Using cylindrical coordinates, the volume of the solid region bounded on the top by the paraboloid z = 12 − x^2 − y^2 and bounded on the bottom by the cone z = x^2 y^2 can be found. The explanation below provides the step-by-step process.
In cylindrical coordinates, we can express the paraboloid and the cone equations as follows:
Paraboloid: z = 12 -[tex]r^2[/tex]
Cone: z = [tex]r^2 cos^2(θ) sin^2(θ)[/tex]
To find the volume of the solid region, we need to determine the limits of integration. The region is bounded by the paraboloid on top and the cone on the bottom. The paraboloid and the cone intersect when 12 - [tex]r^2[/tex] = [tex]r^2 cos^2(θ) sin^2(θ)[/tex]. Simplifying this equation, we get 12 = [tex]r^2[/tex](1 - [tex]cos^2(θ)[/tex] [tex]sin^2(θ[/tex])). Since r is always non-negative, we can rewrite the equation as 12 =[tex]r^2[/tex][tex]sin^2(θ) (1 - sin^2(θ)[/tex]). This equation defines the boundary curve in the polar coordinate plane (r, θ).
To determine the limits of integration for r, we need to find the values of r that satisfy the equation above for each θ. For a fixed value of θ, the equation becomes 12 = [tex]r^2 sin^2(θ) (1 - sin^2(θ))[/tex]. This equation represents a circle with radius [tex]\sqrt(12 sin^2(θ) (1 - sin^2(θ)))[/tex]. Thus, the limits for r are 0 and [tex]\sqrt(12 sin^2(θ) (1 - sin^2(θ)))[/tex].
For the limits of integration for θ, we need to consider the range in which the paraboloid and the cone intersect. The cone is defined in the range 0 ≤ θ ≤ π, and the paraboloid intersects the cone when θ satisfies 12 = [tex]r^2 sin^2(θ) (1 - sin^2(θ))[/tex]. By solving this equation, we find that 0 ≤ θ ≤ π/2.
To calculate the volume, we integrate over the cylindrical coordinates as follows:
V = ∫∫∫ dV
= ∫[0,π/2]∫[0,√[tex](12sin^2(θ)(1-sin^2(θ)))]∫[r^2cos^2(θ)sin^2(θ),12-r^2][/tex] r dz dr dθ
Evaluating this triple integral will yield the volume of the solid region bounded by the given surfaces.
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A radiograph technician has a technique of 30 mAs and 120 kV at 100 cm SID with a 5:1 grid, and produces an intensity of 100 mR. If she wants to keep maintain exposure at: 200 cm, a 10:1 grid, with 138 kV , what should the new mAs be?
The new mAs should be approximately 45 mAs.
The intensity of radiation at a given distance can be calculated using the inverse square law:
Intensity2 = Intensity1 × (Distance1/Distance2)^2
Given:
Intensity1 = 100 mR
Distance1 = 100 cm
Distance2 = 200 cm
Using the above formula, we can calculate the new intensity at 200 cm:
Intensity2 = 100 mR × (100 cm/200 cm)^2
Intensity2 = 100 mR × (1/4)
Intensity2 = 25 mR
To maintain the same exposure at the new distance, the new mAs needs to be adjusted accordingly. We can use the exposure maintenance formula:
mAs2/mAs1 = (kVp2/kVp1) × (Distance1/Distance2)^2 × (Gridratio2/Gridratio1)^2
Given:
mAs1 = 30 mAs
kVp1 = 120 kV
kVp2 = 138 kV
Gridratio1 = 5:1
Gridratio2 = 10:1
Substituting the given values into the formula, we can solve for mAs2:
mAs2/30 mAs = (138 kV/120 kV) × (100 cm/200 cm)^2 × (10/5)^2
mAs2/30 mAs = (1.15) × (0.5)^2 × (4)
mAs2/30 mAs = 1.15 × 0.25 × 4
mAs2/30 mAs = 0.115
Simplifying, we find:
mAs2 = 30 mAs × 0.115
mAs2 ≈ 3.45 mAs
Therefore, the new mAs should be approximately 45 mAs (rounded to the nearest whole number) to maintain exposure at 200 cm with a 10:1 grid and 138 kV.
To maintain exposure at a new distance of 200 cm with a 10:1 grid and 138 kV, the radiograph technician should set the new mAs to approximately 45 mAs. This adjustment takes into account the changes in distance, kV, and grid ratio while ensuring that the radiation intensity remains consistent.
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» 40 students were asked their favourite
shoe colour.
What percentage chose white?
Answer: 40% i think
Step-by-step explanation:
Find a*b. a = (4, 1, 1/5), b = (6,-4, -15)
The calculated product of the vectors a and b is 17
How to calculate the product of the vectorsFrom the question, we have the following parameters that can be used in our computation:
a = (4, 1, 1/5) and b = (6,-4, -15)
The product of the vectors can be calculated using
a * b = Sum of products of each position
Using the above as a guide, we have the following:
a * b = 4 * 6 + 1 * -4 + 1/5 * -15
Evaluate the expression
a * b = 17
Hence, the product of the vectors a and b is 17
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please help ·ω· pppppppllllllllllllllleeeeeeeeeeaaaaaaaassssssssseeeeeeeeee????
Answer:
x=49
Step-by-step explanation:
please help me understand this.Will mark brainlyist to right answer no links!!!
Prove or disprove each of the following statements. (a) Let x be an integer. If 4x² + 3x + 7 is odd, then x must be even. (b) Let A, B and C be sets. If A - C ≤ B - C, then A ≤ B.
The statement is False.
(a) Let x be an integer. If 4x² + 3x + 7 is odd, then x must be even.Statement (a) is false.
Here is the explanation:We know that an integer is odd if and only if it can be represented in the form of 2k + 1, where k is any integer.Let us assume that x is an odd integer. Then, we can write x as 2k + 1, where k is any integer.Substituting the value of x in 4x² + 3x + 7, we get;4x² + 3x + 7 = 4(2k + 1)² + 3(2k + 1) + 7= 4(4k² + 4k + 1) + 6k + 3 + 7= 16k² + 16k + 4 + 6k + 10= 16k² + 22k + 14= 2(8k² + 11k + 7)which is an even integer as it is a multiple of 2.
Therefore, we have proven that if x is odd, then 4x² + 3x + 7 is even.So, we have disproved the statement as it is not true for all integers. It's only true for odd integers only. Therefore, statement (a) is false.
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When the data are interval, the parameters of interest are the population mean μ and the population variance σ².
state whether the following statement is True or False ( if false, motivate your answer).
The statement is true that:
When the data are interval, the parameters of interest are the population mean μ and the population variance σ² .
Given,
Mean and standard variance .
Here,
An interval estimator provides a range of values as an estimate of a parameter, considering uncertainty, while a point estimator provides a single value estimate. The Student's t-distribution is used when the population standard deviation is unknown, and the z-distribution is used when it is known.
Support for the statement:
True: When the data are interval, the parameters of interest are the population mean (μ) and the population variance (σ^2). The population mean represents the average value of the variable of interest in the population, while the population variance measures the variability or spread of the variable in the population.
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NEED HELP!! 50 points!!!
Answer: 3, The initial velocity of the rocket
Answer:
the initial velocity of the rocket
Step-by-step explanation:
what is the solution to the inequality of x+2<5
Answer:
Any number below 3
(2, negative infinity)
Answer:
here's the answer with steps
x + 2 < 5
-2 -2
x < 3
Please help, will mark ya if right
Answer:
It's A I think. I did the math on math.way and it was differeten the the answers you have.
Step-by-step explanation:
help ASAP ill give brainliest
Answer:
9 students is the correct answer :(
HELP ME I NEED HELP!!!
Answer: B
Step-by-step explanation:
4.2 / 0.25 = 21
0.4 · 21 = 8.4
8.4 + 2.5 = 10.9
The Great Knightrola Small Monster Association has four chartered positions: President, Vice President, Treasurer, and Secretary; it also has 12 officers chosen at large. All of these seats must be filled by members of GKSMA, of which there are 50. No member may hold more than 1 position. How many different ways are there to fill the positions
Answer:
Number of different ways of filling the position is 8961115245946500
Step-by-step explanation:
There are total of 4 positions which can be arranged in 4! Ways
Like wise there are 12 positions. Hence the number of arrangements for these 12 positions would be 12!
Overall there are 12+4 = 16 positions in which 50 people needs to be arranged
Number of different ways of filling the position is
50P16 /(12! *4!)
= 8961115245946500
The number of different ways of filling the position is, 8961115245946500
There are total of 4 positions can be arranged in 4! Ways
Similarly there are 12 positions.
Therefore, the number of arrangements for these 12 positions is 12!
When we use the permutation?If there is an arrangement of the data then we use the permutation
Therefore we can say that there are 12+4 = 16 positions in which 50 people needs to be arranged.
The number of different ways of filling the position is,
[tex]50P16 /(12! *4!)[/tex]
By using the calculater you can find,
= 8961115245946500
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Name the coordinates of P if P(0, 4) is rotated 180°
Answer:
P'(0, -4)
Step-by-step explanation:
when a coordinate (x, y) is rotated by 180 degrees, the resulting coordinate will be (-x, -y). Note that each coordinate is negated when rotated about 180 degrees
Given the coordinate P (0, 4). When rotated by 180 degrees, the resulting coordinate will be at P'(0, -4)
Yarely ordered a skateboard. There was a discount of 40% and the original price is
$60. What is the sale price?
Please answer correctly! I will mark you as Brainliest!
Multiply the length by width by the height/3
7 x 6 x 8/3 = 112 in^2
The answer is 112 in^2
Answer:
112
Step-by-step explanation:
Rewrite the equation below in Graphing Form:
2.
y = x2 8x + 31
Answer:
y=x^28x+31
Step-by-step explanation:
Answer: y= x 4 − 6 y = 1 x − 2
Step-by-step explanation:
Write a word problem for 1/3 X 2/3 . Use our definition of
multiplication and math drawings to
determine the answer to the multiplication problem. Explain
clearly.
The pizza is divided into 6 equal pieces in total: 3 initial slices, each further divided into 2 equal pieces.
A pizza is cut into 3 equal slices. Each slice is then cut into 2 equal pieces. How many pieces of pizza are there in total?
To solve this problem, we can use our definition of multiplication. Multiplication is repeated addition. In this case, we are repeatedly adding 2 to itself 3 times. This gives us 2 + 2 + 2 = 6. Therefore, there are 6 pieces of pizza in total.
We can also use math drawings to solve this problem. Here is a math drawing that shows how to find the product of 1/3 * 2/3:
The first row of the math drawing shows 1/3 of a pizza. The second row shows 2/3 of a pizza. The third row shows the product of 1/3 * 2/3. As you can see, the product is 6 pieces of pizza.
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Does the nature of the data allow you describe the difference of the two medians as a multiple of the interquartile range? If so, what is the multiple?
Answer:
1/3
Step-by-step explanation:
From the figure, we know that the median of Amy is 40 and the median of Megan is 35.
Therefore the difference in their medians is = 40 - 35
= 5
We see that Q1 of Megan is 25 and Q3 is 40. Therefore, her interquartile range is given by Q3 - Q1 = 40 - 25
= 15
So, the difference of the medians is the multiple of the interquartile range.
So the multiple is = 5/15
= 1/3 (IQR of Megan)
Answer:
Yes, I can describe the difference of the two medians as a multiple of the interquartile range as shown below.
Step-by-step explanation:
The difference of the medians for the two data sets is 69 − 60 = 9. The interquartile ranges for the data sets are 9 ounces and 8 ounces.
This shows that the difference of the medians is about 1 times, or nearly equal to, the interquartile range of either data set.
For Genivieve, the multiple is 1:
9 = m(9)
m = 1
For Mindy, the multiple is 1.125:
9 = m(8)
m = 1.125
(This is the answer on edumentum)
3x-y=2
x+2y=3
elimination method
Answer:
x=1 and y=1
Step-by-step explanation:
see attachment
$35 sweater ; 18% discount ; 3% tax ; find sales price
Answer: $27.65
Step-by-step explanation:
0.18 = 6.3
Discount = $6.3
0.03 = 1.05
Tax = 1.05
35 - 6.3 - 1.05 = 27.65
HELP HELP HELP HELP HEPL HELP
Answer:
The answer would be Y= 1/4X -3
Step-by-step explanation:
Hope it was helpful .
Good luck ^.^
Find y.
30
20
Need help asap
Answer:
y = 10
Step-by-step explanation:
Using the sine ratio in the right triangle and the exact value
sin30° = [tex]\frac{1}{2}[/tex] , then
sin30° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{y}{20}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
2y = 20 ( divide both sides by 2 )
y = 10
Alex and Jack work for a computer software company. Alex can write a computer program in 24 hours, while
Jack can write it in 16 hours. How long will it take them to write the program together?
Alex and Jack work for a computer software company. Alex can write a computer program in 24 hours, while
Jack can write it in 16 hours. How long will it take them to write the program together?
Answer: 9.6 hours
Two power providers are marketing their electricity plans to residential customers. Everyday Electric Company offers a plan with a base cost of $50 per month plus $0.075 per kilo-watt-hour (kwh) used. Consumer Power Limited offers a plan with a base cost of $30 per month plus $0.125 per kilo-watt-hour (kwh) used. Which graph correctly compares the cost of these competing electricity plans?`
Answer:
D
Step-by-step explanation:
Use the standard normal distribution to find P(Z < -2.33 or z > 2.33). Select one: O 9809 .7888 .0198 O 0606
The probability P(Z < -2.33 or Z > 2.33) is approximately 0.0198.
How to find the probabilitiesTo find the probability P(Z < -2.33 or Z > 2.33) using the standard normal distribution, we can calculate the individual probabilities and then add them together.
Using a standard normal distribution calculator
The value for P(Z < -2.33) is approximately 0.0099.
P(Z < -2.33), which is approximately 0.0099.
we can add the two probabilities together
P(Z < -2.33 or Z > 2.33) = P(Z < -2.33) + P(Z > 2.33)
= 0.0099 + 0.0099
= 0.0198
Therefore, the probability P(Z < -2.33 or Z > 2.33) is approximately 0.0198
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What is
(f + g)(x)?
f(x) = x2 – 36
g(x) = x3 + 2x2 – 10
Enter your answer in standard form in the box
Answer:
x^3+3x^2-46
Step-by-step explanation:
Erika pours 6 cups of milk into 8 glasses. Each glass has the same amount of milk. How many cups of milk are in each glass?
Answer:
each glass will have 0.75 of a cup or 3/4 of a cup * 0.75 and 3/4 is the same*
please answer asap
i will mark brainliest
Solve for x and find the missing angle measurement.
Answer: x=17 and 98
Step-by-step explanation:
Those two angles are supplementary so just add them together and make it equal to 180.
The area of the carpet is 48x^2y+16xy^2, if the width is 8xy inches, what is the length of the carpet?
Answer:
Step-by-step explanation:
(48x^2)(y) + (16x)(y^2).
This is addition, so the distributive property applies. [(48x^2)(y) + (16x)(y^2)]/8xy = (48x^2)(y)/8xy+(16x)(y^2)/8xy = 6x+2y.
Division rules are explained below.
I included the parentheses to emphasize the difference between x^2y, which means x to the 2y, and (x^2)(y), which is x^2 times y.
48/8 = 6x^2 times y divided by 8xy equals x because x^2/x = x^(2-1), because x=x^1, and so, equals x; The same y^2/y. The other 2 y's and 2 x's cancel.
TELL ME IF THIS IS CONFUSING<,BECAUSE I DON'T THINK I EXPLAINED VERY WELL.