Double integration in polar coordinates involves integrating a function over a two-dimensional region in the polar coordinate system. This is done by setting up a double integral in terms of r and θ, and integrating first with respect to r and then with respect to θ.
Double integration in polar coordinates involves integrating a function over a two-dimensional region in the polar coordinate system. The steps to double integrate a function in polar coordinates are as follows:
1. Determine the limits of integration for r and θ. These limits define the region over which the function will be integrated. Typically, the limits are determined by the boundaries of the region in the xy plane.
2. Write the function to be integrated in terms of r and θ. The function must be expressed in polar coordinates for integration in polar coordinates.
3. Set up the double integral by writing the function in polar coordinates, multiplying by the appropriate factors of r and integrating with respect to r first and then θ.
4. Integrate the function with respect to r, using the limits of integration for r determined in step 1.
5. Integrate the result from step 4 with respect to θ, using the limits of integration for θ determined in step 1.
The general form of a double integral in polar coordinates is:
∫∫f(r, θ)r dr dθ
where f(r, θ) is the function to be integrated, and r and θ are the polar coordinates.
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Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice. (a) How many sample points are possible?
(b) What is the probability of obtaining a value of 7?
(c) What is the probability of obtaining a value of 8 or greater?
a) There are 36 sample points that are possible.
b) The probability of obtaining a value of 7 is 1/12.
c) The probability of obtaining a value of 8 or greater is 1/6.
(a) When rolling a pair of dice, each die has six possible outcomes: 1, 2, 3, 4, 5, or 6. The possible outcomes for the pair of dice can be represented by the set of all ordered pairs (a, b), where a and b represent the outcomes of the first and second die, respectively.
Since each die has six possible outcomes, there are a total of 6 x 6 = 36 possible outcomes for rolling a pair of dice.
(b) To obtain a sum of 7, we must have one die showing 1 and the other die showing 6, or one die showing 2 and the other die showing 5, or one die showing 3 and the other die showing 4, for a total of three possible outcomes.
Since each die has six possible outcomes, there are a total of 6 x 6 = 36 possible outcomes. Therefore, the probability of obtaining a sum of 7 is 3/36, or 1/12.
(c) We must have one die showing 2 and the other die showing 6, or one die showing 3 and the other die showing 5, or one die showing 4 and the other die showing 4, or one die showing 4 and the other die showing 5, or one die showing 5 and the other die showing 3, or one die showing 6 and the other die showing 2, for a total of 5 + 1 = 6 possible outcomes.
Therefore, the probability of obtaining a sum of 8 or greater is 6/36, or 1/6.
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The rectangular floor of a classroom is 22 feet in length and 24 feet in width. A scale drawing of the floor has a length of 11 inches. What is the perimeter, in inches, of the floor in the scale drawing?
The scale floor's perimeter is 34 inches.
What is meant by perimeter?The overall length of a form's boundary is referred to as the perimeter of that shape. Adding the lengths of all the sides and edges that surround a form yields its perimeter. It is calculated using linear length units such centimeters, meters, inches, and feet. The perimeter is the space surrounding a shape's edge. Get the perimeter of various forms by multiplying their side lengths.The circumference of an object is known as its perimeter. For instance, your home has a yard that is enclosed. The length of the fence is the perimeter. The length of your fence is 200 feet if the yard is 50 feet by 50 feet.To solve, we can set up fractions:
21/7 = 30/x
Cross multiply and you get:
21x = 210
Divide each side by 21:
x = 10
You can apply the following equation since you're attempting to determine the floor's perimeter:
2w + 2l = p
Substitute to solve:
2 ( width = 10 )
2 ( length = 7 )
2 (7) + 2 (10) = p
14 + 20 = p
34 = p
The complete question is:
The rectangular floor of a classroom is 21 feet in length and 30 feet in width. A scale drawing of the floor has a length of 7 inches. What is the perimeter, in inches, of the floor in the scale drawing?
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A quadrilateral has two angles that measure 232° and 86°. The other two angles are in a ratio of 4:17. What are the measures of those two angles?
The two missing angles of the quadrilateral are 10.5° and 2.47°, calculated from the given angles of 232° and 86°, and a ratio of 4:17.
The angle sum of a quadrilateral is 360°. Therefore, we need to find the sum of the two missing angles.We can start by subtracting the two known angles from the total angle sum and then dividing the difference by the ratio given.
360° - 232° - 86° = 42°
To find the two missing angles, divide the sum by the ratio given.
42° ÷ 4:17
The find the measure of the first angle, divide 42° by 4:
42° ÷ 4 = 10.5°
To find the measure of the second angle, divide 42° by 17:
42° ÷ 17 = 2.47°
Therefore, the two missing angles of the quadrilateral are 10.5° and 2.47°.
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Find the y-Intercept and x-intercept of the line.
5x - 2y = 10
The x-intercept and the y-intercept of the line 5x - 2y = 10 include the following ordered pairs:
x-intercept = (2, 0).
y-intercept = (0, -5).
What is the x-intercept?In Mathematics, the x-intercept can be defined as the point at which the graph of a function crosses the x-coordinate (x-axis) and the value of "y" or y-value is equal to zero (0).
What is y-intercept?In Mathematics, the y-intercept is sometimes referred to as an initial value (base value) and the y-intercept of any graph such as a linear function, generally occur at the point where the value of "x" is equal to zero (x = 0).
In this exercise, we would use an online graphing calculator to plot the linear function in order to determine the x-intercepts and y-intercepts as shown in the image attached below.
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Which shape has two pairs of parallel sides
The shape with two pairs of parallel sides is called a parallelogram.
What is a paralellogram?A parallelogram is a shape that has four sides, in which the two opposite lengths are:
Parallel to each other.Have the same measure.As there are four sides, divided two by two, the shape has two pairs of parallel sides, and is called a parallelogram.
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Please answer both questions in the photo!!! (20p)
Answer:
(2.4, 2)
(4, -3.4)
Step-by-step explanation:
Brainliest Please
How do you solve verifying identity with calculator?
Verifying Identity with a calculator requires an individual to enter specific numerical values into a calculator and then solve a given equation. This can be used to prove the validity of an equation, or the accuracy of a given answer.
The equation is typically written in algebraic form and includes numerical values.Verifying Identity with a calculator requires an individual to enter specific numerical values into a calculator and then solve a given equation. This can be used to prove the validity of an equation, or the accuracy of a given answer. For example, if an individual is asked to verify the identity [tex]sin^2x + cos^2x[/tex]= 1, they will be given numerical values to plug into the equation. For example, if they are given x = 60, they would enter[tex](sin60)^2 + (cos60)^2[/tex] into the calculator and solve. This should result in an answer of 1, proving the equation is accurate.
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Bisectors of triangles
Find RT
The length of RT is 13 units
What is Triangle?A triangle is a three-sided polygon that consists of three edges and three vertices.
As T is the centroid of the given triangle.
TN=TP
x+10=4x-17
Now solve for x
27=3x
x=9
Now TP=4(9)-17
TP=36-17
TP=19
Now from pythagoras theorem we find the length of RT
19²=14²+RT²
361-196=RT²
165=RT²
Take square root on both sides
RT=13
Hence, the length of RT is 13 units
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The measure of an angle is 71.6°. What is the measure of its complementary angle
Answer:
18.4 degrees
Step-by-step explanation:
a complementary angle is 90 degrees and 90 - 71.6 is 18.4. So, the answer is 18.4 degrees.
1001₂ = ______ (base 6)
A. 13
B. 14
C. 15
D. 21
E. 23
The binary number will be equal to 14 in base 6
1001₂ = 14 (base 6)
What is a number base?Base is defined as a set of digits used to represent numbers. There are different number systems which make use of different combinations of digits as the base.
One of the most common which is the binary number system uses only 2 digits, i.e., 0 and 1 to represent numbers, while the Octal number system uses 8 digits, i.e., 0 to 7, to represent numbers.
The most common number system that is in use is the decimal number system that uses the base 10, which includes digits from 0 to 9 for writing numbers.
1001₂ = 1 x 2^3 + 0 x 2^2 + 0 x 2^1 + 1 x 2^0
1001₂ = 1 x 8 + 1 x 1
1001₂ = 8 + 1
1001₂ = 9
Converting 9 into base 6
6 9
6 1 r 3
0 r 1
From downwards pointing up
9 = 13 (base 6)
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to find the value of a numerical or algebraic expression is called?
The process of finding the value of a numerical or algebraic expression is called evaluation.
Evaluation involves substituting the given values of the variables in the expression and then simplifying the expression using the order of operations (PEMDAS).
For example, if we are asked to evaluate the expression 3x + 2y when x=4 and y=5, we would substitute these values to get 3(4) + 2(5), which simplifies to 12 + 10 = 22. Therefore, the value of the expression 3x + 2y when x=4 and y=5 is 22.
In algebra, evaluation is an important process because it allows us to find the value of an expression for specific values of the variables, which can help us to solve equations or inequalities, and to simplify expressions.
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How many solutions does the equation have?
4x + 3 = 4x + 3
Answer:
Infinitely Many
Step-by-step explanation:
Both answer are equal to each other
4. A line segment has two end-points M(3, 7) and N(11, -6). Find the coordinates of the point W that lies on the y-axis such that W is equidistant from M and from N.
would appreciate a step by step answer (quickly please)
The coordinates of point W which is equidistant from M and N ( 7, 1 / 2).
What is coordinate geometry?A coordinate plane is a 2D plane that is formed by the intersection of two perpendicular lines known as the x-axis and y-axis. A coordinate system in geometry is a method for determining the positions of the points by using one or more numbers or coordinates.
Given that the points are M(3, 7) and N(11, -6). The coordinates of the midpoints are,
W = ( x₁ + x₂ ) / 2, ( y₁ + y₂ ) / 2
W = ( 3 + 11 ) / 2, ( 7 - 6 ) / 2
W = 14 / 2, 1 / 2
W = 7, 1 / 2
Hence, the value of the coordinate of the midpoint is ( 7, 1 / 2 ).
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You can work at most 20 hours next week. You need to earn at least $90 to cover
you weekly expenses. Your dog-walking job pays $7.50 per hour and your job as a
car wash attendant pays $6 per hour. Write and graph a system of linear
inequalities to model the situation. Use the graph to list 3 solutions
The system of inequalities is.
x + y ≤ 20
x*7.5 + y*6 ≥ 90
And some solutions can be:
(10, 5)
(11, 5)
(12, 6)
How to write the system of inequalities?Let's define the variables we will be using:
x = number of hours dog-waling
y = number of hours at the car wash.
We know that you can work at most 20 hours, then:
x + y ≥ 20
And you need toearn at least $90, then wecan write:
x*7.5 + y*6 ≥ 90
The system of inequlities is:
x + y ≤ 20
x*7.5 + y*6 ≥ 90
And the graph is the one below. The solutions are all the points in the dark blue region, some solutions we can see there are:
(10, 5)
(11, 5)
(12, 6)
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An architect is drawing a floor plan for the basement of a house. He has drawn the
plans as shown. On the plan, 1/4 inch represents 4 feet in length in the house.
What is the area of the basement?
The area of the basement will be 576 square feet.
What is an area?The area is defined as the space covered by the object in a two-dimensional plane. The area of the rectangle is calculated by the multiplication of the length and the width.
Given that a basement floor plan for a house is being created by an architect. He drew the plans as they are seen. On the plan, one-fourth of an inch corresponds to four feet in the house.
The area of the basement will be calculated as:-
Area = 12 x 6
Area = 72 square inches
Area = 72 x 8 feet
Area = 576 square feet
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Help pleaseee
Solve for g
Answer:
g=4f-k+h
Step-by-step explanation:
f=g-h+k
4
4f=g-h+k
4f-k=g-h
4f-k+h=g
Ben has 300 marbles in a bag.
He gives
65 of the marbles to Sofia
80 of the marbles to Paul
35 of the marbles to Lucas
What fraction of his 300 marbles has Ben given away?
Give your fraction in its simplest form.
Answer:
Ben had given away 3/5 of his fraction.
Step-by-step explanation:
total number of marbles Ben give away,
= 65+80+35
= 180
180/300= 3/5
Solve for x
A. Solve using the change of base formula.
Logby= log y/log b
B. Solve using a different method.
C. verify your solutions by substituting your value back into the original equation.
A) By using the change of base formula, the solution of the expression is,
⇒ x = - 2/3
B) The solution of the expression is,
⇒ x = - 2/3
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ 5³ˣ⁻² = 125²ˣ
Now, We have the change of base formula,
⇒ [tex]log_{b} y = \frac{log y}{log b}[/tex]
Here, The expression is,
⇒ 5³ˣ⁻² = 125²ˣ
⇒ 5³ˣ⁻² = 5⁶ˣ
Take log on both side,
⇒ log 5³ˣ⁻² = log 5⁶ˣ
⇒ (3x - 2) log 5 = 6x log 5
⇒ 3x - 2 = 6x
⇒ - 2 = 6x - 3x
⇒ 3x = - 2
⇒ x = - 2/3
B) By using another method as;
The expression is,
⇒ 5³ˣ⁻² = 125²ˣ
⇒ 5³ˣ⁻² = 5⁶ˣ
By comparing we get;
⇒ 3x - 2 = 6x
⇒ - 2 = 6x - 3x
⇒ 3x = - 2
⇒ x = - 2/3
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I need some help please
Answer:
32
Step-by-step explanation:
Substitute values given into the expression :
3[2(1)+(3)]-4[((1)-(3)]+(3)²
3[5]-(-8)+9
15+8+9 =
32
Hope this helped and have a good day
If the numerator of a fraction is multiplied by 4 and the denominator is reduced by 2,the result is 4 . If the numerator of the fraction is increased by 15 and 2 subtracted from the double of the denominator, the result is 5/4. Find the fraction.
The original fraction described in the question is; 25/31
How to simplify fractions?Let the numerator be x and let the denominator be y.
We are told that the numerator of a fraction is multiplied by 4 and the denominator is reduced by 2,the result is 4. Thus;
4x/(y - 2) = 4
4x = 4y - 8
4x - 4y = -8 -----(1)
The numerator of the fraction is increased by 15 and 2 subtracted from the double of the denominator, the result is 5/4. Thus;
(x + 15)/(2y - 2) = 5/4
4x + 60 = 10y - 10
4x - 10y = - 70 ----(2)
Subtract eq 2 from eq 1 to get;
6y = 62
y = 31/3
Thus;
4x - 4(31/3) = -8
Multiply through by 3 to get;
12x - 124 = -24
12x = 100
x = 25/3
Thus, the fraction is;
(25/3)/(31/3) = 25/31
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The table below shows some inputs and outputs of function f(x). _1 0 2 4 8 f(a) 1 5 13 21 37 Which statement best describes the function? O f(x) is a linear function because z and f(z) are both increasing. O f(2) is an absolute value function because 2 and f(a) are both increasing. O f(z) is a linear function because f(z) is increasing by 4 for each increase in z by 1.
A statement which best describes the function include the following: A. f(x) is a linear function because x and f(x) are both increasing.
What is a linear function?In Mathematics, a linear function can be defined as a type of function whose equation is graphically represented by a straight line on the cartesian coordinate. This ultimately implies that, a linear function has the same slope and it is typically used for uniquely mapping an input variable to an output variable, which both increases simultaneously.
Next, we would determine the slope by using the points contained in the table as follows;
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (5 - 1)/(0 + 1)
Slope (m) = 4.
Slope (m) = (13 - 5)/(2 - 0)
Slope (m) = 4
Slope (m) = (21 - 13)/(4-2)
Slope (m) = 8/2 = 4.
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Shana wants to use all 62 feet of the fencing she has to make a rectangular run for her dog. She decides to make the length of the run 20 feet. She writes and solves the equation 2 l plus 2 w equals 62 to find the width of the run.
Which statements are true of the solution? Check all that apply.
The value of w is 10 feet.
The value of w can be zero.
The value of w cannot be a negative number.
Substitution is used to replace the variable l with a value of 20.
The subtraction property of equality is used to isolate the term with the variable w
Answer:
11 feet long
Step-by-step explanation:
20+20=40
62-40=22
22÷2=11
check
11+11+20+20=62
What is the cube root of 12 167?
From the given information provided, the cube root of 12,167 by prime factorization method is approximately 23.
You can find the cube root of 12,167 using a calculator or by using the prime factorization method.
Using the prime factorization method, we can express 12,167 as the product of its prime factors:
12,167 = 19 × 641
Since the cube root of a product is equal to the product of the cube roots of the factors, we can find the cube root of 12,167 as follows:
∛12,167 = ∛(19 × 641)
∛12,167 = ∛19 × ∛641
∛12,167 = 2.7144 × 8.0186
∛12,167 = 23.1649 (rounded to four decimal places)
Therefore, the cube root of 12,167 is approximately 23.
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The breadth of a rectangle is half of its length. If its perimeter is 76m,find its length and breadth.
The length and breadth of the rectangle are 25.33 meters and 12.67 meters,
What is a rectangle?A rectangle is a quadrilateral having four sides and the sum of the angles is 180 in the rectangle the opposite two sides are equal and parallel and the two sides are at 90-degree angles.
We are given that the breadth of the rectangle is half of its length, so we can write:
B = 0.5L
We are also given that the perimeter of the rectangle is 76m. The perimeter of a rectangle is the sum of the lengths of all four sides, which in this case is:
Perimeter = 2(L + B)
Substituting the value of B from the first equation, we get:
Perimeter = 2(L + 0.5L)
Perimeter = 2(1.5L)
Perimeter = 3L
Now we can solve for L:
3L = 76
L = 25.33
So the length of the rectangle is 25.33 meters.
To find the breadth, we can substitute this value of L back into the first equation:
B = 0.5L
B = 0.5(25.33)
B = 12.67
So the breadth of the rectangle is 12.67 meters.
Therefore, the length and breadth of the rectangle are 25.33 meters and 12.67 meters, respectively.
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2/ac + 1/ab + 3/bc = ?
Answer:
Abc
Step-by-step explanation:
2b + c + 3a = Abc
The population in a small town is increasing annually by 2.8%. What is the quarterly rate of population increase?
Answer:
To find the quarterly rate of population increase, we first need to find the annual rate of population increase in decimal form, which is 2.8% expressed as 0.028.
To find the quarterly rate, we need to divide the annual rate by 4, since there are 4 quarters in a year:
0.028 / 4 = 0.007
Therefore, the quarterly rate of population increase is 0.007 or 0.7%
2. A container holds 5 red pens, 4 purple pens and 3 blue pens. Find the
probability P(red pen, purple pen) without replacement. (use the simplest
fraction)
The probability P(red pen, purple pen) without replacement is 10/33.
What is Probability?It is a branch of mathematics that deals with the occurrence of a random event.
Given that a container holds 5 red pens, 4 purple pens and 3 blue pens.
We need to find the probability P(red pen, purple pen) without replacement.
P(red pen, purple pen) = (number of ways to select a red pen and a purple pen) / (total number of ways to select two pens)
Total number of ways to select two pens=¹²C₂=66 ways.
Number of ways to select a red pen and a purple pen = (number of ways to select a red pen) x (number of ways to select a purple pen from the remaining pens)
Number of ways to select a red pen = 5
Number of ways to select a purple pen from the remaining pens = 4
Therefore, the number of ways to select a red pen and a purple pen without replacement is:
Number of ways to select a red pen and a purple pen = 5 x 4 = 20
P(red pen, purple pen) =20/66
=10/33
Hence, the probability P(red pen, purple pen) without replacement is 10/33.
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What is -8/9 divided 2/3??
Answer:
-1.33333333333
Step-by-step explanation:
Answer:
-1 1/3
Step-by-step explanation:
-12x3^2+36 please ty
Answer:
-72
Step-by-step explanation:
Answer:
-72
Step-by-step explanation:
multiply-12 x3^2 and then add 36
Claire goes to the gym every 4 days and the
park every 6 days. If she does both today,
how many days will pass until the next time
she goes to the gym and park on the same
day?
GCF or LCM? (circle one)
Required, after 12 days Claire will have the day where she has gone to the gym and the park an equal number of times.
What is the least common multiple (LCM)?The least common multiple (LCM) of two numbers represents the smallest multiple that is divisible by both numbers.
Here,
In this case, the LCM of 4 and 6 is 12, which is the smallest number that is divisible by both 4 and 6. This means that after 12 days, Claire will have gone to the gym and the park an equal number of times, and will thus be back at the same point. Therefore, the answer is LCM.
Thus, the required, after 12 days Claire will have the day where she has gone to the gym and the park an equal number of times.
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