaction. The positive slope of the straight line is rate constant k for second order.
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The complete question is given below:
Which one of the following is arithmetic sequence? 15, 12, 9, 6 … 15, 30, 60, 120 … 1, 4, 9, 16 … 1, 8, 27, 64 …
An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed number, called the common difference, to the previous term. For example, the sequence 15, 12, 9, 6, … is an arithmetic sequence because each term is obtained by subtracting 3 from the previous term, which is the common difference. On the other hand, the sequence 15, 30, 60, 120, … is not an arithmetic sequence because the difference between each consecutive pair of terms is not constant. The same is true for the sequences 1, 4, 9, 16, … and 1, 8, 27, 64, …. Therefore, the only arithmetic sequence among the given options is 15, 12, 9, 6, ….
need help please thank you
Answer:
Step-by-step explanation:
They are trying to be tricky :DDD but we got this.
they give you the outside angle at H, but you know both of those two angle also add to 180
180= 131 +∠h
I'm noting that we're solving for the minor angle H of the triangle, the inside angle
49° = ∠h
we also , know that H and J are equal, b/c it's got the red marks , telling us the angles are the same.
use that same strategy to solve,
180 = 2*49 + p
180-98 = p
82 = p
Do you see where I get the equation from? ( 180 = 2*49 + p ) ?
Can someone help me answer this please?
Answer:
B
Step-by-step explanation:
When zero is our denominator the graph is undefined at that point. In the case of b, when x = 1 The denominator is o. That is why there is a striped line at x = -1
what is the coefficient of x in -3 / 7 x
Answer:5103
Step-by-step explanation:We only care about 7xy6, cause it's the only term involving x In your example y = −3, thus 7(−3)6x = 7 ⋅ 729x = 5103x The coefficient of x is 5103
Corey is making a wall-mounted bookshelf. It has 2 shelves, with a third piece of wood intersecting them. To support the shelves, Corey orders some L-brackets to put between the shelves
and the crosspiece. He measures angle 6. How many brackets should he order to have one for every angle that is congruent to angle 6? Explain.
O
2
34
5
6
7 8
A. 2; 27 is congruent to 26.
B. 2; 23 is congruent to 26.
C. 3; 27, 23 are both congruent to 26.
OD. 4; 27, 22, 23 are all congruent to 26.
k
Answer:
B / 2; <3 is congruent to <6
Step-by-step explanation:
To find a congruent angle you must find what can fit inside the giving number the best. If we compare all of the answer is wouldn't make sense, since 7 is greater than 6 it would be wrong. So, A is Incorrect.
C is about 3 and 7. C would fit but 7 wouldn't. So, C is incorrect.
D has both 2 and 3 but also have 4 and 7 which wouldn't match. So, D is incorrect.
( THIS IS WHAT I THINK IS IT)
Answer:
B
Step-by-step explanation:
3 2/5 divided by 4/5
Answer:
17/4
Step-by-step explanation:
3 2/5 divided by 4/5 =
(3 2/5) / (4/5) =
(3 + 2/5) / (4/5) =
(15/5 + 2/5) / (4/5) =
((15+2)/5) / (4/5) =
(17/5) / (4/5) =
5*(17/5) / (5*(4/5)) = ==> multiply 17/5 and 4/5 by 5 to remove fractions
17/4 ==> simplified form
Answer this quick please
The graph of g is the graph of f translated 3 units right. Option C is the answer
How determine how the graphs of f(x) = 2x + 1, and g(x) = 2(x + 3) + 1 compare?In geometry, a translation is a type of transformation that involves moving a geometric figure from one location to another without changing its size, shape, or orientation.
Remember, in the cartesian plane or graph, the movement in x-axis is either to the left (negative) or right (positive).
Since f(x) = 2x + 1, looking at g(x) = 2(x + 3) + 1, you will notice that 3 is added to the value x. This is a movement in the positive x-axis (right translation) and the translation is by 3 units.
Therefore, the graph of g is the graph of f translated 3 units right
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What is the next step for the construction
Answer:Step 6, put the compass on point L
Step-by-step explanation:
Making a small arc from point L will make a little "X" at the point where you should draw the line through.
Answer:
See the answer above
Step-by-step explanation:
Open your eyes and direct your attention to the answer above.
What place value is this
18 thousands x 10 = ?
(T/F) in slicing, if the end index specifies a position beyond the end of the string, python will use the length of the string instead.
True, If the end index for slicing specifies a location after the end of the alphabet, Python will use the list's length instead.
What is slicing?
In Python, there is a concept of string, list and tuples. Sometimes it is required to access part of those string, list and tuples. This is known as slicing.
In case of slicing, suppose the the end index specifies a position beyond the end of the list, Python will return to the default length of the string. So there will be no error if the end index specifies a position beyond the end of the list.
So the above sentence is a true sentence.
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it would take charlene and gina 18 minutes to organize charlene’s comic book collection if they work together. if gina works alone, it would take her 15 minutes longer than it would take charlene working alone. when x represents the number of minutes it takes charlene to organize the comic book collection working alone, the situation is represented by this equation: 1/x + 1/x + 15 = 1/18which value is a solution of the rational equation that must be discarded because of the context?a. x = 0 b. x = -9 c. x = -15 d. x = -30
The value that is a solution of the rational equation that must be discarded because of the context is B. x = 9.
How to illustrate the equation?From the information, it would take Charlene and Gina 18 minutes to organize charlene’s comic book collection if they work together. if gina works alone, it would take her 15 minutes longer than it would take charlene working alone.
The equation can be illustrated as:
1/x + 2/(x + 15) = 1 / 18
The objective is to determine the value which nullify the context. The option will be -9. This is illustrated as:
1 / -9 = 1 / -9 + 15 = 1 / 18
The correct option is B.
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This is do today Pleaseeee helpppp
Answer:
A
Step-by-step explanation:
It is A definitely I'm sure
Calculate the perimeter of the irregular shape please show ur work Bc I have too !!
The perimeter of the irregular shape would be 34 cm.
What is the perimeter of any shape?
In geometry, the perimeter of a shape is defined as the total length of its boundary. The perimeter of a shape is determined by adding the length of all the sides and edges enclosing the shape.
By using the definition of the perimeter we can compute,
Perimeter = 4 + 2 + 2 + 3 + 3 + 6 + 9 + 5
= 34 cm.
Hence, the perimeter of the irregular shape would be 34 cm.
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given the vector a⃗ =3.0i^+1.5j^, find a vector b⃗ that is perpendicular to a⃗ .
1.5i^- 3j^ is the perpendicular vector to the vector 3.0i^+1.5j^
What is perpendicular vector?Whenever there is a 90 degree angle between two vectors, they are said to be perpendicular. Two perpendicular vectors dot product is zero. and dot product is scaler product of 2 vectors.
Let b = xi^+ yj^ be the perpendicular vector to a vector
and we know that 2 vectors are perpendicular if thier dot product is 0
so it implies that
b.a(dot product) =0
=> (3i^+1.5j^) . (xi^+yj^)=0
=> 3xi^.i^+ 3y i^.j^ +1.5xj^.i^+1.5yj^j^=0
=>3x+1.5y=0 ----(i)
becaus i^.j^=0 and j^.i^0
=> x= 1.5 and y=-3 satisfies equation (i).
so the perpendicular vector is 1.5i^- 3j^
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benjamin works as a salesperson at an electronics store and sells phones and phone
accessories. benjamin earns a $10 commission for every phone he sells and a $4
commission for every accessory he sells. On a given day, made a total of
$180 in commission from selling a total of 27 phones and accessories. Graphically
solve a system of equations in order to determine the number of phones sold, x, and
the number of accessories sold, y.
y Click twice to plot each line. Click a line to delete it.
The required number of phones and accessories sold are x = 12, y = 5,
What are simultaneous linear equations?Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.
Here,
Let the number of phones sold be x and the number of accessories sold be y.
According to the question,
He sold a total of 27 phones and accessories,
x + y = 27 - - - - -(1)
And, made a total of $180 in commission,
10x + 4y = 180 - - - - -(2)
Graph both the equation,
From the graph, the solution of the system of equations is given as,
x = 12, y = 5,
Since, when two intersect each other, is the point of a solution of the system of equation.
Thus, the solutions gives the number of phones and accessories sold as x = 12, and y = 5.
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What is 2/5 of 15? help pls
What is 2/5 of 15?
Answer:
2/5 of 15 is 6.
Answer:
6
Step-by-step explanation:
The word of in math refers to multiplication.
We can convert 15 to a fraction as well so we can use them together (15/1).
[tex]\frac{2}{5}[/tex] x [tex]\frac{15}{1}[/tex]
We can then work diagnolly and reduce the 5 and 15 since they are both divisible by 5- this is also known as the GCF or Greatest Common Factor
[tex]\frac{2}{1}[/tex] x [tex]\frac{3}{1}[/tex]
This is the same as 2 x 3 since both numbers divided by 1 are just themselves.
2x3 = 6
please help will give brainliest
The polygon for the given data will be a square. The correct option is E.
What is congruency?In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other. The meaning of congruency is that the two shapes are similar as well as equal in size.
A square is a quadrilateral having all the sides equal and all the angles congruent to each other. The value of all the angles is 90 degrees and the sum of the angles of the square is 360 degrees.
The given data is that all the sides are congruent and all the angles are congruent so the given figure is square.
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a roof has a 5:12 pitch and the roof is 175" horizontally from the ridge to the edge , what is the height of the roof from the lowest point to the highest point?
When a roof has a pitch of 5 : 12 and the roof is 175'' horizontally the height of the roof from the lowest point to the highest point is solved to be 73''
How to determine the roof's heightThe idea of the slope is represented in the form 5: 12 when calculating the height of the roof.
This indicates that there is a 5 increase for every 12 units.
The rise is equal to 5, and the run is 12.
The vertical is the ascent, and the run is the horizontal.
It is possible to compare using the provided ratio as follows.
Let x represent the rise so that
x / 175" = rise / run = 5 / 12
x / 175" = 5 / 12
cross multiplying
175 * 5 = 12x
12x = 875
x = 875 /12
x = 72.92 ''
The height of the roof, measured from its lowest to its highest point, is calculated to be 73 inches.
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FILL IN THE BLANK. two specific types of ____ are e-commerce systems and point-of-sale (pos) systems.
Answer:
Order entry systems
Step-by-step explanation:
Two specific types of order entry systems are e-commerce systems and point-of-sale (pos) systems.
. about 19% of young adults have high blood pressure. consider the proportion of people with high blood pressure in random samples of 45 young adults. what is the standard deviation of this sample proportion?
The standard deviation of the sample proportion is 10.
In math standard deviation is defined as the statistical measure of diversity or variability in a data set.
Here we have given that about 19% of young adults have high blood pressure. consider the proportion of people with high blood pressure in random samples of 45 young adults.
And we need to find the the standard deviation of this sample proportion
Let us assume the expected population standard deviation to be 10, and employing t-distribution to estimate sample size, the study would require a sample size of: 44 to estimate a mean with 67% confidence and a precision of 1.5.
Which means that, if you select a random sample of 44 from a population, and determine the mean to be say y, you would be 67% confident that the mean in the population lies somewhere between y - 1.5 and y + 1.5.
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Beth is making gingerbread men. She uses 2 raisins for eyes and 3 raisins for buttons for each gingerbread man. She buys 4 boxes of raisins, each with 120 raisins in it. How many dozens of gingerbread men can she make?
Answer: 96
Step-by-step explanation: This is how to solve:
120 * 4
= 480
3 + 2 = 5
480 / 5
= 96
First, you need to multiply the boxes bought by the total amount inside the box. (120 * 4)
The, we need to find how many raisins she uses IN TOTAL for each gingerbread man. So, we need to add 3 + 2
This equals 5
So, 120 * 4 = 480
So, in total, she buys 480 raisins. So, to find how much gingerbread she can make, we need to divide 480 by 5. This would get you 96. I hope this helps ;)
show your work please
Answer:
Correct Answer is A
Step-by-step explanation:
5×14 = 70
using the addition law of indices
(x¹. x⁷) ( y⁰. y⁵)
Add the powers
Therefore: x^(1+7) = x⁸
y^(0+5) = y⁵
Answer: 70x⁸y⁵