The length of the curve r(t) over the interval 0 ≤ t ≤ 2π is approximately 285.97 units.
The length of the curve given by the vector-valued function r(t) over the interval [a, b] is given by the formula:
L = ∫[a,b] ||r'(t)|| dt
where r'(t) is the derivative of r(t) with respect to t and ||r'(t)|| is its magnitude.
In this case, we have:
r(t) = ⟨4cos(5t), 4sin(5t), t^(3/2)⟩
r'(t) = ⟨-20sin(5t), 20cos(5t), (3/2)t^(1/2)⟩
||r'(t)|| = √( (-20sin(5t))^2 + (20cos(5t))^2 + ((3/2)t^(1/2))^2 )
||r'(t)|| = √( 400sin^2(5t) + 400cos^2(5t) + (9/4)t )
||r'(t)|| = √( 400 + (9/4)t )
So the length of the curve over the interval [0, 2π] is:
L = ∫[0,2π] √( 400 + (9/4)t ) dt
Making the substitution u = 20t^(1/2)/3, we get:
du/dt = 10t^(-1/2)/3
dt = (3/10)u^(-1/2) du
When t = 0, u = 0, and when t = 2π, u = 20√(π)/3. Substituting these values and simplifying, we get:
L = ∫[0,20√(π)/3] √( 1 + u^2 ) du
Using the substitution x = sinh(u), we get:
dx/dt = cosh(u)
dt = dx/cosh(u)
When u = 0, x = 0, and when u = 20√(π)/3, x = sinh(20√(π)/3). Substituting these values and simplifying, we get:
L = ∫[0,sinh(20√(π)/3)] √( 1 + sinh^2(x) ) dx
L = ∫[0,sinh(20√(π)/3)] cosh(x) dx
Using the formula for the integral of cosh(x), we get:
L = sinh(sinh(20√(π)/3)) - sinh(0)
L ≈ 285.97
Therefore, the length of the curve r(t) over the interval 0 ≤ t ≤ 2π is approximately 285.97 units.
Learn more about length of the curve here
https://brainly.com/question/31376454
#SPJ11
In the triangle below, what is the measure (in degrees) of angle X?
66°
55°
X
Answer:
59
Step-by-step explanation:
Angle degrees in a triangle add up to 180
66+55+x=180
121+x=180
x=59
Answer
59°
Step-by-step explanation:
A triangle's angles add up to 180°.
So to find the last angle of the shape, you need to add the given angles and then subtract the answer from 180° looking like this:
66° + 55° = 121°
180° - 121° = 59°
Fifty people enter a contest in which three identical prizes will be awarded. In how many different ways can the prizes be awarded to the entrants
There are 117600 ways for the distribution of the prizes.
According to the statement
We have given that the number of people enter in a contest is 50.
And
Number of prizes given identically is 3.
And we have to find that the by how many ways can be prizes be awarded.
So,
Provided that no one can receive more than one prize,
It means The first prize can be given to any of the 50 people, the second to any of the remaining 49, and the third to the remaining of the 48,
Here we use Permutation
Then the possibilities become is
50*49*48= 117600
So there are above written ways to distribute the prizes.
There are 117600 ways for the distribution of the prizes.
Learn more about Permutation here https://brainly.com/question/4658834
#SPJ4
An arithmetic sequence is a linear function whose domain is restricted to the set of non-negative integers.
"An arithmetic sequence is a linear function whose domain is restricted to the set of non-negative integers."
The general form an arithmetic sequence is:
[tex]a_{n} = a_{1} + (n- 1) d[/tex]
The function is linear. The sequence consists of either numbers that are increasing or decreasing based on the value of d, the common difference.So, the statement provided is True.
Learn more about arithmetic sequence
brainly.com/question/10396151
#SPJ4
Alex has 84 quarters and nickels worth a total of $17.80.
Which equation can be used to find q, the number of quarters Alex has?
The equation which can be used to find the number of quarters and nickels is; n + q = 84, 0.05n + 0.25q = 17.80.
Quarters and nickelsTotal coins = 84Total value = $17.80let
Number of quarters = qNumber of nickels = nn + q = 84
0.05n + 0.25q = 17.80
From equation (1)
n = 84 - q
Substitute into (2)
0.05n + 0.25q = 17.80
0.05(84 - q) + 0.25q = 17.80
4.2 - 0.05q + 0.25q = 17.80
- 0.05q + 0.25q = 17.80 - 4.2
0.20q = 13.60
q = 13.60 / 0.20
q = 68
Substitute q = 68 into
n + q = 84
n + 68 = 84
n = 84 - 68
n = 16
Therefore, the number of quarters and nickels are 68 and 16 respectively.
Learn more about quarters and nickels:
https://brainly.com/question/17127685
#SPJ1
How many three-digit numbers use only the digits 0,1 and 5
The number of three digit numbers which use only the digits 0,1, and 5 is; 18.
How many three digit numbers use 0,1 and 5?Since, it follows from the task content that three digit numbers are to be considered, Hence, it follows that the choice for the first digit is between 1 and 5, for the second number and third number, the choice lies within all three numbers.
On this note, we have;
= 2× 3 × 3.
= 18.
Read more on combinations;
https://brainly.com/question/11732255
#SPJ1
Laurel appears at Comedy Cafe at a random time between 1:00 and 2:00. Hardy turns up at a random time between 2:00 and 2:30. The probability that Laurel waits for no more than 40 minutes for Hardy can be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m n$.
Answer:
i need more info please
Step-by-step explanation:
x/4 ÷ 2/3 = 1/5
solve for x
The value of x from the equation x/4 ÷ 2/3 = 1/5 is 8/15
Fractionx/4 ÷ 2/3 = 1/5
x/4 × 3/2 = 1/5
(x × 3) / (4 × 2) = 1/5
3x / 8 = 1/5
cross product3x × 5 = 8 × 1
15x = 8
x = 8/15
Therefore, value of x from the equation x/4 ÷ 2/3 = 1/5 is 8/15
Learn more about fraction:
https://brainly.com/question/11562149
#SPJ1
rationalize the denominator
PLS HELP
Answer:
x = 5, y = sqrt6, z = 3
Step-by-step explanation:
[tex]\frac{10}{\sqrt{6} } \\\\\frac{(10)(\sqrt{6} )}{(\sqrt{6} )(\sqrt{6}) } \\\\\frac{10\sqrt{6} }{6} \\\\or\\\\\frac{5\sqrt{6} }{3}[/tex]
❄ Hi there,
let us revise the way we rationalize the denominator before we get down to the actual process.
__________
Suppose you had a problem, and you did all the steps, and you arrived at the following answer: [tex]\sf{\dfrac{10}{\sqrt6}}[/tex].
Well, this answer is not in its simplest form, because it has an irrational number in its denominator.
So what we do is we get rid of that irrational number in the denominator by multiplying the fraction's top & bottom by [tex]\sqrt6[/tex]:
[tex]\sf{\dfrac{10\sqrt6}{\sqrt6\times \sqrt6}}[/tex]
Now we simplify
[tex]\sf{\dfrac{10\sqrt6}{6}}[/tex]
Reduce
[tex]\sf{\dfrac{5\sqrt6}{3}}[/tex]
- - - - - - - -
please helppp I really don't understand this lol
Answer:
Khan Academy.
Step-by-step explanation:
You can go on khan academy and search up "percentages and numbers." The videos should immediately pop up. there are also lessons if you don't want to risk getting your problem wrong. If you are still confused and do not understand please just comment and I will respond.
The Random Variable is normally distributed with mean = 98 and standard deviation =
20.41.
On the sampling distribution with n = 81, for what value of the sample mean (X) would have
10% of the possible sample means less than it? Round your answer to 2 decimal places.
If mean is 98, standard deviation is 20.41, n=81 then sample mean will be 131.57445 which would have 10% of the possible mean less than it.
Given that mean is 98,standard deviation is 20.41,n=81 and significance level be 10%.
We are required to find the value of sample mean.
Because the value of n is greater than 30 so we will use z statistics in the problem.
We know that,
Z=(X-μ)/σ in which μ is population mean, σ is standard deviation.
Confidence level=1-0.01=0.09 means 90%.
Z value for 90% confidence level be 1.645.
we have to put the value of Z=1.645 , μ=98,σ=20.41 to get the value of X.
1.645=(X-98)/20.41
1.645*20.41=X-98
33.57445=X-98
X=98+33.57445
X=131.57445
Hence if mean is 98, standard deviation is 20.41, n=81 then sample mean will be 131.57445 which would have 10% of the possible mean less than it.
Learn more about z test at https://brainly.com/question/14453510
#SPJ1
Which set of numbers can represent the side lengths, in millimeters, of an obtuse triangle?
8, 10, 14
9, 12, 15
10, 14, 17
12, 15, 19
Answer:
8, 10, 14
Step-by-step explanation:
1. 8² + 10² = 164 and 14² = 196, so this works.
2. 9² + 12² = 15² = 225, so this is a right triangle.
3. 10² + 14² = 296 and 17² = 289, so this is an acute triangle.
4. 12² + 15² = 369 and 19² = 361, so this is an acute triangle.
Answer:
a
Step-by-step explanation:
Explain why the triangles are similar, and then find the values of x and y.
The values of x and y from the figure is 9 and 5 respectively
Similarity theorem of triangleTwo triangles are similar if the ratio of the similar sides is equal to a constant. From the given diagrams;
x/(8+4) = 6/8
x/12 =6/8
8x = 6 * 12
8x = 72
Divide both sides by8 to have:
8x/8 = 72/8
x = 9
Similarly;
10+y/x = 10/6
10+y/9 = 10/6
10+y/9 = 5/3
Cross multiply
3(10+y) = 5(9)
30 + 3y = 45
3y = 45-30
3y = 15
y = 5
Hence the values of x and y from the figure is 9 and 5 respectively
Learn more on similar triangles here: https://brainly.com/question/14285697
#SPJ1
A linen napkin, in the shape of a square, has sides that are 10 inches long.
If the napkin is folded along the diagonal, whats the length of the diagonal?
round to the nearest tenth.
Answer: A=187.5,2
Step-by-step explanation:
At the movie theatre, child admission is and adult admission is . On Wednesday, tickets were sold for a total sales of . How many child tickets were sold that day
The child tickets were sold that day is 82.
Given the cost of child admission $6.30 and the cost of adult admission $9.80 and on Wednesday 144 tickets were sold for a total sales of $1124.20.
Let the child tickets are x and adult tickets are y.
On Monday, 144 tickets were sold for a total sales.
x+y=144 ........(1)
Total sales = $1124.20
Need to find the child tickets sold on Wednesday.
Total sales = 6.30x+9.80y
Total sales = $1124.20
So,6.30x+9.80y=1124.20 .......(2)
Find the value of y from equation (1), we get
x+y-x=144-x
y=144-x
Now, we will substitute the value of y in equation (2) to find the value of x, we get
6.30x+9.80(144-x)=1124.20
Apply the distributive property that is a(b+c)=ab+ac, we get
6.30x+9.80×144-9.80x=1124.20
6.30x+1411.2-9.80x=1124.20
Combine the variable terms on left side, we get
1411.2-3.5x=1124.20
subtract 1411.2 from both sides, we get
1411.2-3.5x-1411.2=1124.20-1411.2
-3.5x=-287
Divide both sides with -3.5x, we get
-3.5x/(-3.5)=-287/(-3.5)
x=82
Therefore, 82 child tickets were sold on Wednesday when cost of child admission $6.30 and the cost of adult admission $9.80.
Learn more about variable from here brainly.com/question/14786534
#SPJ4
Sofa co is selling sofas with a 33% discount off the original price to reduce its stock. Calculate the discounted price of a sofa originally selling for £999.00. Please show your working.
pls help me..
look at the pic below!!
Answer:
y = 3(x - 2)² - 5==================
Given equationy = 3x² - 12x + 7Convert this to vertex formy = a(x - h)² + kFollow the stepsy = 3x² - 12x + 7 =
3(x² - 4x) + 7 = Factor out 3
3(x² - 4x + 4) - 3*4 + 7 = Complete the square
3(x - 2)² - 5 Simplify
Answer:
y = 3(x -2)² -5
Step-by-step explanation:
It looks like you want to write the standard-form quadratic ...
y = 3x² -12x +7
in vertex form.
The process is similar to that of "completing the square."
StepsFactor out the leading coefficient
Do this for the variable terms only.
y = 3(x² -4x) +7
Identify the x-term coefficient
The x-term coefficient (inside parentheses) divided by 2 is the opposite of the x-coordinate of the vertex. It is the constant term in the squared binomial of the vertex form. Square this value.
x coefficient: -4
binomial constant: -4/2 = -2
square of this: (-2)² = 4
Add this inside parentheses
To keep the overall equation unchanged, you must also subtract an equal amount outside parentheses.
y = 3(x² -4x +4) +7 -3(4)
Simplify to vertex form
Write the trinomial in parentheses as a square, and simplify the y-coordinate of the vertex.
y = 3(x -2)² -5
According to a secret government report, 1/3 of
all the jelly beans in the world are red. If that
1/3 amounts to 38,567,942 red jelly beans, how
many jelly beans are there in total?
Proportionately, the total jelly beans in the world will be 115,703,826.
What is a proportion?A proportion refers to a fraction of a whole.
A proportion is a ratio or a number relationship between two or more values or variables.
The proportion of a whole value can be represented as a fraction, decimal, or percent.
Data and Calculations:Red jelly beans in the world = 1/3
And 1/3 = 38,567,942
1/1 = 115,703,826 (38,567,942/1/3)
Thus, if 1/3 of the jelly beans in the world is 38,567,942, then the whole jelly beans will be 115,703,826.
Thus, proportionately, the total jelly beans in the world will be 115,703,826 because 1/3 of 115,703,826 is 38,567,942.
Learn more about proportions at https://brainly.com/question/1496357
#SPj1
The volume of a cylinder is 100. Another cylinder has twice the height but half the base radius. What is the volume of the second cylinder
Answer:
50 units³Step-by-step explanation:
Cylinder volume formula:
V = πr²hGiven:
The first cylinder, has base radius - r and height - h. Its volume is:
V₁ = πr²h = 100The second cylinder has:
R = r/2 and H = 2hIts volume is:
V₂ = πR²H = π(r/2)²*(2h) = πr²h/2 = V₁/2Substitute the value of the first cylinder:
V₂ = V₁/2 = 100/2 = 50Which equation gives the lines show on Graph y= 2x+4
y= -1/2x-4
y=1/2x+4
y=2x-4
A private tutoring company charges $20 for a 1-hour session. Currently, 300 students are each tutored for an hour each week. Since the company is losing money, the owner has decided to increase the price. For each 50¢ increase, she estimates that 5 fewer students will participate. If the company needs to bring in $6,210 per week to stay in business, what price must be charged for a 1-hour tutoring session to produce this amount of revenue?
The price that must be charged for a 1-hour tutoring session to produce this amount of revenue is 24.5 dollars
How to solve Word ProblemThe first thing to do with word problem is to understand the question. After that, make analysis of the question.
It is given that a A private tutoring company charges $20 for a 1-hour session. Currently, 300 students are each tutored for an hour each week. Since the company is losing money, the owner has decided to increase the price. For each 50¢ increase, That is $20.5 for each students.
If she estimates that 5 fewer students will participate, that is, 295 students, and the company needs to bring in $6,210 per week to stay in business.
Let us assume that 1 hour per day. And there are 7 days in a week.
Divide the money by 7 to know the value per day
6210/7 = 887.14
Now, divide the amount per day by 250 students to know the amount per student per hour
887.14/295 = 3.007
The price that must be charged for a 1-hour tutoring session to produce this amount of revenue should be 20.5 + 4 = 24.5 dollars
Therefore, each student will pay $24.5 per hour
Learn more about Word Problem here: https://brainly.com/question/13818690
#SPJ1
A civil engineer is designing a public parking lot for the new town hall. The number of cars in each row should be six less than the number of rows in the lot. The mayor has requested that the new lot should hold twice as many cars as the current parking lot, which has space for 56 cars.
Determine which equation the civil engineer can use to find the number of rows, x, he should include in the new parking lot.
A.
6x2 − 56 = 112
B.
x2 − 6 = 112
C.
x2 − 6x = 112
D.
2x2 − 12x = 112
The equation which the civil engineer can use to find the number of rows is 2x² - 12x = 112. Option D
How to determine the equation
Let the number of rows be x
Number of cars in each row = x - 6 = 56
Number of rows for parking lot = 2x
Then,
Number of cars for new parking lot = 2x ( x - 6 = 56)
Expand the expression,
2 ( x - 6 = 56)
2x² - 12x = 112
Thus, the equation which the civil engineer can use to find the number of rows is 2x² - 12x = 112. Option D
Learn more about word problems here:
https://brainly.com/question/1781657
#SPJ1
Q.estimate the area of each polygon in square units.
i cant understand the question can someone help me and provide detailed explanation with the solution. please provide reasons so i can grasp the concept.
The area of the polygon is 12 square units. In mathematics and everyday life, an estimate is an approximate calculation or guess that is close to the actual value but may not be exact.
To make an estimate or guess, you can square off the parallelogram and create a rectangle and add your squares (or multiply 3x4) to = 12": This part explains a method for estimating the area of a parallelogram. It suggests that if you take a parallelogram and "square it off" (convert it into a rectangle by drawing a perpendicular line), you can then find the area of the rectangle, which is easier to calculate. In this case, the dimensions of the rectangle are given as 3 units for the base and 4 units for the height, and the estimated area is 12 square units.
You can use the formula for the exact area using base x height (3x4) you get the same answer of 12 square units": This part emphasizes that the estimation method used in step 2 (multiplying 3 by 4) produces the same result as the exact formula for finding the area of a parallelogram, which is base times height. In this case, both the estimation and the formula give an area of 12 square units.
To know more about polygon:
https://brainly.com/question/33245559
#SPJ3
HELP IM BEING TIMED I NEED THE ANSWER QUICKLY AS POSSIBLE!!
What is the range of the absolute value function below?
According to the graph
It's a Translated modulus functionIt has vertex at (4,1)Maximum value is 1
So
Range is
f(x)≤1Option C
Order the following expressions from least to greatest
(Quick Please)
What is the area of this triangle?
_units²
The area of the triangle is 4 units²
What is the area of a triangle?The area of a triangle is the half the base multiplied with the height of that triangle
Area = 1/ 2 × b × h
From the figure given,
base = 7 - 3 = 4 units
height = 4 - 2 = 2 units
Area = 1/ 2 × 4 × 2
Area = 1/ 2 × 8
Area = 4 units²
Thus, the area of the triangle is 4 units²
Learn more about a triangle here:
https://brainly.com/question/1475130
#SPJ1
Which of the following is a geometric sequence?
A. 7, 4, 1, -2, …
B. 2, 4, 6, 8, …
C. 1, 1/2, 1/4, 1/8, …
D. 1, 1, 2, 3, 5, …
Answer: C
Step-by-step explanation:
Each term is half the previous term.
WILL GIVE BRAINLIEST
The number of lattes sold daily by two coffee shops is shown in the table.
Shop A Shop B
55 36
52 40
50 34
47 39
51 44
48 41
53 40
53 38
Based on these data, is it better to describe the centers of distribution in terms of the mean or the median? Explain.
Mean for both coffee shops because the data distribution is symmetric
Median for both coffee shops because the data distribution is not symmetric
Mean for shop B because the data distribution is symmetric; median for shop A because the data distribution is not symmetric
Mean for shop A because the data distribution is symmetric; median for shop B because the data distribution is not symmetric
Answer:
It will be a more accurate thing to put it into the mean.
Step-by-step explanation:
Mainly, there is a lot of numbers in the data. So the reason Mean for shop A because the data distribution is symmetric; median for shop B because the data distribution is not symmetric.
Hoped this helped. Please mark me brainliest!
Mediciones recientes del fondo del mar que se extiende a lo largo de las crestas a medio océano muestran que la deriva continental es del orden de 4 cm. Suponga que este valor ha sido constante a través del tiempo y considere la distancia actual entre Sudamérica y África como 5000 millas. ¿Cuánto tarda Aproximadamente pangea en fragmentarse?
Usando proporciones, hay que la pangea tardra aproximademente 20,116,750 años para fragmentar-se.
¿Qué es una proporción?Una proporción es una fracción de la cantidad total, e puede ser encontrada aplicando una regla de três.
La deriva continental es del orden de 4 cm al año. La distancia actual entre Sudamérica y África es de 5000 millas. En cm, esta distancia es la seguiente:
5000 millas = 5000 x 160934 cm = 80,467,000 cm.
Asi, el tiempo que levará para la pangea fragmentar-se es:
t = 80,467,000/4 = 20,116,750 años
Puede-se aprender mas a cerca de proporciones es brainly.com/question/26950887
#SPJ1
If p(a) = 2/3, p(b) = 4/5, and p(anb) = 3/5, what is P( AuB) ( the n is the arc shape direction, so is the u)
By the inclusion/exclusion principle,
[tex]P(A\cup B) = P(A) + P(B) - P(A\cap B)[/tex]
[tex]P(A\cup B) = \dfrac23 + \dfrac45 - \dfrac35[/tex]
[tex]P(A\cup B) = \dfrac{10}{15} + \dfrac{12}{15} - \dfrac9{15}[/tex]
[tex]P(A\cup B) = \boxed{\dfrac{11}{15}}[/tex]
If a person stands 40 ft from a tree and the angle of elevation made from their eyes (5 ft above the ground) to the top of the tree is 47°, how tall is the tree? Round your answer to the nearest foot.
Using the given information, the height of the tree is 48 ft
TrigonometryFrom the question, we are to determine the height of the tree
In the given diagram, the height of the tree is (x + 5) feet
First, we will determine the value of x
tan 47° = x / 40
x = 40 × tan47°
x = 42.89 ft
∴ The height of the tree = (42.89 + 5) ft
Height of the tree = 47.89 ft
Height of the tree ≈ 48 ft
Hence, the height of the tree is 48 ft.
Learn more on Trigonometry here: https://brainly.com/question/15821537
#SPJ1