calculus lll
Use a double integral to find the area of the region Rwhere R is bounded by the circle r = 8sine (a) 647 (b) 1676 (c) 870 (d) 327
Answers
The correct option is (c) 870.
The given curve can be represented as 8sin θ = r or r = 8sin θ. We can also say that θ is from 0 to π since the curve represents half of a circle, and it is symmetric around the y-axis.
Therefore, the area of the region R can be found by integrating the double integral over the region R.∫(θ=0 to π)∫(r=0 to 8sin θ) r dr dθ∫(θ=0 to π)∫(r=0 to 8sin θ) r dr dθ= ∫(θ=0 to π) [1/2 * r²] r=8sin θ to r=0 dθ= ∫(θ=0 to π) 1/2 * (8sin θ)² dθ= 32 ∫(θ=0 to π) sin² θ dθ
Now we need to use the double angle identity sin² θ = (1 - cos 2θ)/2, and put sin² θ in the form of cos 2θ.
∫(θ=0 to π) sin² θ dθ= 32 ∫(θ=0 to π) (1 - cos 2θ)/2 dθ= 32 [(θ/2) - (1/4) sin 2θ] θ=0 to π= 32 [π/2 - 0] - 0= 16π The area of the region R is 16π. Therefore, the correct option is (c) 870.
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Related Questions
I need help please
Answers
Answer:
Its out of A and C srry couldn't help much
Use the number line to answer the following 2 22 questions. How many groups of 5 3 3 5 start fraction, 5, divided by, 3, end fraction are in 1 11? groups Evaluate. 9 ÷ 5 3 = 9÷ 3 5 =9, divided by, start fraction, 5, divided by, 3, end fraction, equals
Answers
The value of 9 ÷ 5 3 (9 divided by 5 divided by 3) is 1 remainder 1.To determine how many groups of 5 3 3 5 (5 divided by 3) are in 1 11, we can use the number line to visualize the division.
Starting at 0 on the number line, we can count in increments of 5 3 3 5 (5 divided by 3) until we reach or exceed 1 11.
On the number line, we can observe that there are two groups of 5 3 3 5 (5 divided by 3) that fit into 1 11. This means that 1 11 can be divided into two groups of 5 3 3 5.
Therefore, the answer to the question "How many groups of 5 3 3 5 (5 divided by 3) are in 1 11?" is 2.
Now, evaluating the expression 9 ÷ 5 3 (9 divided by 5 divided by 3), we perform the division from left to right:
9 ÷ 5 = 1 remainder 4
Then, taking the remainder 4 and dividing by 3:
4 ÷ 3 = 1 remainder 1.
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Factor:9/100 x^2 - 4/25
Answers
Answer:
\(\frac{9}{100}x^2-\frac{4}{25}=(\frac{3}{10}x-\frac{2}{5})(\frac{3}{10}x+\frac{2}{5})\)Explanation:
Given the expression:
\(\frac{9}{100}x^2-\frac{4}{25}\)This can be written as a difference of two squares. But before that, we can rewrite the expression as:
\((\frac{3}{10}x)^2-(\frac{2}{5})^2\)So, as a different of two squares, we write:
\((\frac{3}{10}x-\frac{2}{5})(\frac{3}{10}x+\frac{2}{5})\)triangle is an isosceles right triangle in the unit circle. a circle with center a at the origin of an x y plane. explain why . use the pythagorean theorem to explain why .
Answers
The Pythagorean Theorem is used to show that the hypotenuse has a length of sqrt(2).
In a unit circle, the radius is always equal to 1 unit. Now, consider an isosceles right triangle with two equal sides of length 1 unit
By the Pythagorean Theorem, the length of the hypotenuse (c) of this triangle can be found as:
\(c^2 = 1^2 + 1^2\)
\(c^2 = 2\)
\(c = sqrt(2)\)
Now, let's consider a circle centered at the origin with a radius of sqrt(2) units. Any point on this circle has coordinates (x, y) such that:
\(x^2 + y^2 = (sqrt(2))^2\)
\(x^2 + y^2= 2\)
This equation represents the unit circle, and any point on the isosceles right triangle we considered earlier also satisfies this equation. Therefore, the isosceles right triangle is inscribed in the unit circle.
In summary, the isosceles right triangle is inscribed in the unit circle because its hypotenuse has a length of sqrt(2) units, which satisfies the equation of the unit circle \((x^2 + y^2 = 1)\). The Pythagorean Theorem is used to show that the hypotenuse has a length of sqrt(2).
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7. The floor area of a passage is 2,4 m². The floor has to be tiled with square tiles of which the length of the sides is 200 mm. How many tiles do you need to cover the floor?
Answers
The number of tiles needed to cover the floor is 60 tiles.
Given Information,
The floor area of the passage = 2.4 m²
Length of a side of square tile = 200mm
Length of the side is in mm which needs to be converted in its standard form which is meters.
Therefore, the length of side = 200/1000 m
= 0.2 m
Area of the square tile = 0.2 x 0.2
= 0.04 m²
Let the number of tiles be x.
∴ Number of tiles = \(\frac{Area of passage}{Area of each tile}\)
x = 2.4/0.04
x = 60
∴ The number of tiles needed to cover the floor is 60 tiles.
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Help me with this assignment it’s due today
Answers
The exact value of side lengths of triangle are d= \(\frac{8}{\sqrt{3} }\) and h=\(\frac{5\sqrt{2} }{2}\)
Describe Triangle?A triangle is a closed, two-dimensional shape that consists of three straight sides and three angles. It is one of the basic shapes in geometry and is often used in various mathematical and scientific contexts.
The three sides of a triangle are usually named using lowercase letters a, b, and c, and the three angles are named using uppercase letters A, B, and C, with the opposite angles and sides having the same letter. The sum of the angles in a triangle is always 180 degrees, and this property is known as the Angle Sum Theorem.
Triangles can be classified based on their side lengths and angles. Based on the side lengths, triangles can be classified as:
Scalene triangle: A triangle in which all three sides have different lengths.
Isosceles triangle: A triangle in which two sides have the same length, and the third side has a different length.
Equilateral triangle: A triangle in which all three sides have the same length.
Let's start with the first triangle. We can use the trigonometric ratios for a 30-60-90 triangle:
sin(60) = opposite / hypotenuse = d / (2d) = \(\frac{1}{2}\)
cos(60) = adjacent / hypotenuse = 4 / (2d) = \(\frac{1}{2\sqrt{3} }\)
Solving for d:
d = 2 * sin(60) = 2 *\(\frac{1}{2}\) = 1
2d = 4 / cos(60) = \(\frac{4}{\frac{1}{2\sqrt{3} }}\) = \(\frac{8}{\sqrt{3} }\)
So d = 1 and 2d = \(\frac{8}{\sqrt{3} }\) in the first triangle.
For the second triangle, we can use the trigonometric ratios for a 45-45-90 triangle:
sin(45) = opposite / hypotenuse = \(\frac{h}{5}\)
cos(45) = adjacent / hypotenuse = \(\frac{h}{5}\)
Since sin(45) = cos(45) = \(\frac{1}{\sqrt{2} }\), we can solve for h:
h = 5 * sin(45) = 5 * \(\frac{1}{\sqrt{2} }\) = \(\frac{5\sqrt{2} }{2}\)
So h = \(\frac{5\sqrt{2} }{2}\) in the second triangle.
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the original price P of a shirt less $5 discount
Answers
Answer:
P - 5
Step-by-step explanation:
hello
original price is P
$5 discount means that the final price is P - 5
hope this helps
Answer:
p-5
Step-by-step explanation:
If 5^3b-1=5^b-3, what is the value of b?
Answers
Answer: -1
Step-by-step explanation:
\(5^{3b-1}=5^{b-3}\\\\3b-1=b-3\\\\2b-1=-3\\\\2b=-2\\\\b=\boxed{-1}\)
consider a function z(x, y) and its partial derivative (∂z/∂y)x. can this partial derivative still be a function of x?
Answers
A function z(x, y) and its partial derivative (∂z/∂y)x is aslo a function.
Function:
In set math, function refers an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable)
Given,
Consider a function z(x, y) and its partial derivative (∂z/∂y)x.
Here we need to identify that the partial derivative still be a function of x.
In order to find the solution for this function, we must know the definition of partial derivative,
The definition of partial derivative is, "partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant."
Here the partial derivative of a function f with respect to the differently x is variously denoted by f’x, fx, ∂xf or ∂f/∂x.
The symbol ∂ is partial derivative.
So, as per the definition of the partial derivative, the given partial derivative (∂z/∂y)x is also a function.
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How many roots does this function have?
y = 8x4 - 5x3 + 2x2 - 8x + 7
no links please
Answers
Answer:
there is no root
Step-by-step explanation:
there is no root
Where do I add parentheses to make the equation true: 6 x 8 – 4 = 24
Answers
Answer:Yes
Step-by-step explanation: Without parentheses, this equation equals 44. Put the parentheses around 8-4 to make the equation true
Please help on both! Will give brainliest!!
Answers
1. Show that the function
g
(
x
)
=
x
−
2
5
g(x)=
5
x−2
is the inverse of f(x) = 5x + 2.
Step 1: The function notation f(x) can be written as a variable in an equation. Is that variable x or y?
____
Write f(x) = 5x + 2 as an equation with the variable you chose above. (2 points)
Answers
To find the inverse function, we need to switch x and y and solve for y. So, g(x) = (x-2)/5 is the inverse of f(x) = 5x+2.
The variable in the equation is y.
y = 5x + 2
Now, we need to switch x and y and solve for y.
x = 5y + 2
x - 2 = 5y
y = (x - 2)/5
So, the inverse function of f(x) is g(x) = (x-2)/5. To show that this is the inverse of f(x), we need to verify that g(f(x)) = x and f(g(x)) = x for all values of x.
g(f(x)) = g(5x+2) = (5x+2-2)/5 = x
f(g(x)) = f((x-2)/5) = 5(x-2)/5 + 2 = x - 2 + 2 = x
Therefore, g(x) = (x-2)/5 is the inverse function of f(x) = 5x+2.
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what is the area for a radius of 8 mm
Answers
Answer:
A= 201.06
Step-by-step explanation:
A = pi r^2
A = pi x 8 x r^2 = 201.06
Find the equation of the linear function represented by the table below in slope-intercept form.
Answers
Answer:
y=-2x+5
Step-by-step explanation:
For every increase in x, y decreases by 2
so the slope is -2
You can solve for the y-intercept by going backward, 1-1=0, 3+2=5
the y-intercept is (0,5)
Using the slope intercept equation format, the answer is y-2x+5
14 problems, solve all correctly to be awarded brainiest. Tip: None of the variables have a listed value, so just add or subtract them.Do not assume for example that y=1 because it does not. If you do not answer them all, your question will be reported as incomplete and removed.
YOU MUST SHOW YOUR WORK
[Find the sum]
1. (5 - t) + (3t + 2)
2. 2(-5y +6) + (2y -8)
3. 5(0.3s - 2) + 2(5 - 3.4)
4. 1/5(-15w - 20) + 1/2(3 - 4w)
5. (7k + 9) + (4k - 3)
6. 4(2.5g - 4) + 3(1.2g - 4)
7. 1/3(6p - 3) + 1/7(7p + 14)
8. 1/8(16k - 24) + 1/5(2 + 10k) {K is not 1000. K is a variable.}
[Find the difference]
9. (8 - u) - (5u +1)
10. (3h + 4) - 6(5 - 1.4h)
11. 1/4(16j - 12) - 1/9(18j + 45)
12. (2x + 7) - 6(5x - 8)
13. (12 + 7.2b) - 3(0.9b - 4)
14. 2/3(12n + 6) -1/5(10n - 2)
Answers
Answer:
1. 2t+7
2. -8y + 4
3. 3/2s-6.8
4. -5w-5/2
5. 11k+6
6. 13.6g-28
7. 3p+1
8. 4k-13/5
9. -6u+7
10. 11.4h-26
11. 2j-8
12. -28x - 55
13. 4.5b + 24
14. 6n + 22/5
Step-by-step explanation:
1. (5 - t) + (3t + 2)
Let's add the variables first. -t + 3t = 2t. Now do 5+2 for 7, and you will get 2t+7
2. 2(-5y +6) + (2y -8)
First, distribute the left side. (-10y + 12) + (2y - 8)
Then, add the variables -10 + 2 = -8y
Now, add the constants 12+ - 8 = 4
-8y + 4
3. 5(0.3s - 2) + 2(5 - 3.4)
Distribute first, on both sides, and you will get (3/2s-10)+(10-6.8)
There are no other variables, so we just add the constants.
-10+10-6.8 = -6.8. So we get
3/2s-6.8
4. 1/5(-15w - 20) + 1/2(3 - 4w)
Distribute both sides, this will give (-3w-4)+(3/2-2w)c.
Now add the variables. -3w+-2w = -5w
Then do -4+3/2 for -5/2.
-5w-5/2
5. (7k + 9) + (4k - 3)
Add the variables, 7k+4k=11k, then Add the constants. 9+-3= 6.
11k+6
6. 4(2.5g - 4) + 3(1.2g - 4)
Distribute and you will get (10g-16)+(18/5g-12).
Add the variables for 68/5g (or 13.6g)
Add the constants, -16+-12 = -28.
13.6g-28
7. 1/3(6p - 3) + 1/7(7p + 14)
Distribute first. (2p-1)+(p+2)
Add variables, 2p+p = 3p
Add constants, -1 + 2 = 1
3p+1
8. 1/8(16k - 24) + 1/5(2 + 10k)
Distribute first.
(2k - 3) + (2/5 +2k)
Add variables, 2k+2k = 4k
Add constants, -3+2/5 = -13/5
4k-13/5
9. (8 - u) - (5u +1)
Subtract the variables.
-u-5u =-6u
Subtract constants, 8-1 = 7.
-6u+7
10. (3h + 4) - 6(5 - 1.4h)
Distribute first, (3h+4)-(30-8.4h)
Subtract variables. 3h- -8.5h = 11.4h
Subtract constants. 4-30 = -26
11.4h-26
11. 1/4(16j - 12) - 1/9(18j + 45)
Distribute, (4j-3)-(-2j+-5)
Subtract variables, 4j-2j = 2j
Subtract constants, -3 - 5 = -8
2j-8
12. (2x + 7) - 6(5x - 8)
Distribute first. (2x+7)-(30x-48)
Subtract variables, 2-30 = -28x
Subtract constants, 7- - 48 = 55
-28x - 55
13. (12 + 7.2b) - 3(0.9b - 4)
Distribute,
(12+7.2b) - (2.7b-12)
Subtract variables. 7.2b-2.7b = 4.5b
Subtracts constants. 12- - 12 = 24
4.5b + 24
14. 2/3(12n + 6) -1/5(10n - 2)
Distribute,
(8n + 4) - (2n - 2/5)
Subtract variables, 8n-2n = 6n
Subtracts constants, 4 - - 2/5 = 22/5
6n + 22/5
Sorry this took so long, showing your work for simplification is tedious.
4. Which r - value suggests a strong
positive correlation?
r= 0.1847
r= -0.1847
r= 0.9974
r= -0.9974
Answers
Answer:
It would be r= 0.9974.
The closer the r value is to 1 or -1 the stronger it is.
Positive and negative are just determining what "side" its on.
Let me know if this helps!
A bakery sold apple pies for $12 and blueberry pies for $14 on Saturday sold a total of 41 pies and collected a total of $536 how many apple pies did you sell and how many blueberry pies do they sell
Answers
Answer:
22 blueberry pies and 19 apple pies.
Step-by-step explanation:
This can be solved using a system of equations. Your equation for amount of pies sold is x + y = 41, with x representing apple pies and y representing blueberry pies, and 41 being the total amount of pies sold. The other equation is for the amount of dollars earned from selling the pies, which is:
12x + 14y = 536 $12 per apple pie, $14 per blueberry pie, and a total of $536.
Let's first solve for x. The new equation would be x = 41 - y. Now, we plug this equation into the second equation.
12(41 - y) + 14y = 536 Here we substituted x for the equation we made.
429 - 12y +14y = 536
-12y +14y = 44 Here we moved the 429 to the other side by subtracting it from the left and right.
2y = 44
y = 22. Now we know that there was a total of 22 blueberry pies sold. Now we plug 22 pies back into the very first equation to find x, the amount of apple pies.
x + 22 = 41
x = 19. Now we know there was a total of 19 apple pies sold.
Hope this helps!
Factor the difference of perfect squares. 4x^2 - 1
Answers
Answer:
(2x+1)(2x-1)
Step-by-step explanation:
4x^2-1
(2x)^2-1
(2x)^2-1^2
(2x+1)(2x-1)
simplify
(8p^6)^1/3
simplifyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
Answers
Answer:
\(2p^2\)
Step-by-step explanation:
Step 1: Apply the exponentiation property:
\((8p^6)^\frac{1}{3} = 8^\frac{1}{3} * (p^6)^\frac{1}{3}\)
Step 2: Simplify the cube root of 8:
The cube root of 8 is 2:
\(8^\frac{1}{3} =2\)
Step 3: Simplify the cube root of \((p^6)\):
The cube root of \((p^6)\) is \(p^\frac{6}{3} =p^2\)
Step 4: Combine the simplified terms:
\(2 * p^2\)
So, the simplified expression is \(2p^2\).
The original price of a marker is $5.00 What will the new price be after a
15% increase ? *
Answers
Answer:
5.75
Step-by-step explanation:
for a 15% increase, you multiply 5.00 x 1.15. you then get 5.75. hope this helps!
Answer:$5.75
Step-by-step explanation:
solving multistep equations-24=1-x+6x
Answers
The given expression is
-24=1-x+6x
-24 = 1+6x-x
-24 = 1+ 5x
-24 - 1 = 5x
-25 = 5x
\(\begin{gathered} x=\frac{-25}{5}=\text{ -5 } \\ x=\text{ -5} \end{gathered}\)So the answer is -5.
Use Liam DeWitt's Flash Card statement and the credit calendar for Exercises a-e.
A) what is Liam’s average daily balance
B) what is Liam’s monthly periodic rate?
C) what is Liam’s finance charge?
D) what is Liam’s new balance?
E) what is Liam’s available credit?
Answers
The answers for this task are gotten in most cases by Careful Observation of Liam DeWitt's Bank Account Information.
What is a Bank Account Information?
A bank account information refers to all the details present on or which can be deduced from one's statement or record of banking activities..
Liam's Average Daily Balance is: Total sum of balance each day divided by the number of days. That is
(250+87+1300+470-200+34.76+102.71+45.90+84.60)/9
ADB= $241.66
B) Liam's Monthly Periodic Rate (MPR) is the Annual Interest Rate divided by the number of periods. In this case, that will be:
19.8%/12
MPR = 1.65%
C) Liam's Finance Charge is (Average Daily Balance * APR)/365.
That is (241.66 * 19.8)/365 = 13.11%
D) Liam's New Balance is calculated by removing new inflow from old balance. That is
(250+87.60+1,300+470.63+34.76+102.71+45+848.60)-3,240.5
= $-101.20
E) Liam's Available Line of Credit is clearly stated as $4,000.
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Two-fifths of one less than a number is less than three-fifths of one more than that number. What numbers are in the solution set of this problem?
O xx-5
0 x>-5
O x>-1
O x<-1
Answers
Answer:
x > -5
Step-by-step explanation:
\( \frac{2}{5} (x - 1) < \frac{3}{5} (x + 1)\)
\(2(x - 1) < 3(x + 1)\)
\(2x - 2 < 3x + 3\)
\( - 5 < x\)
\(x > - 5\)
6x + 4x - 1 = 2(5x + 4)
6x + 4x - 1 =
• 5x +
• 4
X – 1 = X + 8
10x -x-1 = 10x - X + 8
Answers
Answer: no solutions
Step-by-step explanation:
One earthquake has MMS magnitude 3.9. If a second earthquake has 120 times as much energy (earth movement) as the first, find the magnitude of the second quake.
Answers
Answer:
The right answer is "5.9". A further solving is provided below.
Step-by-step explanation:
The given values are:
Initial Magnitude,
M₁ = 3.9
Let Second magnitude be M₂.
Now,
⇒ \(M_2-M_1=log(\frac{l_2}{l_1} )\)
On substituting the values, we get
⇒ \(M_2-3.9 = log (120)\)
On putting the value of "log 120", we get
⇒ \(M_2-3.9=2.079\)
On adding "3.9" both sides, we get
⇒ \(M_2-3.9+3.9=2.079+3.9\)
⇒ \(M_2=5.9\)
do parallel lines have tha same slope
Answers
Answer:
The slopes of parallel lines are equal.
Step-by-step explanation:
Two lines are parallel if their slopes are equal and they have different y-intercepts.
The economy of Rayton can be described by the following statistics:
Current real GDP = $10,000
Full employment real GDP = $12,000
Tax revenue = $3000
Government transfer payments = $1000
Government spending = $3000
a) In what stage of the business cycle is the Rayton economy? Explain.
b) What is the current budget balance in Rayton? Show your work.
c) Senator Fullington, an influential politician in Rayton, has argued that the budget must be
balanced. How could this be accomplished? How would balancing the budget affect real GDP in
Rayton?
Answers
Based on the current real and the full employment real GDP, Rayton's economy is at a Recession stage.
The current budget balance in Rayton is -$1,000. The budget can be balanced by reducing government spending or increasing taxes. Both of these would decrease real GDP.
What does the economic data on GDP show?The fact that the current real GDP is less than the full employment GDP shows that the nation is at a recession stage where it is underperforming.
What is the current budget balance?This can be found as:
= Government revenue(taxes) - government spending
= 3,000 - 1,000 - 3,000
= -$4,000
In order to make a balanced budget where government revenue will equal government spending, the government either needs to reduce spending or increase revenue ( taxes ).
Increasing taxes or reducing spending, will lead to less money available for spending in the economy which will lead to a decrease in real GDP.
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Please help me with this and also remember your answer has to be step by step
The two figures above are congruent. Find the measure of the angle that isn't labeled on either figure.
Answers
The measure of the angle that isn't labeled are:
∠B = 100
∠K = 100
∠L = 41
∠M = 86
What are corresponding angles?The angles that are in the same position on two parallel lines intersected by a transversal line on the parallel lines.
Corresponding angles are equal.
We have,
Two congruent figures.
Each figure's total sum of the angles must be 360.
The figures are congruent so the corresponding angles must be equal.
From the first figure we get,
∠A = 133 ( Corresponding angles )
∠A + ∠B + ∠C + ∠D = 360
133 + ∠B + 41 + 86 = 360
∠B = 360 - 260
∠B = 100
Now,
∠k = ∠B = 100
∠L = ∠C = 41
∠M = ∠D = 86
Thus,
The measure of the angle that isn't labeled are:
∠B = 100
∠K = 100
∠L = 41
∠M = 86
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Work out 15% of £40 pls pls pls
Answers
After answering the provided question, we can conclude that Therefore, equation 15% of £40 is £6.
What is equation?In mathematics, an equation is a statement that states the equivalence of two expressions. An equation consists of two sides separated by an algebraic equation (=). For instance, the argument "2x + 3 = 9" states that the sentence "2x + 3" equals the number "9". The purpose of solving equations is to find the value or amounts of the variable(s) that always allow the equation to be true. Equations can just be simple or complicated, regular or nonlinear, and contain one or more variables. For example, in the equation "x2 + 2x - 3 = 0," the variable x is raised to the second power. Lines are used extensively in mathematics, including algebra, calculus, and geometry.
To work out 15% of £40, y
15% = 15/100 = 0.15
15% of £40 is:
0.15 x £40 = £6
Therefore, 15% of £40 is £6.
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