Write the equation in spherical coordinates. (a) 4z2 = 5x2 + 5y2 (b) x2 + 3z2 = 4
Answers
The transformation from the Cartesian coordinate system (x,y,z) to the spherical coordinate system (r,θ,φ) is given by
x = rsinφcosθ
y = rsinφsinθ
z = rcosφ
Part a.
The given equation is 4z² = 5x² + 5y².
In spherical coordinates, obtain
4r²cos²φ = 5r²sin²φ cos²θ + 5r²sin²φ sin²θ
= 5r²sin²φ(cos²θ + sin²θ)
= 5r²sin²φ
4/5 = tan²φ
tanφ = 2/√5 = (2/5)√5
Answer: tanφ = (2/5)√5
Part b.
The given equation is x² + 3z² = 4
In spherical coordinates, obtain
r²sin²φ cos²θ + 3r²cos²φ = 4
Answer: r²(sin²φ cos²θ + 3cos²φ) = 4
In spherical coordinates, equations in the form of [tex]4z^2 = 5x^2 + 5y^2[/tex] can be written as [tex]4ρ^2cos^2(θ) = 5ρ^2sin^2(θ)[/tex], and equations in the form of x^2 + 3z^2 = 4 can be written as ρ^2sin^2(θ) + 3ρ^2cos^2(θ) = 4.
Explanation:(a) In spherical coordinates, the equation [tex]4z2 = 5x2 + 5y2[/tex] can be written as [tex]4ρ2cos2(θ) = 5ρ2sin2(θ)[/tex]. Here, ρ represents the distance from the origin to the point and θ represents the angle measured from the positive x-axis to the point in the xy-plane.
(b) For the equation x2 + 3z2 = 4, we can rewrite it in spherical coordinates as[tex]ρ2sin2(θ) + 3ρ2cos2(θ) = 4.[/tex]
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Related Questions
You swim 121 out of 1,000 meters. How can you write this as a decimal
Answers
0.121
The number in decimal form will be 0.121 meters.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that you swim 121 out of 1,000 meters. The decimal for the number will be written by dividing the number by 1000.
Decinmal form = 121 / 1000
Decimal form = 0.121 meters
Therefore, the decimal form of the number will be 0.121 meters.
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∠DFG and ∠JKL are complementary angles. m∠DFG = x + 5, and m∠JKL = x − 1. Find the measure of each angle.
Answers
∠DFG and ∠JKL are complementary angles
Measure of each angle
[tex]m<DFG=48\\m<JKL= 42[/tex]
Given :
∠DFG and ∠JKL are complementary angles. m∠DFG = x + 5,
and m∠JKL = x − 1
When two angles are complementary then the sum of angles is 90 degrees
∠DFG and ∠JKL are complementary angles.
So, [tex]m<DFG+m<JKL =90[/tex]
Now replace the expressions and solve for x
[tex]x+5+x-1=90\\2x+4=90\\2x=90-4\\2x=86\\x=43[/tex]
Now replace x with 43 and find out each angle
[tex]m<DFG= x+5\\m<DFG= 43+5=48\\\\m<DFG=48\\m<JKL=x-1=43-1=42[/tex]
[tex]m<DFG=48\\m<JKL= 42[/tex]
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Solve for x.
−5x−4(x−6)=−3
Answers
-5x -4x + 24 = -3 Subtract 24 from both sides, collect like terms
-9x = -27 Divide both sides by -9
x= 3
-5x -(4x-6) = -3
-5x -4x + 24 = -3
-9x = -27
x= 3
Multiple choice strategy: some students have suggested that if you have to guess on a multiple-choice question, you should always choose
c. carl, the student, wants to investigate this theory. he is able to get a sample of past tests and quizzes from various teachers. in this sample there are 110 multiple-choice questions with four options (a, b, c, d). the distribution of correct answers from this sample is given in the frequency table below. correct answer frequency a 22 b 26 c 38 d 24
Answers
Come up with the answer in your head before looking at the possible answers, this way the choices given on the test won't throw you off or trick you.
Eliminate answers you know aren't right.
Read all the choices before choosing your answer.
If there is no guessing penalty, always take an educated guess and select an answer.
Don't keep on changing your answer, usually your first choice is the right one, unless you misread the question.
In "All of the above" and "None of the above" choices, if you are certain one of the statements is true don't choose "None of the above" or one of the statements are false don't choose "All of the above".
In a question with an "All of the above" choice, if you see that at least two correct statements, then "All of the above" is probably the answer.
A positive choice is more likely to be true than a negative one.
Usually the correct answer is the choice with the most information.
Students with better study methods and strategies score higher on their exams.Everyone is different, different methods work for different people the following are only suggestions on improving upon your current studying techniques.It is best to review the material right after class when it's still fresh in your memory.Don't try to do all your studying the night before the test. Instead space out your studying, review class materials at least several times a week, focusing on one topic at a time.
Have all of your study material in front of you: lecture notes, course textbooks, study guides and any other relevant material.
Find a comfortable and quiet place to study with good lighting and little distractions
(try avoiding your own bed, it is very tempting to just lie down and take a nap).
Start out by studying the most important information.
Learn the general concepts first, don't worry about learning the details until you have learned the main ideas.
Take notes and write down a summary of the important ideas as you read through your study material.
Take short breaks frequently. Your memory retains the information that you study at the beginning and the end better than what you study in the middle.
Space out your studying, you'll learn more by studying a little every day instead of waiting to cram at the last minute. By studying every day, the material will stay in your long-term memory but if you try to study at the last moment, the material will only reside in your short-term memory that you'll easily forget.
Make sure that you understand the material well, don't just read through the material and try to memorize everything.
If you choose to study in a group, only study with others who are serious about the test.
Test yourself or have someone test you on the material to find out what your weak and strong areas are. You can use the review questions at the end of each chapter or practice tests the teacher may give out as well as other materials.
Listening to relaxing music such as classical or jazz on a low volume can relieve some of the boredom of studying.
Don't study later than the time you usually go to sleep, you may fall asleep or be tempted to go to sleep, instead try studying in the afternoon or early evening. If you are a morning person try studying in the morning.
At Crescent High School, 120 students plan on going to an in-state college and 70 students plan on going to an out-of-state college. What is the ratio of students planning on going to an in-state college to students planning on going to an out-of-state college?
Answers
In this problem, it's asking for the # of students planning to go to an instate college to the # of students planning to go to an out of state college.
So, 120 : 70
Even though it doesn't tell you to simplify, it will never hurt you, so always do it.
Both 120 and 70 can be divided by 10, so the simplified answer would be 12:7
Hope this helps!
Hi, How do you find the first and second derivatives of the function.
y=(x^2-7/63x) (x^4+1/x^3)
I think for the first derivative dy/dx it's 2/63x-3/63x^-3+4/9x^-5 but I'm not sure, and I have no clue for the second derivative d^2y/dx^2.
Answers
The first hand derivative of the function is [tex]6x^5 - \frac{35}{63}x^4 - \frac{1}{x^2} - \frac{4}{63x^3}[/tex] and the second derivative is [tex]30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]. To find the first derivative, apply the product rule to the given function. Then, differentiate the first derivative to obtain the second derivative. Simplify each step carefully.
To find the first derivative of the given function [tex]y = \left( x^2 - \frac{7}{63}x \right) \left( x^4 + \frac{1}{x^3} \right)[/tex], we'll use the product rule, which states that if [tex]y = u(x) \cdot v(x)[/tex], then [tex]y' = u' \cdot v + u \cdot v'[/tex].
First, define u(x) and v(x) as following:
[tex]u(x) = x^2 - \frac{7}{63}x = x^2 - \frac{1}{9}x[/tex][tex]v(x) = x^4 + \frac{1}{x^3}[/tex]Compute u'(x):
[tex]u'(x) = 2x - \frac{1}{9}[/tex]Compute v'(x):
[tex]v'(x) = 4x^3 + (-3)x^{-4} = 4x^3 - \frac{3}{x^4}[/tex]Apply the product rule: [tex]y' = u' \cdot v + u \cdot v'[/tex]
Thus,
[tex]y' = \left(2x - \frac{1}{9}\right)\left( x^4 + \frac{1}{x^3} \right) + \left( x^2 - \frac{1}{9}x \right) \left( 4x^3 - \frac{3}{x^4} \right)[/tex]Simplify this expression step-by-step to find the first derivative.
[tex]y' = \left(2x - \frac{1}{9}\right)\left( x^4 + \frac{1}{x^3} \right) + \left( x^2 - \frac{1}{9}x \right) \left( 4x^3 - \frac{3}{x^4} \right)[/tex][tex]y'[/tex] [tex]&= \left(2x \cdot x^4 + 2x \cdot \frac{1}{x^3} - \frac{1}{9} \cdot x^4 - \frac{1}{9} \cdot \frac{1}{x^3} \right)[/tex][tex]&\quad + \ \left( x^2 \cdot 4x^3 - x^2 \cdot \frac{3}{x^4} - \frac{1}{9}x \cdot 4x^3 + \frac{1}{9}x \cdot \frac{3}{x^4} \right)[/tex][tex]y'[/tex] [tex]&= 2x^5 + \frac{2}{x^2} - \frac{1}{9}x^4 - \frac{1}{9x^3} \\[/tex] [tex]&\quad + \ 4x^5 - \frac{3}{x^2} - \frac{4}{9}x^4 + \frac{1}{3x^3}[/tex][tex]y'[/tex] [tex]&= 6x^5 - \frac{1}{x^2} - \frac{5}{9}x^4 + \frac{2}{9x^3}[/tex][tex]y'[/tex] [tex]&= 6x^5 - \frac{5}{9}x^4 - \frac{1}{x^2} + \frac{2}{9x^3}[/tex]To find the second derivative, differentiate the first derivative, carefully differentiating each term:
[tex]y'' &= \frac{d}{dx}\left( 6x^5 \right) - \frac{d}{dx}\left( \frac{5}{9}x^4 \right) - \frac{d}{dx}\left( \frac{1}{x^2} \right) + \frac{d}{dx}\left( \frac{2}{9x^3} \right) \\[/tex][tex]y''[/tex] [tex]&= 30x^4 - \frac{5}{9} \cdot 4x^3 - \left( -2x^{-3} \right) + \left( -\frac{2}{9} \cdot 3x^{-4} \right) \\[/tex][tex]y''[/tex] [tex]&= 30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]So, for the function [tex]y = \left( x^2 - \frac{7}{63}x \right) \left( x^4 + \frac{1}{x^3} \right)[/tex], we have:
First derivative [tex](y')[/tex] [tex]&= 6x^5 - \frac{5}{9}x^4 - \frac{1}{x^2} + \frac{2}{9x^3}[/tex]Second derivative [tex](y'')[/tex] [tex]&= 30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]120 × a = 2889 what is the product
Answers
2889 divided by 120 to get your answer.
24.075
How would I find a given triangle where: A=0.1, B=1 and c=9? Thanks in advance.
C=
a=
b=
Answers
[tex]\bf \cfrac{sin(B)}{b}=\cfrac{sin(C)}{c}\implies \cfrac{sin(1)}{b}=\cfrac{sin(\pi -1.1)}{9} \\\\\\ \boxed{\cfrac{9sin(1)}{sin(\pi -1.1)}=b} \\\\\\ \cfrac{sin(A)}{a}=\cfrac{sin(C)}{c}\implies \cfrac{sin(0.1)}{a}=\cfrac{sin(\pi -1.1)}{9} \\\\\\ \boxed{\cfrac{9sin(0.1)}{sin(\pi -1.1)}=a}[/tex]
make sure your calculator is in Radian mode.
Jeff invests an amount at 4% interest compounded annually. After 3 years, he has $1687.30. What was the original amount Jeff invested?
$1500
$1200
$1450
$750
Answers
:
To find the effect of 4% annual interest over 3 years, take 1.04 and raise it to the third power:
1.04³ = 1.125
Divide the final amount, 1687.30 by 1.125 to find the principal amount:
1687.30 ÷ 1.125 = ~1500
Answer is A.
John throws a rock straight down with speed 12 m/s from the top of a tower. the rock hits the ground after 2.37 s. what is the height of the tower? (air resistance is negligible)
Answers
The tower is 28.44 meters tall.
A 27 oz bottle of a new soda costs $2.25. What is the unit rate, rounded to the nearest tenth of a cent?
Answers
1 / x =
Divide 2.25 by 27 to get
.08 dollars per ounce
The unit rate is the cost per ounce of soda. By dividing the total cost by the total ounces, we get the price per ounce in dollars ($0.08333), and converting this to cents gives us $8.3 cents per ounce.
Explanation:The term unit rate refers to a rate in which the second term is 1. In this case, we want to find out how much 1 ounce of soda costs.
First, you want to divide the total cost of the bottle by the total ounces in the bottle. So you divide $2.25 by 27. The answer you get is the price of one ounce of soda in dollars. When calculating it, you get approximately $0.08333.
To get the rate in cents, convert the dollars to cents by multiplying by 100 (since 1 dollar is 100 cents). The answer ($8.33) is the cost of one ounce to the nearest tenth of a cent.
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Two numbers total 53 and have a difference of 25. Find the two numbers.
Answers
and
have a difference (x - y) of (=) 25.
So, x + y = 53 and x - y = 25.
Solve one equation for a variable. We'll solve for x in the first equation.
x + y = 53 Subtract y from both sides
x = 53 - y
Substitute x for (53 - y) in the second equation.
x - y = 25 Substitute (53 - y) in the x spot
53 - y - y = 25 Combine like terms (-y and -y)
53 - 2y = 25 Subtract 53 from both sides
-2y = -28 Divide both sides by -2
y = 14
Now, find x by plugging this y-value (14) into the first equation.
x + y = 53 Substitute y for 14
x + 14 = 53 Subtract 14 from 53
x = 39
Check both of your answers by plugging them into the original equations.
x + y = 53 Plug in your numbers (39 and 14)
39 + 14 = 53 Add
53 = 53
x - y = 25 Plug in your numbers (39 and 14)
39 - 14 = 25 Subtract
25 = 25
So, your answers are x = 39 and y = 14 .
A 20-foot cable hangs from the top of a building that is 50 feet high. if the cable weighs 2 pounds per foot, how much work is done in lifting up the entire cable to the top of the building?
Answers
40*50=2000
2000/60=33.33
about 33 hours by lifting by hand
To find the work done in lifting the entire cable to the top of the building, multiply the weight of the cable by the height of the building.
Explanation:To calculate the work done in lifting the entire cable to the top of the building, we need to find the weight of the cable. The weight of the cable can be found by multiplying the length of the cable (20 feet) by its weight per foot (2 pounds per foot). Therefore, the weight of the cable is 20 feet × 2 pounds per foot = 40 pounds.
The work done in lifting the cable to the top of the building is equal to the product of the weight of the cable and the height of the building. So, the work done is 40 pounds × 50 feet = 2000 foot-pounds.
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In an x-y plot of an experiment what is usually plotted on the x axis?
a. the independent variable, which is the parameter that was manipulated.
b. th
Answers
A rectangular swimming pool has a volume capacity of 2160 cubic feet. Water is being added to the pool at a rate of about 20 cubic feet per minute. Determine about how long it will take to fill the pool completely if there were already about 1200 gallons of water in the pool.
Answers
Answer:
Step-by-step explanation:
y=mx+b
2160=20x+60.417)
2160= 20x+160.417
2160-160.417=20x+160.417-160.417
1999.583=20x
1999.583/20=20x/20
x= 99.98 min
Total time taken to fill the pool is 100 minutes
Given:
Amount of water already in pool = 1200 gallon
Total capacity of pool = 2160 cubic feet
Rate of water supply = 20 cubic feet per minute
Find:
Total time taken to fill the pool
Computation:
1 cubic feet = 7.5 gallon water
Amount of water already in pool = 1200 / 7.5
Amount of water already in pool = 160 cubic feet
Remain volume = 2160 - 160
Remain volume = 2000 cubic feet
Total time taken to fill the pool = 2000 / 20
Total time taken to fill the pool = 100 minutes
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Estimate instantaneous rate of change of the point indicated p(x) = 3x^2-5, x = 2
Answers
Ordering Least to greatest 2 9/11, 4/5, 2.91, 0.9
Answers
You would like to join a fitness center. Fit-N-Trim charges $80 per month. Fit-For-Life charges a one tim e membership fee of $75 and $55 per month. The lines shown in the graph represent the cost y to attend each fitness center for x months. Are the lines parallel?
Answers
y=55x+75
They are not parallel
Lakeya makes cupcakes to sell at her friend's lemonade stand. She sold 24 cupcakes and sold them for $3 each. Write a number sentence using the variable m to represent the total amount of money she made.
Answers
because she sold 24 cupcakes for $3 then you would multiply 24 and 3 to get the amount of money she made
Which of the following is rational
A. 3*pi
B. 2/3+9.26
C. sqrt of 45 + sqrt of 36
D.14.3... + 5.78765239...
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What is 164% of 25? I have no idea and im in the middle of a test XDDD
Answers
One side of a rectangle is 3 inches shorter than the other side, and the perimeter is 54 inches. Which of the following equations could be used to determine the dimensions of the rectangle
Answers
Side B: x-3
Perimeter: 2(l+w)
54"=2(x+x-3)
54"=2(2x-3)
54"=4x-6
54-6=4x
48/4=x
12=x
Therefore the longer side is 12" and the shorter side of the rectangle is (12-3)= 9"
Hope I helped :)
How many times does 1/2 fit into 30
Answers
30 ÷ [tex] \frac{1}{2} [/tex]
To divide by a fraction, multiply by its reciprocal (find reciprocal by flipping the fraction)
Ex. 30 * 2
Multiply 30 by 2 to get 60.
marissa sprinted for 48 yards. how many feet did she sprint.
Answers
HELP! solve for p p+10/p-7=8/9
Answers
use the cross method solve the equation.
9P+90=8P-56
9P-8P=-56-90
P=-146
Hope this helps.
a line that intersects the points (-23,-17) and (-13,-7), find the slope, now fill in the missing inputs to the slope-intercept formula -17=1(?)+b or (?)=(-13)+b
Answers
slope = (y2-y1) / (x2-x1)
The given points are (-23,-17) and (-13,-7), therefore:
y2 = -7
y1 = -17
x2 = -13
x1 = -23
Substitute with these values to get the slope as follows:
slope (m) = (-7--17) / (-13--23) = 1
The formula of the slope-intercept is:
y = mx+b where m is the slope and b is the intercept.
Using the point (-23,-17), the formula would be:
-17 = 1(-23) + b
Using the point (-13,-7), the formula would be:
-7 = 1(-13) + b
A point is located in a polar coordinate system by the coordinates r = 2.8 m and θ = 18°. find the x- and y-coordinates of this point, assuming that the two coordinate systems have the same origin.
Answers
lisha is making headbands using ribbon. she would like to make 12 head bands. Each one requires 15.5 inches of ribbon. She estimates that she will need to buy 160 inches of ribbon is her estimate reasonable? Explain your reasoning and lol I'm not in college i just did that
Answers
15.5 * 12 = 186
Her estimate is not reasonable.
186-160 = 26
She still needs 26 more ribbons.
A college freshmen must take a science course, a humanities course and a mathematics course. If she may select any of 6 science courses, any of 4 humanities and any of 4 mathematics courses, how many ways can she arrange her program? Make a tree diagram and list the possibilities.
Answers
what is the digit in the tenths place 15.16 ?