The cylindrical coordinates of the point (-2, 2, 2) are (r, θ, z) = (√8, 3π/4, 2).
The cylindrical coordinates of the point (-9, 9√3, 6) are (r, θ, z) = (18√3, -π/3, 6).
(a) To change the point (-2, 2, 2) from rectangular to cylindrical coordinates, we use the formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
z = z
Substituting the given values, we get:
r = √((-2)^2 + 2^2) = √8
θ = arctan(2/(-2)) = arctan(-1) = 3π/4 (since the point is in the second quadrant)
z = 2
(b) To change the point (-9, 9√3, 6) from rectangular to cylindrical coordinates, we use the formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
z = z
Substituting the given values, we get:
r = √((-9)^2 + (9√3)^2) = √(729 + 243) = √972 = 6√27 = 18√3
θ = arctan((9√3)/(-9)) = arctan(-√3) = -π/3 (since the point is in the third quadrant)
z = 6
(c) To express the region E in cylindrical coordinates, we need to find the limits of integration for r, θ, and z. Since the region is given by the inequalities:
x^2 + y^2 ≤ 9
0 ≤ z ≤ 4 - x^2 - y^2
In cylindrical coordinates, the first inequality becomes:
r^2 ≤ 9
or
0 ≤ r ≤ 3
The second inequality becomes:
0 ≤ z ≤ 4 - r^2
The limits for θ are not given, so we assume θ varies from 0 to 2π. Therefore, the region E in cylindrical coordinates is:
0 ≤ r ≤ 3
0 ≤ θ ≤ 2π
0 ≤ z ≤ 4 - r^2
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The conversion from rectangular to cylindrical coordinates are
(-2, 2, 2) ⇒ (2√2, -π/4, 2).
(-9, 9√3, 6) ⇒ (18, -π/3, 6).
How to find the coordinatesTo change from rectangular to cylindrical coordinates we use the formula below
r = √(x² + y²)
θ = arctan(y / x)
z = z
a
Using the given values
r = √((-2)² + 2²) = √(4 + 4) = √8 = 2√2
θ = arctan(2 / -2) = arctan(-1) = -π/4 (since x and y are both negative)
z = 2
hence in cylindrical coordinates, the point (-2, 2, 2) can be represented as (2√2, -π/4, 2).
b)
Using the given values (-9, 9sqrt(3), 6)
r = √((-9)² + (9√3)²) = √(81 + 243) = √324 = 18
θ = arctan((9√3) / -9) = arctan (-√3) = -π/3 radian
z = 6
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Jack is in a helicopter and looking down at two friends’ homes. The angle of depression to the first home is 72°. The angle of
depression to the second home is 49°. If the homes are 420m apart, what is the distance between the helicopter and the first
home?
The distance between the helicopter and the first home is 369. 87 m
How to solve the distance using the angle of depression?The distance between the helicopter and the first home can be found using sine law,
Therefore,
180 - 72 - 49 = 59 degrees.
Hence,
420 / sin 59 = A / sin 49°
where
A = distance form the first home to the helicopter.
cross multiply
420 sin 49 = A sin 59°
A = 420 sin 49 / sin 59
A = 316.978023694 / 0.8571673007
A = 369.869339199
A = 369. 87 m
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The average age of the members of a country club is 54 years old. If there are ten 36-year-olds, fifty 60-year-olds, and twenty other members all of the same age, what is the age of the twenty other members
The age of twenty other members is 48 years.
Given that the average age of the members of a country club is 54 years old and if there are ten 36-year-olds, fifty 60-year-olds, and twenty other members all of the same age.
The average is defined as the sum of a set of values divided by n, where n is the total set of values. A mean is another name for an average.
The average age is given by=(Sum of ages)/(Number of members (n))
Let x be the age of other twenty members.
Given A=54 years and n=10+50+20=80
So, now, we will substitute the values in the formula, we get
[tex]\begin{aligned}54&=\frac{10\times 36+50\times 60+20\times x}{10+50+20}\\ 54&=\frac{360+3000+20x}{80}\\ 4320&=3360+20x\end[/tex]
Further, subtract 3360 from both sides, we get
4320-3360=3360+20x-3360
960=20x
Furthermore, we will divide both sides with 20, we get
960/20=20x/20
48=x
Hence, the age of the twenty other members when average age of the members of a country club is 54 years old is 48 years.
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What is the solution to 9|x + 8| > 45?
The solution to the inequality is x > -3 or x < -13
How to solve the inequality?The inequality is given as:
9|x + 8| > 45
Divide through by 5
|x + 8| > 5
Split the inequality
x + 8 > 5 or x + 8 < -5
Solve for x
x > -3 or x < -13
Hence, the solution to the inequality is x > -3 or x < -13
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NEED HELP ASAP
Factor the trinomial: x² +22x + 72. Explain your steps.
I got you, bro
Answer:
(x+4)(x+18)
Explanation
Let's factor x2+22x+72
x2+22x+72
The middle number is 22 and the last number is 72.
Factoring means we want something like
(x+_)(x+_)
Which numbers go in the blanks?
We need two numbers that...
Add together to get 22
Multiply together to get 72
Can you think of the two numbers?
Try 4 and 18:
4+18 = 22
4*18 = 72
Fill in the blanks in
(x+_)(x+_)
with 4 and 18 to get...
(x+4)(x+18)
Answer:
(x+4)(x+18)
Answer:
(x+4)(x+18)
Step-by-step explanation:
Using the factorisation method :
We need 2 numbers that multiply to give +72 and add to give +22 :
To find these numbers we write out all the factors of 72 :
1, 72
2 , 36
3 , 24
4 , 18 ----> These must be the numbers as 4+18 = 22
Now we split +22x into +4x and +18x :
x² + 4x + 18x + 72
Factor the first 2 terms together and the last two terms :
x(x+4) + 18(x+4)
Keep 1 bracket and terms outside the brackets collect them into one:
(x+4)(x+18)
This is our final answer
Hope this helped and have a good day
Mary won $200,000 in a state lottery. she first paid income tax of 30% on the winnings. she invested some of the rest at 1.5% and some at 4%, earning $4,350 interest per year. how much did she invest at each rate?
Marry rate of investment is 4%.
Lets solve it through simple steps,
30% 0f $200,000 = $60,000
$200,000-$60,000 = $140,000
Rate 1.5%
Rate 4% = (140,000-x)
Interest earned = $4,350
1.5x+4(150,000-x)=4350*100
1.5x+600,000-4x=435,000
1.5x-4x=435,000-600,000
-2.5x=-165,000 /-2.5
x=66000
$ 66,000 1.5%
Balance = $74,000 at 4%
Hence the rate is 4%
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During a pumpkin launching contest, two pumpkins are launched at the same time. the path of pumpkin a is modeled by the equation y = –0.01x2 0.32x 4. the path of pumpkin b is modeled by the equation y = –0.01x2 0.18x 5. if x represents the distance the pumpkin traveled in feet and y represents height of the pumpkin in feet, what does the intersection point of the equations represent?
The distance at which pumpkins are the same height.
What do you mean by X and Y coordinates?
The X and Y coordinates are an address that help locate a point in two dimensions of space. The coordinates (x, y) are used to represent any point in the coordinate plane. The x value indicates the point's position in regard to the x-axis, and the y value indicates its position in relation to the y-axis.
SolutionBoth equations' graphs would be at an intersection, which would indicate that their x and y coordinates are identical. The same y coordinate would indicate being at the same height at that precise distance since y refers to the pumpkin's height.
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Answer:
D). the horizontal distance of the pumpkins from the launch site when their heights are the same
Step-by-step explanation:
I took the flipping test nickle
Find the common ratio for the geometric sequence defined by the formula: an=40(2‾√)n−1 a n = 40 ( 2 ) n − 1
The ratio of the geometric sequence 40[tex]2^{n-1}[/tex] is 2.
Given that geometric sequence is 40*[tex]2^{n-1}[/tex] and we have to find the common ratio of all the terms.
Geometric sequence is a sequence in which all the terms have a common ratio.
Nth termof a GP is a[tex]r^{n-1}[/tex] in which a is first term and r is common ratio.
Geometric sequence=40*[tex]2^{n-1}[/tex]
We have to first find the first term, second term and third term of a geometric progression.
First term=40*[tex]2^{1-1}[/tex]
=40*[tex]2^{0}[/tex]
=40*1
=40
Second term=40*[tex]2^{2-1}[/tex]
=40*[tex]2^{1}[/tex]
=40*2
=80
Third term=40*[tex]2^{3-1}[/tex]
=40*[tex]2^{2}[/tex]
=40*4
=160
Ratio of first two terms=80/40=2
Ratio of next two terms=160/80=2
Hence the common ratio of geometric sequence is 2.
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Find the interest on 16,100 for 4 years and 6months at 10% compounded quarterly.
25,110.51 is the total amount accrued over the course of 4.5 years, principle plus interest, on a principal of 16,100.00.
Compound interest calculationWe may employ the compound interest future value formula provided by:
[tex]A=P(1+r/n)^{nt}[/tex]
Where A is the final amount, P stands for the current value,
r=interest rate, n represents how many times interest is compounded annually, and t represents the number of years.
In this case,
P= 16100
r = 10%
t = 4 years and 6 months=4.5 years
n=4
So, A=[tex]16100(1+0.1/4)^{4*4.5}[/tex]
A= [tex]16100(1+0.025)^{18}[/tex]
A= 25110.51
So, 25,110.51 is the total amount accrued over the course of 4.5 years, principle plus interest, on a principal of 16,100.00 at a rate of 10 percent per year compounded four times each year.
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write the slope-intercept form of the equation of each line
1. 9x -7y = -7
2. 11x -4y = 32
3. 11x -8y =-48
One ticket to a ride of the merry-go-round at the Sunday Fair costs $2.
Jenny and her friends have $36 with them. If, after buying tickets to the merry-go-round, they want to be left with no less than $15 , find the maximum number of tickets that they can buy.
Considering the definition of an inequality, the maximum number of tickets that they can buy is 10.
Definition of inequalityAn inequality is the existing inequality between two algebraic expressions, connected through the signs:
greater than >.less than <.less than or equal to ≤.greater than or equal to ≥.An inequality contains one or more unknown values called unknowns, in addition to certain known data.
Solving an inequality consists of finding all the values of the unknown for which the inequality relation holds.
Maximum number of tickets that they can buyIn this case, you know that
One ticket to a ride of the merry-go-round at the Sunday Fair costs $2.Jenny and her friends have $36 with them.After buying tickets to the merry-go-round, they want to be left with no less than $15.So, they want to spend on the purchase of tickets for the merry-go-round a value less than or equal to $36 - $15= $21.
Being "x" the maximum number of tickets that they can buy, the inequality that expresses the previous relationship is
2x≤ 21
Solving:
x≤ 21÷2
x≤ 10.5
Then, the maximum number of tickets that they can buy is 10.
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Which inequality can be used to explain why these three segments cannot be used to construct a triangle? ac ab > cb ac cb < ab ac cb > ab ac ab < cb
The response option that demonstrates why the three segments cannot be combined to form a triangle is AC + CB< AB.
What is triangle inequalities?According to the triangle inequality, the length of any two sides of a triangle added together must be longer than the length of the third side. Since a straight line is the shortest distance between two places, this conclusion follows. If the triangle isn't degenerate, the inequality is strict (meaning it has a non-zero area).
Which inequality explains why a triangle cannot be made from the three segments?Since the length of any two sides of a triangle added together is longer than the length of the third side, it follows from the triangle inequalities theorem that this is the case.
As a result, the necessary inequality is AC + CB AB because the sum of sides AC + CB is less than AB.
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I understand that question you are looking for is:
Which inequality can be used to explain why these three segments cannot be used to construct a triangle?
A. AC + AB > CB
B. AC + CB < AB
C. AC + CB > AB
D. AC + AB < CB
5[tex]5x-3y=34\\x-2y=4[/tex]
a nickel has 5 grams what is the mass of a nickel in milligrams
The mass of the same nickel which is 5 grams in milligrams is 5,000 milligram.
Weight1 gram = 1000 milligramWeight of a nickel = 5 gram
Weight of same nickel in milligram = 5 grams × 1000
= 5,000 milligram
Therefore, the mass of the same nickel is 5,000 milligram
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The blue dot is at what value on the number line?
Answer:
blue dot have value of -6 in y inverse axis
PLEASE HELP ASAP!!!!!!!
Given TU is parallel to XW
Prove XW = 8
Which step is missing?
The missing step in the proof is: D. Statement: XW/UT = XV/UV, Reason: Corresponding sides of similar triangles are proportional.
What are Similar Triangles?When two triangles are proven to be similar to each other, their corresponding angles would be congruent while their corresponding sides will be proportional to each other.
In the proof given, both triangles are proven to be similar by the AA similarity theorem, therefore their corresponding sides will be proportional. This implies that: XW/UT = XV/UV.
The answer is D.
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#1) a normal distribution has a mean of 101 and a standard deviation of 4.2. what is the z-score of 105?
Step-by-step explanation:
z = (specific score - mean) / standard deviation
in our case
z = (105 - 101)/4.2 = 4/4.2 = 0.952380952... ≈ 0.95
as z-tables usually round the z-scores to hundredths.
The z-score of the normal distribution is 0.95.
What is normal distribution?A probability distribution which is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution.
It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean. The normal distribution shows as a "bell curve" on a graph.
Z-score: A Z-score is just a metric that quantifies how closely a value relates to a mean of a set of values. Standard deviations from of the mean are used to measure Z-score.
A Z-score of zero means the data point is actually score is the same as the mean score.
The formula for z-score is-
[tex]Z=\frac{x-\mu}{\sigma}[/tex]
Z = standard score
x is observed value = 105
[tex]\mu[/tex] is mean of the sample 101.
[tex]\sigma[/tex] is standard deviation of the sample 4.2.
Substitute all the values in the z-score formula;
z = ( 105 - 101)/4.2
z = 0.9523..
z = 0.95
Therefore, the z-score the given normal distribution is 0.95.
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There are 24 students in a class. if 16 are boys, what fraction of the students are boys?
Answer:
2/3 is the fraction of students who are boys.
Step-by-step explanation:
First of all, a fraction consists of two parts: The denominator and the numerator. The denominator is the whole, while the number is a part of the whole (and sometimes above).
1. Since in your question, there are 24 students in total in the class, the denominator will be 24.
2. You said that 16 of them are boys, so 16 is the numerator.
3. In a fraction, it will be 16/24.
4. 16/24 can be simplified into 2/3 since both 16 and 24 can be divided by 8.
So our final answer is 2/3.
N=
Help me please thanks :))
The measure of n of the sector with the given area is: 100°
What is the Area of a Sector?The area of a sector of a circle = ∅/360 × πr².
We are given that:
Area of the sector = 40π
Radius (r) = 12 units
∅ = n°
Plug in the values and solve for n
40π = n/360 × π(12²)
40π = n/360 × π144
Divide both sides by 144π
40π/144π = n/360 × 144π/144π
40/144 = n/360
40(360) = n(144)
14,400/144 = n
n = 100°
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mAFD=90 mAFB=31 Find mCFE
The value of angle CFE in the diagram attached is 51 degrees.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
From the diagram:
∠AFB = ∠CFD; ∠BFC = ∠DFE
Hence:
∠AFC = ∠CFE
∠AFD = ∠AFC + ∠CFD = ∠AFC + ∠AFB
90 = ∠AFC + 31
∠AFC = 51° = ∠CFE
The value of angle CFE in the diagram attached is 51 degrees.
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Charlie deposits $600 into an account that pays simple interest at a rate of 2% per year. How much interest will he be paid in the first 3 years?
Answer:
Step-by-step explanation:
rounded to the nearest cent his answer would be $212.24
Given the sequence 4, 8, 16, 32, 64, ..., find the explicit formula.
Answer:
[tex]\sf 2^{(n+1)}[/tex]
Step-by-step explanation:
Explicit formula is used to represent all the terms of the geometric sequence using a single formula.
[tex]\sf \boxed{\bf t_n=ar^{(n-1)}}[/tex]
Here, a is the first term.
r is the common ratio.
r = second term ÷ first term
4, 8,16,32,64,.....
a = 4
r = 8 ÷4 = 2
[tex]\sf t_n =4*2^{(n-1)}[/tex]
[tex]\sf = 4*2^n * 2^{(-1)}\\\\ = 4*2^n*\dfrac{1}{2}\\\\ = 2*2^n[/tex]
[tex]\sf = 2^{(n+1)}[/tex]
1 x/3 - 3/4 = 5/12
solve for x!
The value of x is 7/2
How to solve for x?The equation is given as:
x/3 - 3/4 = 5/12
Multiply through by 12
4x - 9 = 5
Add 9 to both sides
4x = 14
Divide by 4
x = 7/2
Hence, the value of x is 7/2
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From Andys house to Billys hometown you can travel by 3 roads. And to get from Billys hometown to Willies's house you can travel by 5 roads. How many possible wats are there to travel from andys house to willies house?
From Dan's ranch one road is built to get to Andy's house and two roads are built to get to Willies house (see previous problem). How many way are there now to get from Andys house to willies house?
Andy to get to Willie's house = 15 ways
Andy to get to Willie's house = 45 ways
How to determine the waysFrom the information given, we have that from
Andy's house to Billy's hometown = 3 roads
Billy's hometown to Willies house = 5 ways
To travel from Andy's house to Willies house, we should multiply the number of ways
= 3 × 5
= 15 ways
Dan's ranch to Andy's house = 1 road
To Willie's house = 2 roads
For Andy to get to Willie's house, we have;
= 3 × 15
= 45 ways
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If 0 is greater than f but less than or equal to 90, and cos(22f-1) = sin(7f+4), what is the value of f?
The value of f from the given expression is 3
Sine and cosine functionGiven the expression below
cos(22f-1) = sin(7f+4),
This can also be expressed
cos(22f-1) = cos(90-7f-4)
cos(22f-1) = cos(86-7f)
22f - 1 = 86 - 7f
Collect the like terms
22f+7f = 86 + 1
29f = 87
f = 3
Hence the value of f from the given expression is 3
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Someone help me solve these two problems having trouble solving them
The first equation is 5x² - 20x + 27 and the second equation will be y = 3x² + 6x + 15.
How to compute the equation?It should be noted that an equation simply means a formula that's used to express the equality of two expressions that are given in Mathematics.
It should be noted that before solving the equations, it's important to expand the functions that are squared and then solve accordingly. This will be illustrated below.
The first equation that's given is:
y = 5(x - 2)² + 7
= 5(x - 2)(x - 2) + 7
= 5(x² - 2x - 2x + 4) + 7
= 5(x² - 4x + 4) + 7
= 5x² - 20x + 20 + 7
= 5x² - 20x + 27
The second equation given is illustrated as:
y = 3(x + 1)² + 12
y = 3(x + 1)(x + 1) + 12
y = 3(x² + 2x + 1) + 12
y = 3x² + 6x + 3 + 12
y = 3x² + 6x + 15
In conclusion, writing the equations in form of y = ax² + bx + c will be y = 5x² - 20x + 27 and y = 3x² + 6x + 15.
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can someone please help mee
Answer:
I think that this answer is 22
The sum of two numbers is 140, and their difference is 20. What are the two numbers?
Answer:
60 and 80.
Step-by-step explanation:
If the numbers are x and y:
x + y = 140
x - y = 20 Adding:
2x = 160
x = 80
So y = 140 - 80 = 60.
Answer:
80 and 60
Step-by-step explanation:
Expressing the Word Problem mathematically.
Let the first number be X.
Let the second number be Y.
From the question,
[tex]x + y = 140 - - - - - (1) \\ x - y = 20 - - - - - (2) \\ From \: equation \: (2) \\ x = 20 + y - - - - - - (3) \\ Substitute \: equation \: 3 \: into \: 1 \\ 20 + y + y = 140 \\ 20 + 2y = 140 \\ 2y = 140 - 20 \\ 2y = 120 \\ Dividing \: bothsides \: by \: 2 \\ \frac{2y}{2} = \frac{120}{2} \\ y = 60 \\ Substitute \: y \: into \: quation \: 3 \\ x = 20 + 60 \\ x = 80 \\ Therefore, \: x = 80 \: and \: y = 60[/tex]
The two numbers are 80 and 60
The probability of event X is 30%. The probability of event Y is 40%. If events X and Y are mutually exclusive, what is the probability of event X or Y
The probability of event X or Y is P = 0.7, or, in percentage form, the probability is 70%.
How the get the probability of event X or Y?
First, two events are mutually exclusive if these events can't happen at the same time.
This means that if happens X, with a 30% of probability, then Y can't happen.
If happens Y, with a 40% of probability, then X can't happen.
Then the probability of X or Y (the probability that one of these two events happens, not both) is just the addition of the two individual probabilities.
Then we have:
P(X or Y) = 0.3 + 0.4 = 0.7
Or, in percent form, we get:
P(X or Y) = 70%
The probability of event X or Y is just 70%.
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Match the vocabulary word with the correct definition.
1. a line segment where two faces meet
edge
2. a three-dimensional figure with one circular base
dimensions
3. to separate something into its basic parts
area
4. units used to measure the volume of a three-dimensional figure
cube
5. a three-dimensional figure with two parallel, congruent, circular bases and a curved surface
decompose
6. a three-dimensional figure made of six congruent squares
composite figure
7. a geometric figure that is made up of two or more basic shapes
base
8. the length of a plane figure; or, a special face of a solid figures
cone
9. the measurement of the space inside a plane figure
cylinder
10. measurements indicating the size and shape of an object, such as length and width
cubic units
Answer:
1. edge
2. cone
3. decompose
4. cubic units
5. cylinder
6. cube
7. composite figure
8. base
9. area
10. dimensions
Graph the quadratic function by transforming the graph of f(x) = x^2. Give the minimum or maximum
vertex value and the equation for the axis of symmetry.
Answer: the maximum is (-2,5 )and the equation for the axis of symmetry is x = -2
Explanation if you want:
for the min and max: when a quadratic has a maximum point, that is the largest point that it will ever be at. When it has a minimum, that is the smallest.
the +5 tells us that we have an upward movement of 5. The +2 means we actually move to the point -2. bc: (x+2)=0 = -2so, from our initial function, those are the changes.for the axis of symmetry: basically, this is the line that splits the graph into two equal parts.
it is the line that goes through the vertex (which we already know is (-2,5).there u go