The length of the path described by the parametric equations x = cos(3t) and y = sin(3t) for 0 ≤ t ≤ π/2 is 3(π/2).
To find the length of the path, we need to use the formula for arc length:
L = integral from a to b of √(dx/dt)² + (dy/dt)² dt
where a and b are the starting and ending values of t.
Here, we have x = cos(3t) and y = sin(3t). Therefore,
dx/dt = -3sin(3t) and dy/dt = 3cos(3t)
Now, we can substitute these into the formula for arc length:
L = integral from 0 to π/2 of √((-3sin(3t))² + (3cos(3t))²) dt
L = integral from 0 to π/2 of √(9sin²(3t) + 9cos²(3t)) dt
L = integral from 0 to pi/2 of 3 dt
L = [tex]3[t]_0^{(\pi/2)[/tex] = 3(pi/2)
The length of the path described by the parametric equations x = cos(3t) and y = sin(3t) for 0 ≤ t ≤ π/2 is 3(π/2).
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The length of the path described by the parametric equations x = cos(3t) and y = sin(3t), for 0 ≤ t ≤ π/2, is given by the integral of the square root of the sum of the squares of the derivatives of x and y with respect to t.
Using the Pythagorean identity sin²θ + cos²θ = 1, we can simplify the length integral as follows:
L = ∫[0,π/2] √((dx/dt)² + (dy/dt)²) dt
L = ∫[0,π/2] √((-3sin(3t))² + (3cos(3t))²) dt
L = ∫[0,π/2] √(9sin²(3t) + 9cos²(3t)) dt
L = ∫[0,π/2] √9(dt)
L = 3 ∫[0,π/2] dt
L = 3[t] [0,π/2]
L = 3(π/2 - 0)
L = 3π/2
Therefore, the length of the path described by the given parametric equations for 0 ≤ t ≤ π/2 is 3π/2 units.
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Complete Question
the length of the path described by the parametric equations x=cos3t and y=sin3t , for 0≤t≤π2 is given by .
You want to retire on the equivalent of $50,000 per year in today's money. inflation is expected to be 3%. you will retire in 30 years. you will earn 10% annually on your investments from not till retirement and you will earn 8% during retirement (which is expected to last 20 years). how much do you need to save per month in order to fund this retirement?
$662.18 needs to be saved per month.
$90,000 now at 3% inflation will become 90000*1.03^30 after 30 years.
i.e. $218,454 is required in retirement years (for 20 years)
Calculating the PV of these payments at 7% is calculated as below.
FV = 0
1/y = 7%
n=20; PMT = 218454; calculate PV = $2,314,305
Now we want to get $2,314,305 at the start of retirement by saving per year. Enter the following in the financial calculator to get the per month amount.
FV = 2314305; PV = 0; n= 12*30; 1/y = 12%/12; calculate PMT = $662.18
So $662.18 needs to be saved per month.
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1) (x-y)(y-2)(x+2) / 2x^2+4yz+6 =
2) 4(x+y)(x-y)(y-x) / t+d+m =
3) ( √25+√9 / 2 ) + √21675 =
4) (9x^2+3x+27)(3+6y)(4+6y) / 1+2a+38+4c+2000z =
5) (x^2+6x+5) / (x+2)(x+6) =
^^^Simplify
6) (xy/m) + (xm/y) - (m/x) = 63 x=
Isolate
Could anyone help me with these?
See below for the solution to each expression
How to solve the expressions?Expression 1
The expression is given as:
(x-y)(y-z)(x+z)/(2x^2+4yz+6)
Expand the numerator
(x^2 - y^2 -xz + yz)(x + z)/(2x^2+4yz+6)
Further, expand the numerator
(x^3 - xy^2 - x^2z + xyz + x^2z - y^2z - xz^2 + yz^2)/(2x^2+4yz+6)
Evaluate the like terms
(x^3 - xy^2 + xyz - y^2z - xz^2 + yz^2)/(2x^2+4yz+6)
Hence, the equivalent expression of (x-y)(y-z)(x+z)/(2x^2+4yz+6) is (x^3 - xy^2 + xyz - y^2z - xz^2 + yz^2)/(2x^2+4yz+6)
Expression 2
The expression is given as:
4(x+y)(x-y)(y-x)
Apply the difference of two squares
4(x^2 - y^2)(y - x)
Expand the expressions in the brackets
4(x^2y - x^3 - y^3 + xy^2)
Open the bracket
4x^2y - 4x^3 - 4y^3 + 4xy^2
Hence, the equivalent expression of 4(x+y)(x-y)(y-x) is 4x^2y - 4x^3 - 4y^3 + 4xy^2
Expression 3
The expression is given as:
(√25+√9/2) + √21675
Take the square roots of 25, 9 and 21675
(5 +3/2) + 5√867
Evaluate the sum
13/2 + 5√867
Hence, the equivalent expression of (√25+√9/2) + √21675 is 13/2 + 5√867
Expression 4
The expression is given as:
(9x^2+3x+27)(3+6y)(4+6y) /1+2a+38+4c+2000z
Factor out 3 and 2 from the numerator
3(3x^2 +x + 9) * 3(1 + 2y) * 2(2 + 3y)/1+2a+38+4c+2000z
This gives
18(3x^2 +x + 9)(1 + 2y)(2 + 3y)/1+2a+38+4c+2000z
The expression cannot be further simplified.
Hence, the equivalent expression of (9x^2+3x+27)(3+6y)(4+6y) /1+2a+38+4c+2000z is 18(3x^2 +x + 9)(1 + 2y)(2 + 3y)/1+2a+38+4c+2000z
Expression 5
The expression is given as:
(x^2+6x+5)/(x+2)(x+6)
Factorize the numerator
(x + 1)(x + 5)/(x + 2)(x + 6)
Hence, the equivalent expression of (x^2+6x+5)/(x+2)(x+6) is (x + 1)(x + 5)/(x + 2)(x + 6)
Expression 6
6) (xy/m) + (xm/y) - (m/x) = 63
Factor out x
x(y/m + m/y) - m/x = 63
Evaluate the sum
x(y^2 + m^2)/y - m/x = 63
Multiply through by xy
x^2(y^2 + m^2) - my = 63xy
Add my to both sides
x^2(y^2 + m^2) = 63xy + my
The above implies that the variable x cannot be isolated from the equation
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Find the measure of the arc or central angle indicated. Assume that lines which appear to be diameters are actual diameters.
Answer:
? = 120°
Step-by-step explanation:
the central angle is the same as the arc that subtends it , that is
? = 120°
Tas
The three points marked on the graph are
the vertices (corners) of a parallelogram.
10
9
y
8
7
6
5
4
3
2
1
A
1 2
C
2
B
3 4 5 6 7 8 9
10
What are the coordinates of the fourth
vertex?
Answer: (8,10)
Step-by-step explanation:
To go from A to C, we go right 2 units and up 6 units.
So, if we let the fourth vertex be D, then to go from B to D, we must also go right 2 units and up 6 units from B.
Therefore, the answer is (8, 10).
A random sample of n = 4 scores is obtained from a normal population with m = 30 and s = 8. what is the probability that the sample mean will be smaller than m = 22?
the probability that the sample mean will be smaller than m = 22 is 0.02275.
For the given question,
A z-score measures exactly how many standard deviations a data point is above or below the mean. It allows us to calculate the probability of a score occurring within our normal distribution and enables us to compare two scores that are from different normal distributions.
Here sample size, n = 4
mean, μ = 30
standard deviation, σ = 8
The probability that the sample mean will be smaller than m = 22 is
z = [tex]\frac{m-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]
⇒ z =[tex]\frac{22-30}{\frac{8}{\sqrt{4} } }[/tex]
⇒ z = [tex]\frac{22-30}{\frac{8}{2} }[/tex]
⇒ z = [tex]-\frac{8}{4}[/tex]
⇒ z = -2
Refer the Z table for p value,
Thus for P(z < -2) is 0.02275.
Hence we can conclude that the probability that the sample mean will be smaller than m = 22 is 0.02275.
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Tom has x bags of apples. he buys 8 times more and sells 12. write the amount he has left as an algebraic expression:
Answer:
Step-by-step explanation:
x times 8 +12
8x+x=9x-12
Answer = 9x-12
WILL. GIVE BRAINLIEST IF RIGHT!!
Answer:
The class and exam medians are almost the same.
Step-by-step explanation:
The median of a box-and-whisker plot is the line inside the box.
• As we can see, the medians of class and median are very similar, both around 85. This is why the first option is correct.
•The second option is incorrect because the exam median is not much higher than the class median; the difference is very small.
• The third option is incorrect because exam does not have the lowest median. In fact, exam median is slightly higher than class median.
• The fourth option is incorrect because outliers do not affect the median.
1)Convert the following binary numbwers into decimal number(only a,b,c,f,i)
#1
(11)_2(1×2⁰+1×2¹)_10(1(1)+1(2))_10(1+2)_10(3)_10#b
(110)_2(0+1×2¹+1×2²)_10(2+4)_10(6)_10#c
(111)_2(1×2⁰+1×2¹+1×2²)_10(1+2+4)_10(7)_10#f
(10011)_2(1×2⁰+1×2¹+0+0+1×2⁴)_10(1+2+16)_10(19)_10#i
(10110101)_2(1×2⁰+0+1×2²+0+1×2⁴+1×2⁵+0+1×2⁷)_10(1+4+16+32+128)_10(181)_10Answer:
a) 3₁₀
b) 6₁₀
c) 7₁₀
f) 19₁₀
i) 181₁₀
Step-by-step explanation:
Binary to Decimal Conversion (Positional Notation Method)
Multiply each digit by the base (2) raised to the power dependent upon the position of that digit in the binary number. Sum all the values obtained for each digit.Express the number as a decimal number by placing subscript 10 after it.For a binary number with 'n' digits:
The right-most digit is multiplied by 2⁰The left-most digit is multiplied by [tex]\sf 2^{n-1}[/tex]For example, to convert the binary number 111001₂ into a decimal:
[tex]\begin{array}{ c c c c c c}1 & 1 & 1 & 0 & 0 & 1\\\downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0\\\end{array}[/tex]
Multiply each digit by the base (2) raised to the power as indicated above and sum them:
[tex]=(1 \times 2^5)+(1 \times 2^4)+(1 \times 2^3)+(0 \times 2^2)+(0 \times 2^1)+(1 \times 2^0)[/tex]
[tex]= 32+16+8+0+0+1[/tex]
[tex]= 57[/tex]
Finally, express as a decimal number ⇒ 111001₂ = 57₁₀
Question (a)
[tex]\begin{aligned}\implies 11_2 & = (1 \times 2^1)+(1 \times 2^0)\\& = 2+1\\& = 3\end{aligned}[/tex]
Therefore, 11₂ = 3₁₀
Question (b)
[tex]\begin{aligned}\implies 110_2 & = (1 \times 2^2)+(1 \times 2^1)+(0 \times 2^0)\\& = 4+2+0\\& = 6\end{aligned}[/tex]
Therefore, 110₂ = 6₁₀
Question (c)
[tex]\begin{aligned}\implies 111_2 & = (1 \times 2^2) +(1 \times 2^1)+(1 \times 2^0)\\& =4+2+1\\& = 7\end{aligned}[/tex]
Therefore, 111₂ = 7₁₀
Question (f)
[tex]\begin{aligned}\implies 10011_2 & =(1 \times 2^4)+(0 \times 2^3)+ (0 \times 2^2) +(1 \times 2^1)+(1 \times 2^0)\\& =16+0+0+2+1\\& = 19\end{aligned}[/tex]
Therefore, 10011₂ = 19₁₀
Question (i)
[tex]\phantom{)))}10110101_2 \\\\=(1 \times 2^7)+(0 \times 2^6)+(1 \times 2^5)+(1 \times 2^4)+(0 \times 2^3)+ (1 \times 2^2) +(0 \times 2^1)+(1 \times 2^0)\\\\=128+0+32+16+0+4+0+1\\\\= 181[/tex]
Therefore, 10110101₂ = 181₁₀
"The average speed of a vehicle and the distance it covers in a period " Is this a direct proportion
Answer:
Yes
Step-by-step explanation:
As the Average speed increases, the amount of distance covered will increase. As the average speed decreases, the amount of distance traveled will decrease as well over a certain period.
15 athletes are assigned to one of three teams. Each team will have five athletes. The teams are different, so it matters who gets assigned to which team. How many ways are there to assign the athletes to the teams
Answer:
There is only one way to assign the athletes to the team by giving 5 athletes to each team
HELP!!!!!!QUICK!!!!!!!
Which expression is equivalent to startfraction 3 x superscript negative 6 baseline y superscript negative 3 baseline over 15 x squared y superscript 10 baseline endfraction? assume x not-equals 0, y not-equals 0.
The correct expression is 1 Over 5 x Superscript minus 8 Baseline y Superscript minus 13 Baseline endFraction.
Given expression [tex](3x)^-6*y^-3/ 15x^2*y^10[/tex]
Lets simplify the equation:
[tex]= > (3x)^-6*y^-3/ 15x^2*y^10\\=1/5 x^-8*y^-13[/tex]
Expression:
An expression is a sentence with a minimum of two numbers or variables and at least one math operation. This math operation can be addition, subtraction, multiplication, or division. The structure of an expression is: Expression is (Number/Variable, Math Operator, Number/Variable)
Hence the correct expression is 1 Over 5 x Superscript minus 8 Baseline y Superscript minus 13 Baseline EndFraction.
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How many minutes are equal to 2 hours 20 minutes? how many minutes are equal to 2 hours 20 minutes? 100 120 220 240 none of these
Answer:
None of the answers
Step-by-step explanation:
There are 60 minutes in one hour so to find the number of minutes in two hrs we multiply 2 by 60: 2*60 = 120 minutes
We then add the extra 20 minutes: 120+20 = 140 minutes
None of the answers include 140 minutes.
Find the polar coordinates in radians from the given rectangular coordinates (-11, -3).
the polar coordinates are:
(√130, 285.3°)
How to find the polar coordinates?For rectangular coordinates (x, y), the polar coordinates are:
[tex]R = \sqrt{x^2 + y^2}[/tex]
For the angle θ we need to notice that if both coordinates are negative, like in this case, we would be on the fourth quadrant, then:
θ = arctan(y/x) + 270°
In this case, we have:
x = -11
y = -3
Then:
[tex]R = \sqrt{(-11)^2 + (-3)^2} = \sqrt{130} \\\\\theta = arctan(-3/-11) + 270\° = 285.3\°[/tex]
Then the polar coordinates are:
(√130, 285.3°)
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I need help solving this problem
Answer:
Please see the picture below.
Step-by-step explanation:
what is 23,578 times 39,344
Answer:927652832
Step-by-step explanation:
using long multiplication step by step multiply.
SOMEONE PLEASE HELP!!! Find the area of the sector. Round your answer to the nearest tenth.
Answer:
398.2 cm^2
Step-by-step explanation:
[tex]\frac{270}{360}[/tex] * [tex]169\pi[/tex] ≈398.2 cm^2
Select the correct answer from the drop-down menu.
Compare the steps for constructing a perpendicular line to line ABthrough point C using a reflective device and using paper.
.C
A
When using paper to construct the perpendicular line, the paper is folded so that
line ABwill lie exactly on top of each other.
When using a reflective device to construct a line perpendicular to line AB, set one end of the device to coincide with
then swing the other end of the device so the reflection of line ABcoincides with
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and the two sides of
and
The C is perpendicular to AB
When using paper, the paper perpendicular te construct to is folded so that AB is divided equally and the two sides of. line AB will lie exactly on top of each other.
When using a reflective device to construct a live perpendicular to line AB, set One end of the device to coincide with point C, and then swing the other end of the device so that the reflection of line AB coincides with the midpoint of AB
The question is incomplete hence the complete question is shown here
Select the correct answer from the drop-down menu. Compare the steps for constructing a perpendicular line to line through point using a reflective device and using paper. , When using paper to construct the perpendicular line, the paper is folded so that and the two sides of line will lie exactly on top of each other. When using a reflective dewice to construct a line perpendicular to line , set one end of the device to coincide with , and then swing the other end of the device so the reflection of line coincides with.
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Answer:
I toke the test, I think the first one is A and B coincide
Step-by-step explanation:
pls help me on this one :((
Answer:
The answer is B) b
Step-by-step explanation:
According to the Trapezoid area formula, the area of the trapezoid is [(2d+a+b) × c] ÷ 2, however the area also equals (d+b) × c.
So d+(a+b)/2=d+b
(a+b)/2=b
a+b=2b
a=b
hope this helps <3
how to simplify cos(6x) + 1
Can you help me,pls
Answer:
x^4 +2 +x^-4
Step-by-step explanation:
The square of a binomial is a form worthy of memorization.
(a +b)² = a² +2ab +b²
ApplicationHere, you have a binomial with a=x² and b=1/x². Using these values in the pattern, we have ...
[tex]\left(x^2+\dfrac{1}{x^2}\right)^2=(x^2)^2+2(x^2)\dfrac{1}{x^2}+\left(\dfrac{1}{x^2}\right)^2\\\\=\boxed{x^4+2+\dfrac{1}{x^4}}[/tex]
Jake started biking to the coffee shop traveling 8 mph, after some time the bike got a flat so Jake walked the rest of the way, traveling 4 mph. If the total trip to the coffee shop took 9 hours and it was 60 miles away, how long did Jake travel at each speed
Jake travel to the coffee shop by bike with speed of [tex]8mph[/tex] is [tex]56mile[/tex] and when bike got a flat so Jake walked [tex]4mile[/tex]
How to find the speed by biking and walking ?
let [tex]b[/tex] = time spent biking
then [tex]8-b[/tex] =time spent walking
Write distance equation
[tex]dist=speed *time[/tex]
So bike distance [tex]+[/tex] walk distance [tex]=60mile[/tex]
[tex]8b+4(8-b)=60\\8b+32-4b=60\\4b=60-32\\b=28/4\\\\b=7hrs[/tex]
Then [tex]8-7=1 hrs[/tex] spent walking.
For cross check the answer find that actual distance for each
[tex]8(7)=56mi\\4(1)=4mi\\-------\\total distance=60mile[/tex]
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The area of Marc's square garden is 225 ft².
What are the dimensions of his garden?
F. 5 ft by 45 ft
H. 15 ft by 15 ft
G. 12 ft by 12 ft
J. not given
Answer:
H. 15 ft by 15 ft
Step-by-step explanation:
area of square = side²
225 ft² = side²
side = √(225 ft²)
side = 15 ft
Answer: H. 15 ft by 15 ft
The expression 40x²-65x + 50 represents the sum of the interior angles of a regular pentagon in degrees. If the
interior angles of the pentagon are equal, which expression represents the measure of two angles?
Answer:
[tex]16x^2-26x+20[/tex]
Step-by-step explanation:
Since it represents the sum of interior angles, the angle is essentially: [tex]angle + angle + angle + angle + angle[/tex], and there are 5 angles since a pentagon has 5 interior angles. To find what one interior angle is, we simply divide the polynomial by 5, to do this we can just divide each coefficient by 5 (distributing the division): [tex]\frac{40x^2-65x+50}{5} = \frac{40x^2}{5} - \frac{65x}{5} + \frac{50}{5}[/tex] this simplifies to: [tex]8x^2-13x+10[/tex]. This represents what one of the interior angles is. Since we want an expression measuring two of the angles, we simply multiply this polynomial by 2, to do this we just multiply each coefficient by 2 (distributing the 2): [tex]2(8x^2-13x+10) = 16x^2-26x+20[/tex] which is the solution
Please i really need help
for both qeutions
100 points
Answer:
7. a) slope = 5
7. b) slope = 1/2
8. slope = 1
Step-by-step explanation:
slope = rise/run
rise is the vertical change, run is the horizontal change
7. a)
Pick any two points on your line. (0,0) and (1,5) are easy to work with.
The vertical change between the two points is 5
The horizontal change between those points is 1
so slope = 5/1 = 5
7. b)
Again, you can use (0,0) as one of your points. The point (2,4) is good as the second.
slope = 4/2 = 1/2
8. We don't have a graph, but we are given two points to work with, (1,4) and (5,8). Use the formula given...
m = slope = (y2-y1) / (x2-x1)
= (8 - 4) / (5 - 1)
= 4/4
= 1
I hope that helped!
Which of the statements below is true for the following set of numbers?42,10,36,51,70,28
Answer:
statement 1(one)
Step-by-step explanation:
One cupcake weighs 3 1/2 ounces how many cupcakes are there in 28 ounce package
Using division, the 8 cupcakes are there in 28 ounce package if one cupcake weighs 3 1/2.
According to the question,
One cupcake weighs 3 1/2 ounces. In order to find the number of cupcakes are there in 28 ounce package.
1 cupcake = 7/2 ounces
x cupcakes = 28 ounces
28 = 7/2 * x [cross multiplication]
x = (28*2)/7 [Using division]
x = 8
Hence, using division, the 8 cupcakes are there in 28 ounce package if one cupcake weighs 3 1/2.
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A sector has a radius of 12 centimeters and an angle 65 degrees. find its arc length
If the radius of sector is 12 centimeters and angle of 65 degrees then the arc's length is 13.6 cm.
Given a sector has a radius of 12 cm and an angle 65 degrees.
We have to calculate arc's length.
We know that arc's length is equal to Θr where Θ is in radians.
We have been given angle in degrees ,so we have to convert degrees in radians.
Angle=65*π/180=13π/36
length of arc =Θr
Putting the values in the formula above :
length =13π/36*12
=13*3.14/3
=40.82/3
=13.6
Hence if the radius of sphere is 12 cm and angle 65 degrees then the length of arc of sphere is 13.6 cm.
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HELP! what is this? what’s the answer?!
D. not continuous at X=0
if you trace the function you'll find a jump in the graph. To be continuous the line can't have any breaks in it. In this example, it's discontinuous when X=0
10 points to you if you help me
and brainlyist
Answer:
Step-by-step explanation:
For the frequency distribution, count the numbers that satisfy each category.
79 to 83: 4 (81, 82, 83, 81)
84 to 88: 3 (87, 88, 86)
89 to 93: 7 (92, 93, 93, 93, 90, 90, 91)
94 to 98: 1 (97)
For the historagram, raise the bars at the bottom of the histogram to the numbers above (4, 3, 7, 1; in this order)