A solid lies above the cone z = x2 + y2 and below the sphere x2 + y2 + z2 = z. write a description of the solid in terms of inequalities involving spherical coordinates
The solid lying above the cone z = x^2 + y^2 and below the sphere x^2 + y^2 + z^2 = z, in spherical coordinates, is described by the inequalities 0 ≤ ρ ≤ 2 cos φ (W.r.t the sphere) and φ ≥ π/4 (W.r.t the cone), with 0 ≤ θ ≤ 2π (full revolution for θ).
Explanation:In spherical coordinates, we represent a point in space using three values: ρ (the distance from the origin), φ (the angle measured from the positive z-axis down to the line connecting the origin and the point), and θ (the angle measured in the x-y plane from the positive x-axis to the projection of the line segment from the origin to the point).
The given cone z = x2 + y2 in spherical coordinates becomes ρ cos φ = ρ2 sin2 φ, which simplifies to tan φ = 1/ρ or φ = π/4. This is because for a cone with vertex at the origin, φ is constant. So, our first inequality is φ ≥ π/4.
The sphere's equation x2 + y2 + z2 = z becomes ρ2 = ρ cos φ, which further simplifies to ρ = 2 cos φ. So, the second inequality is ρ ≤ 2 cos φ, which bounds ρ from above by the sphere.
In summary, the solid is described by the inequalities 0 ≤ ρ ≤ 2 cos φ and φ ≥ π/4, with 0 ≤ θ ≤ 2π (since θ revolves full circle in spherical coordinates).
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The solid can be described using the inequalities: 0 ≤ r ≤ cos(θ), 0 ≤ θ ≤ 2π, and 0 ≤ ϕ ≤ π. These inequalities define the limits for the radius, polar angle, and azimuthal angle in spherical coordinates.
Explanation:To describe the solid in terms of inequalities involving spherical coordinates, we need to find the limits for the radius, polar angle, and azimuthal angle.
Considering the given information, the solid lies above the cone z = x2 + y2, which implies that the z-coordinate ranges from 0 to r2.
As for the sphere x2 + y2 + z2 = z, we can rewrite it in spherical coordinates as r2 = r cos(θ) or r = cos(θ).
Therefore, the solid can be described using the following inequalities in spherical coordinates:
0 ≤ r ≤ cos(θ) 0 ≤ θ ≤ 2π 0 ≤ ϕ ≤ π Learn more about Solid described in spherical coordinates here:
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A wheel turns 1,800 revolutions per minute. How fast does it turn in radians per second?
A wheel turn 60π radians per second.
What is Division method?Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications. For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.
Given that;
A wheel turns 1,800 revolutions per minute.
Now,
Since, A wheel turns 1,800 revolutions per minute.
Hence, Number of revolution in one seconds = 1800 / 60
= 30
We know that;
⇒ 1 revolution = 2π radian
So, Number of revolution in radian per second = 30 × 2π
= 60π
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Doors for a small cabinets are 11.5 inches long. Doors for the large cabinets are 2.3 times as long as the doors for the small cabinets. How many large doors can be cut from a board that is 10.5 feet long?
In the game of blackjack played with one deck, a player is initially dealt 2 different cards from the 52 different cards in the deck. a winning "blackjack" hand is won by getting 1 of the 4 aces and 1 of 16 other cards worth 10 points. the two cards can be in any order. find the probability of being dealt a blackjack hand. what approximate percentage of hands are winning blackjack hands?
We are given here that a blackjack hand consists of:
1 of the 4 aces = 4 / 52
1 of the 16 cards worth 10 points (10, jack, queen, king) = 16 / 52
So assuming that cards are dealt without replacement, therefore the probability of getting a blackjack hand is:
P = 1st is ace * 2nd is 10 pt card + 1st is 10 pt card * 2nd is ace
P = (4 / 52) * (16 / 51) + (16 / 52) * (4 / 51)
P = 0.04827 = 4.83%
Therefore there is a 4.83% probability to get a blackjack hand.
Final answer:
The probability of being dealt a blackjack hand with a single deck is approximately 0.0483, which translates to about 4.83% of the hands.
Explanation:
To find the probability of being dealt a blackjack hand in a single-deck game, we need to calculate the chances of drawing an ace and a 10-point card (10, Jack, Queen, or King) in any order. There are 4 aces and 16 cards worth 10 points in a deck of 52 cards. The probability of drawing an ace first is 4/52, and then drawing a 10-point card is 16/51, giving us (4/52)*(16/51). Conversely, if a 10-point card is drawn first (16/52) followed by an ace (4/51), the probability is (16/52)*(4/51). Adding the two probabilities gives us the chance of a blackjack in either order:
P(Blackjack) = (4/52)*(16/51) + (16/52)*(4/51) = (64/2652) + (64/2652) = 128/2652 ≈ 0.0483
The approximate percentage of hands that are winning blackjack hands is about 4.83%.
a garden table and a bench cost $1,000 combined the cost of a garden table is three times the cost of the bench what is the cost of the bench.
Jack has 702 acres of land which requires 1.2 acre-feet of water to grow crops successfully. Currently it cost 12.95 per acre-foot to purchase water. How much will it cost to water all his crops
A) 9,090.90
B) 10,909.08
C) 15,540.00
D) none
What is the matter with this number: .000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
Is it real?
Let c be parametrized by x = et sin (10t) and y = et cos (10t) for 0 ≤ t ≤ 2. find the length l of
c. l = 2 correct: your answer is correct.
To find the length of the curve c parameterized by given equations, we differentiate these equations with respect to t, find the magnitude of the derivative vector, and integrate this magnitude from t = 0 to t = 2.
Explanation:Finding the Length of Parametrized CurveTo find the length L of a curve c parametrized by x = et sin(10t) and y = et cos(10t) for 0 ≤ t ≤ 2, we use the formula for the arc length of a curve given in parametric form. The formula is L = ∫ |c'(t)| dt, where |c'(t)| represents the magnitude of the derivative of the parametric equations. The integral is evaluated over the given interval of t.
Step-by-step ExplanationFirst, compute the derivatives of the parametric equations: dx/dt and dy/dt.Next, find the magnitude of the velocity vector |c'(t)| = sqrt((dx/dt)^2 + (dy/dt)^2).Finally, integrate |c'(t)| from t = 0 to t = 2.Using this approach, we find that the length L of the curve c over the interval from 0 to 2 is indeed 2, as verified by the calculus of parametric functions.
3x-2=x
show work and check
Round 10,386.145 to the nearest tenth
The rectangular sandbox at the local community park has a width of 24.5 meters and its length is 31.7 meters. What is the perimeter, in meters, of the rectangular sandbox?
To calculate the perimeter of the rectangular sandbox, use the formula P = 2l + 2w, where l is the length and w is the width. For the given dimensions, 31.7 meters in length and 24.5 meters in width, the perimeter is 112.4 meters.
Explanation:The question asks us to calculate the perimeter of a rectangular sandbox. The formula to calculate the perimeter of a rectangle is 2 times the length plus 2 times the width, often written as P = 2l + 2w.
Given the dimensions of the sandbox, the length (l) is 31.7 meters, and the width (w) is 24.5 meters.
We calculate the perimeter as follows:
Perimeter = 2 × Length + 2 × WidthPerimeter = 2 × 31.7 m + 2 × 24.5 mPerimeter = 63.4 m + 49.0 mPerimeter = 112.4 metersTherefore, the perimeter of the rectangular sandbox is 112.4 meters.
A heap of rubbish in the shape of a cube is being compacted into a smaller cube. given that the volume decreases at a rate of 4 cubic meters per minute, find the rate of change of an edge, in meters per minute, of the cube when the volume is exactly 125 cubic meters.
Using implicit differentiation, it is found that the rate of change of an edge is of -0.0533 meters per minute.
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The volume of a cube of edge e is given by:
[tex]V = e^3[/tex]
In this problem, the volume is of 125 m³, thus, we solve the above equation to find the length of an edge, in metres.
[tex]V = e^3[/tex]
[tex]125 = e^3[/tex]
[tex]e = \sqrt[3]{125}[/tex]
[tex]e = 5[/tex]
Now, for the rate of change, we need to apply the implicit differentiation, thus:
[tex]V = e^3[/tex]
[tex]\frac{dV}{dt} = 3e^2\frac{de}{dt}[/tex]
[tex]\frac{dV}{dt} = 3(5)^2\frac{de}{dt}[/tex]
[tex]\frac{dV}{dt} = 75\frac{de}{dt}[/tex]
Volume decreases at a rate of 4 cubic meters per minute, thus:
[tex]\frac{dV}{dt} = -4[/tex]
The rate of change of an edge is [tex]\frac{de}{dt}[/tex]. Then:
[tex]-4 = 75\frac{de}{dt}[/tex]
[tex]\frac{de}{dt} = -\frac{4}{75}[/tex]
[tex]\frac{de}{dt} = -0.0533[/tex]
The rate of change is of -0.0533 cubic meters per minute.
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3.1666666667 as a fraction
If 5x=3x-8, evaluate 4x+2
Isaac drinks 8 glasses of water each day. He says he will drink 2,920 glasses of water in a year that has 365 days. Is the exact answer reasonable
multiply 365 by 8
365 * 8 = 2920
so yes it is reasonable
What are fissures and how are theu created by land subsidence?
In 34 pound of a spice mix, there is 5/6 cup of cinnamon. How much cinnamon does the spice mix contain per pound? a 5/8 cup b 9/10 cup c 1 1/9 cups d 1 3/5 cups
Answer:
c. 1 1/9 cups
Step-by-step explanation:
To find cups per pound, divide cups by pounds.
(5/6 cup)/(3/4 pound) = (5/6)/(3/4) cup/pound
= (5/6)×(4/3) cup/pound . . . . . . . invert the denominator and multiply
= 20/18 cup/pound = 10/9 cup/pound = 1 1/9 cup/pound
What is the cytoplasm? A. a fluid in which organelles are suspended B. a type of organelle that assembles proteins C. the exterior envelope surrounding the nucleus D. the inner surface of the cell's wall
Karen is riding her bike at 4 miles per hour she wants to show this on a graph what should she draw
20 randomly selected statistics students were given 15 multiple-choice questions and 15 open-ended questions - all on the same material. the professor was interested in determining which type of questions the students scored higher. this experiment is an example of a one sample test of means. a two sample test of means. a paired t-test. a test of proportions.
solve the system of linear equations. separate the x- and y- with a coma.
6x=-14-8y
-12x=20+8y
Answer:
-16 81 19
Step-by-step explanation:
a recipe makes a total of 5 cups a serving of pudding is 3/4 cup how many servings of pudding does the recipe make?
To find the number of servings of pudding the recipe makes, divide the total number of cups by the serving size. The recipe makes approximately 6.67 servings of pudding.
Explanation:To find the number of servings of pudding the recipe makes, divide the total number of cups by the serving size. In this case, the recipe makes 5 cups and each serving is 3/4 cup. So, to find the number of servings, divide 5 cups by 3/4 cup: 5 cups ÷ (3/4 cup) = 5 cups × (4/3 cup) = 20/3 servings
Therefore, the recipe makes approximately 6.67 servings of pudding.
A person standing 20 feet from a street light casts a 10 foot shadow. How many times taller is the streetlight than the person? Assume the triangles are similar.
What are the coordinates of the vertex for f(x) = x2 + 4x + 10?
Answer:
The coordinates of the vertex are [tex] (-2, -6)[/tex]
Step-by-step explanation:
The graph of this function is a parabola that opens up in the Cartesian plane. We can find like this:
[tex]y = (x ^ 2 + 4x +4) +6 = (x + 2) ^ 2 +6[/tex], where, [tex](y - 6) = (x + 2) ^ 2[/tex]
In the canonical equation of the parabola:
[tex](y - k) = (x + h) ^ 2[/tex], with the point (h, k) as the vertex, [tex]h = -2[/tex] and [tex]k = 6[/tex].
Conclusion: the coordinates of the vertex are [tex](-2, -6)[/tex]
The department store where you work is having a sale. Every item is to be marked down 15% What will the sale price of a $152 coat be
multiply 152 by 15%
152 * 0.15 = 22.80 ( amount of discount
152 - 22.80 = 129.20 ( sale price)
Julian is 10 years younger the Thomas. The sum of their ages is 74. What is Thomas’s age?
Thomas = x
Julian = x-10
x +x-10 = 74
2x-10 = 74
2x = 84
x = 84/2 = 42
Thomas is 42
Julian is 32
A family has five children. the probability of having a girl is 1/2. what is the probability of having at least 4 girls?
Can someone help me with this?
Every week, Mr. Kirkson uses 316 gallon of water to water every 13 square foot of his garden. How many gallons of water does Mr. Kirkson use per square foot to water his garden each week? Enter your answer in the box as a fraction in simplest form.
Is It A Fraction If So Then The Answer Is [tex]\frac{9}{16}[/tex]
If Not Then The Other Guy Is Right
HoPe It HeLpS PlEaSe MaRk Me ThE BRAINlYEST!!!!!!!!!!!!
What is the solution of the equation f(x) = g(x) ?
A. x = -4
B. x = -2
C. x = 2
D. x = 4