Using polar/cylindrical coordinates, we take
x = r cos(t )
y = r sin(t )
z = z
so that
dV = dx dy dz = r dr dt dz
The solid is given by the set
D = {(x, y, z) : x ² + y ² ≤ 1 and 0 ≤ z ≤ 4 - 2x - y}
or in cylindrical coordinates,
D = {(r, t, z) : 0 ≤ r ≤ 1, 0 ≤ t ≤ 2π, and 0 ≤ z ≤ 4 - 2r cos(t ) - r sin(t )}
Then the volume of the solid is
[tex]\displaystyle \iiint_D \mathrm dV = \iiint_Dr\,\mathrm dr\,\mathrm dt\,\mathrm dz \\\\ = \int_0^{2\pi} \int_0^1 \int_0^{4-2r\cos(t)-r\sin(t)} r\,\mathrm dz\,\mathrm dr\,\mathrm d t \\\\ = \int_0^{2\pi}\int_0^1 (4r-2r^2\cos(t)-r^2\sin(t))\,\mathrm dr\,\mathrm dt \\\\ = \int_0^{2\pi} \left(2 - \frac23 \cos(t) - \frac13 \sin(t)\right) \,\mathrm dt \\\\ = \boxed{4\pi}[/tex]
The volume of the given solid below the plane 2x + y + z = 4 and above the disk x2 + y2 ≤ 1 is 4π.
What is the volume of the solid in polar coordinates?The volume of the solid in polar coordinates can be calculated by the double integral in polar coordinates.
[tex]V = \iint_R f(x,y) \,dA.\\[/tex]
Similarly, by polar coordinates,
x = r cost
y = r sint
z = z
we get, dV = dx dy dz = r dr dt dz
The solid is given
D = {(x, y, z) : x ² + y ² ≤ 1 and 0 ≤ z ≤ 4 - 2x - y}
or in polar coordinates,
D = {(r, t, z) : 0 ≤ r ≤ 1, 0 ≤ t ≤ 2π, and 0 ≤ z ≤ 4 - 2r cos(t ) - r sin(t )}
The volume of the solid is
[tex]\rm \int \int\int\ dV = \int \int\int\ rdr dt dz\\ \rm = \int\limits^{2\pi}_0\int\limits^1_0 \int\limits^{4-2rcost-rsint}_0 rdr dt dz\\\rm =\int\limits^{2\pi}_0\int\limits^1_0 {4r-2r^{2} cost-r^{2} sint} dr dt\\\rm =\int\limits^{2\pi}_0( 2-2/3cost-1/3sint) dt\\\rm =4\pi[/tex]
Hence, The volume of the given solid below the plane 2x + y + z = 4 and above the disk x2 + y2 ≤ 1 is 4π.
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Plz can anyone slove 3 & 4 questions plzz
Answer:
For question 3 its definitely 5^15 and I searched 4 so I thinks its 71 or 161? sorry I don't know angles but 3 is correct
Answer:
[tex] {5}^{4 + 8 + 3} = {5}^{15} \\ \\ 90 - 19 = 71[/tex]
How do you write 3 3/4 as a decimal?
Pls help me I really need this today. If you need the snack price is here!
Dried Fruit $4.25 per pound
Mixed Nuts $6.75 per pound
Trail Mix $3.50 per pound
9514 1404 393
Answer:
Yes, Yolanda bought enough
Step-by-step explanation:
a. Read and understand the question, including what is given and what is asked for. Here, we have ...
Given: mixed nuts $6.75 per pound, $21.60 spent on mixed nuts; dried fruit $4.25 per pound, $11.90 spent on dried fruit.
Find: is the total weight of purchased mixed nuts and dried fruit at least 6 pounds?
__
b. Select appropriate relations to create a path from what is given to what is asked for. Here, we have ...
weight = cost/(cost per pound)
total weight = weight of mixed nuts + weight of dried fruit
__
c. Use the relations with the given information to come to a conclusion that will answer the question.
weight of mixed nuts = $21.60/($6.75/lb) = 3.2 lb
weight of dried fruit = $11.90/($4.25/lb) = 2.8 lb
total weight = (3.2 lb) +(2.8 lb) = 6.0 lb
Yolanda wanted to buy a total of 6 pounds for the party, and she bought 6 pounds. Yolanda bought enough for the party.
convert the following into decimal 111.01
Assuming the given number is in base 2, we have
111.01₂ = 2² + 2¹ + 2⁰ + 2⁻²
111.01₂ = 4 + 2 + 1 + 0.25
111.01₂ = 7.25
Please help i really need it, i will give brainliest to the right answers
Monique is taking part in a chili cooking contest. She has to buy enough supplies to make a very large batch of chili. The table shows the prices of some of the ingredients at the grocery store.
Answer:
1. 5
2. 1.29
3. 9.90
Step-by-step explanation:
They are buying 5 of each thing, so multiply it by 5.
The diced tomatoes cost $1.29.
When you solve the equation, you get 9.9. Convert that to money, and you have $9.90
8.24 ➗ 2 = 4.12 ones ➗ 2
Answer:
2.06
Step-by-step explanation:
Which statement de ribes the system of equations?
[x+2y = 2
[x-2y=-2
Answer:
x+2y=2 x-2y=-2. It has infinitely many solutions. It has no solution. It has one solution
3x + (2 + x) + 5 x 2 as a product of two factors
Answer:
4x+12
I think that's it............
[tex]\sqrt{1.69}[/tex]
1.3 Is that the answer you looking for
Simone works at an assembly plant she made 490 items in five days how many items did she average per day?
Answer:
98 per day
Step-by-step explanation:
490 / 5
= 98
A home has a triangular backyard. The second angle of the triangle is 4º more than the
first angle. The third angle is 21° more than three times the first angle. Find the angles of
the triangular yard
Answer:
31, 114 and 35
Step-by-step explanation:
I set up the equation as such:
(3x + 21) + (x + 4) + x = 180
then I combined like terms:
5x + 25 = 180
And then solving for x I get:
5x = 155
x = 31
And then plugged the solved x value into the terms to give:
(3(31) + 21) = 114
(31 + 4) = 35
alongside the found angle of 31.
Answer:
[tex]31^{\circ}, 35^{\circ}, 114^{\circ}[/tex]
Step-by-step explanation:
Let the first angle be [tex]x[/tex]. Then, the second angle is [tex](x+4)^{\circ}[/tex] and the third angle is [tex](3x+21)^{\circ}[/tex].
We know that the sum of the angles in a triangle will always be [tex]180^{\circ}[/tex]. So, we want to solve the equation for x: [tex](x)+(x+4)+(3x+21)=180[/tex].
Combining like terms on the left side gives [tex]5x+25=180[/tex].
Subtracting [tex]25[/tex] from both sides gives [tex]5x=155[/tex].
Dividing both sides by [tex]5[/tex] gives [tex]x=31[/tex].
So, the first angle is [tex]31^{\circ}[/tex].
The second angle is [tex]x+4=31+4=35[/tex], so the second angle is [tex]35^{\circ}[/tex].
The third angle is [tex]3x+21=3(31)+21=93+21=114[/tex], so the third angle is [tex]114^{\circ}[/tex].
So, the angles are [tex]\boxed{31^{\circ}, 35^{\circ}, 114^{\circ}}[/tex] and we're done!
The check that these angles work is left as an exercise to the reader.
A plumber has a 36-in. pipe. He must cut into two pieces so that one piece is 14 inches longer than the other. How long is each piece?
Evaluate.
(1/5+1/2)^2+3/10⋅5
Enter your answer as a mixed number in simplest form by filling in the boxes.
Answer:
1/5+1/2²+3/10•5=1/7²+15/100
The resultant of the expression (1/5+1/2)^2+3/10⋅5 in mixed number will be 1 99/100.
What is a number system?The number system is a way to represent or express numbers.
Any of the multiple sets of symbols and the guidelines for utilizing them to represent numbers are included in the Number System.
As per the given expression,
(1/5+1/2)^2+3/10⋅5
⇒ (1/5+1/2)²+ 15/10
Take LCM in the bracket as 10
⇒ [(2 + 5)/10]² + 15/10
⇒ (7/10)² + 15/10
⇒ 49/100 + 15/10
Take LCM as 100
⇒ (49 + 150)/100 = 199/100
⇒ 1 99/100
Hence "The resultant of the expression (1/5+1/2)^2+3/10⋅5 in mixed number will be 1 99/100".
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What is the next number in the sequence 3,5,8,12,17?
Answer:
it would be 23
Step-by-step explanation:
5-3=2
5-8=3
12-8=4
17-12=5
23-17=6
A polynomial of degree at least 3 where one or more of the roots are fractions
#1 p(x)=(x-3)*(x-5)*(x-1)=x3-9x2+23x-15
You try the next one...a fractional root would look like (x-3/2)
Using the Factor Theorem, it is found that an example of a polynomial of degree at least 3 where one or more of the roots are fractions is given by:
[tex]f(x) = x^3 - \frac{7}{2}x^2 + \frac{1}{2}x - 1[/tex]Factor Theorem: The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
In which a is the leading coefficient.In this problem:
A polynomial of degree 3 will be built, hence it will have 3 roots.A leading coefficient of a = 1 is supposed.Two non-fractional roots, which are x = 1 and x = 2, hence [tex]x_1 = 1, x_2 = 2[/tex].The fractional root is [tex]x_3 = \frac{1}{2}[/tex].Hence:
[tex]f(x) = (x - x_1)(x - x_2)(x - x_3)[/tex]
[tex]f(x) = (x - 1)(x - 2)\left(x - \frac{1}{2}\right)[/tex]
[tex]f(x) = (x^2 - 3x + 2)\left(x - \frac{1}{2}\right)[/tex]
[tex]f(x) = x^3 - \frac{7}{2}x^2 + \frac{1}{2}x - 1[/tex]
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PLEASE HELP ITS MATH THANK YOUUUU
Answer:
4 Kilometers
Step-by-step explanation:
if 2 centimeters = 1 kilometer than 8 centimeters = 4 kilometers
Why was Yellowstone National Park created?
A. to set aside land for animals as more people moved west
B. to protect bears from humans and other predators
C. to keep different types of animals together in one place
D. to stop people from moving further west
Answer:
Step-by-step explanation:
The answer is A. to set aside land for animals as more people moved west
Determine the midpoint of segment CD
Answer:
9
Step-by-step expation:
Tell me pls but pls no files
Answer:
I think its A
Step-by-step explanation:
one of the root of the quadratic equation x²-(4+p)x+12=0 is three time the other find the root of the equation
Answer:
it should be p=14,x=16,12
The roots of the equations are 2 and 6.
What is a quadratic equation?The equation of the form ax² + bx +c is known as a quadratic equation.
Given that, the quadratic equation is x² - (4 + p)x + 12= 0.
Let the roots of the quadratic equation be a and b.
If the coefficient of x² is 1 in a quadratic equation, then the coefficient of x is equal to the sum of the roots and the constant term is equal to the product of the roots.
Therefore,
a + b = -(4 + p) (1)
ab = 12 (2)
Given that, one root is three times the other, so, let a = 3b and substitute it into equation (2)
3b(b) = 12
3b² = 12
b² = 4
b = 2
Therefore, the value of a is:
a= 3(2)
a = 6
Hence, the roots of the equations are 2 and 6.
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A recycling plant recycled 2 tons of cans yesterday. This is a third of their usual amount How much does the plant usually recycle?
25 points :)
At midnight in Alaska, the temperature in Sitka is 29.7 degrees Fahrenheit.
Each hour after that the temperature decreases by 1.23 degrees until the sun comes up.
What is the temperature 2 hours after midnight?
Temperature
Level : JHSthe temperature in Sitka 2 hours after midnight is :
= 29.7°F + (2 × 1.23°F)
= 29.7°F + 2.46°F
= 32.16°F
So, the temperature in Sitka 2 hours after midnight is 32.16°F
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Answer:
27.24° FStep-by-step explanation:
Initial temperature = 29.7° FDecrease rate = 1.23° F / hAfter 2 hours the temperature is:
29.7° F - 2*1.23° F = 27.24° F4 X 4=16
4 X 5=20
What percent increase?
Step-by-step explanation:
20-16/16 x100
4/16 X 100 = 25%
Helloooo could someone plz help me on this ?
Answer:
15 cubic units
Step-by-step explanation:
If you add up the blocks you get 15
question 10. please i didnt listen to the lecture please help
Answer:
I wrote the answers in the screenshot
Find the greatest common factor (GCF) of 68 and 76.
A. 4
B. 6
C. 8
D. 2
Answer:
Its A
Step-by-step explanation:
Answer:
A. 4
Step-by-step explanation:
The ratio of Kelly's money to Vera’s money was 7:3. After Kelly spent $35, she had $9 more than Vera. How much money did each girl have at first?
Your travel guide contains a grid map of London, with each unit on the grid representing 0.5 kilometers. If Buckingham Palace is located at (−2,2) and St. Paul's Cathedral is located at (−2,−3), what is the direct distance (not walking distance, which would have to account for bridges and roadways) between the two landmarks in kilometers? Round your answer to two decimal places, if necessary.
Answer:
5.00
Step-by-step explanation:
The direct distance between Buckingham Palace and St. Paul's Cathedral is 5 kilometers.
Given the coordinates of Buckingham Palace as (-2, 2) and St. Paul's Cathedral as (-2, -3).
To find the direct distance between two points on a grid, we can use the distance formula, which is derived from the Pythagorean theorem. The distance formula in a 2-dimensional Cartesian plane is:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
We can calculate the distance between them as follows:
x₁ = -2, y₁ = 2 (Coordinates of Buckingham Palace)
x₂ = -2, y₂ = -3 (Coordinates of St. Paul's Cathedral)
Distance = √((-2 - (-2))² + (-3 - 2)²)
= √(0² + (-5)²)
= √(0 + 25)
= √25
= 5 kilometers
So, the distance is 5 kilometers.
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Hi anyone help me with this it’s for my daughter and i am not good with mathematics
Answer: Ma'am It says that 39-14=25 is the example problem
Write an equation of the line that passes through the point (6,2) and is perpendicular to the line y=3x+3
Answer:
y = -[tex]\frac{1}{3}[/tex] + 4
Step-by-step explanation:
first of all we know the slope of the equation which is 3x. then we know that we are supposed to write an equation that is perpendicular to y = 3x + 3. so since it is supposed to be perpendicular we have to make the current slope opposite.
so slope for our new equation will be -[tex]\frac{1}{3}[/tex]x.
now since we know our slope we use point-slope form and plug everything in.
point-slope form: y1-y=m(x1-x)
plug it in:
y - 2 = -[tex]\frac{1}{3}[/tex] (x - 6)
= y - 2 = -[tex]\frac{1}{3}[/tex]x +2
= y = -[tex]\frac{1}{3}[/tex] + 4
