The length of the curve [tex]r(t) = 4t i + 8t^{3/2} j + 9t^{2}k, 0\leq t\leq 1[/tex] is 13.
The given curve is defined by the parametric equation [tex]r(t) = 4t i + 8t^{3/2} j + 9t^{2}k, 0\leq t\leq 1[/tex].
The length of a curve is given by the formula
⇒ [tex]|| r(t) || = \int\limits^a_b {\sqrt{{(x')^{2}} + (y')^{2} + (z')^{2} } } \, dt[/tex]
where the variables x', y', z' are defined as
[tex]x' = \frac{d(4t)}{dt} = 4 \\y' = \frac{d(8t^{3/2})}{dt} = 12t^{1/2} \\x' = \frac{d(9t^{2})}{dt} = 18t\\[/tex]
Hence, substituting the values in the formula for the length
⇒ [tex]|| r(t) || = \int\limits^0_1 {\sqrt{{(4)^{2}} + (12t^{1/2})^{2} + (18t)^{2} } } \, dt[/tex]
⇒ [tex]|| r(t) || = \int\limits^0_1 {\sqrt{{16 + 144t + 324t^{2}} } \, dt[/tex]
The given definite integral is in the form of
[tex]\int\limits^a_b {\sqrt{P(x)}} \, dx = \int\limits^a_b {\sqrt{{ax^{2} + bx + c}} } \, dx[/tex]
where P(x) is a polynomial of degree 2.
Writing the polynomial P(x) as
⇒ [tex]\sqrt{P(x)} = \sqrt{a}\sqrt{{(x + b_1)^{2} + c_1}} dx[/tex]
⇒ [tex]\int\limits^a_b {\sqrt{P(x)}} \, dx = \sqrt{a}\int\limits^a_b {\sqrt{{(x + b_1)^{2} + c_1}}} \, dx[/tex]
where [tex]b_1, c_1[/tex] are defined as
⇒ [tex]b_1= \frac{a}{2b} \\c1 = \frac{c}{a} - b_1^{2}[/tex]
Therefore, the solution will be
⇒ [tex]\int\limits^a_b {\sqrt{{(x + b_1)^{2} + c_1}}} \, dx = \frac{(b_1 + x)\sqrt{P(x)} + c_1log{(b_1 + x + \sqrt{P(x)}}}{2}[/tex]
Hence, solving the integral
⇒ [tex]|| r(t) || = \int\limits^0_1 {\sqrt{{16 + 144t + 324t^{2}} } \, dt[/tex]
we get the value as 13.
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x+6/-2= 5
what is x
Answer:
Step-by-step explanation:
x - 3 = 5
(x -3) +3 = 5+3
x - 3 + 3 = 8
so,
x = 8
Factor t12 – r3s15. (t4 – rs5)(t8 – rs5t4 – r2s10) (t4 – rs5)(t8 rs5t4 r2s10) (t9 – rs12)(t18 rs12t9 r2s24) (t4 rs5)(t8 – rs5t4 r2s10)
The factor is [tex](t^4+rs^5)(t^8-rs^5t^4+r^2s^{10})[/tex].
Option (D) is correct.
What is factorization?
Factorization or factoring is defined as the breaking or decomposition of an entity (for example a number, a matrix, or a polynomial) into a product of another entity, or factors.
Given:
[tex]t^{12}-r^{3}s^{15}[/tex]
For factorizing the above expression is
now, using the formula
(a³ - b³) = ( a+ b)( a² -ab + b²)
Then,
[tex]=(t^4)^{3}-(rs^{5})^{3}\\=(t^4+rs^5)((t^4)^2-(t^4\timesrs^5)+(rs^5)^2)\\=(t^4+rs^5)(t^8-rs^5t^4+r^2s^{10})[/tex]
Hence, the correct factorization is [tex](t^4+rs^5)(t^8-rs^5t^4+r^2s^{10})[/tex].
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Answer:
B
Step-by-step explanation:
If you have an average of 31 and you have the numbers 63,53,20,10, what is the missing value?
Answer: 9
Step-by-step explanation:
Let's start with what we know. If we have the data set:
63, 53, 20, 10, x
The average is:
(63+53+20+10+x)/5 = 31
We multiply both sides by 5
(65+53+20+10) = 155
146 + x = 155
x = 9
Select the correct answer. Circle F is represented by the equation (x + 6)2 + (y + 8)2 = 9. What is the length of the radius of circle F? A. 3 B. 9 C. 10 D. 81
Considering a circle whose equation is (x + 6)² + (y + 8)² = 9. The radius of the circle is 3 units
How to determine the radius of the circleInformation given in the question
Circle F is represented by the equation (x + 6)² + (y + 8)² = 9
Equation of a circle is given as
(x - h)² (y - k)² = r²
The equation of the circle is written in standard form is sown above comparing the equation with the given question gives
rewriting the equation
(x + 6)² + (y + 8)² = 9
(x + 6)² + (y + 8)² = 3²
comparing shows that r = 3
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Urgent help guysssss
Answer: A
Step-by-step explanation:
PLEASE HELP MEEEEEEEEEEEEEEE
Answer:
Step-by-step explanation:
(-1 - 3)/(3 - 0) = -4/3
y - 3 = -4/3(x - 0)
y - 3 = -4/3x + 0
y = -4/3x + 3
Find the length of the diagonal triangle using pythagoras theorem.given / = 8cm b=6cm
Answer: 10 cm
Step-by-step explanation:
Given, length=8cm
breadth=6 cm
Pythagoras theorem states that sum of square of base and square of perpendicular height is equal to square of hypotnyse.
Thus, we can write:
Considering, base=length=8cm
Perpendicular=breadth=6cm
Therefore, length of diagonal =√(length∧2+breadth∧2)
=√(8∧2+6∧2) cm
= 10 cm
Therefore the length of diagonal = 10cm.
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1. Is the relation on the right a function? Explain.
2. Can you state the domain and range of a relation that is not a function? If yes, what is the domain and range of the relation on the right?
3. Angelo states that all functions are relations but not all relations are functions. Do you agree with Angelo? Support your answer using an example.
1) The relation in the right is not a function.
2) domain
D: {-60, 20, 70}
range
R: {-6, 3, 2, 5}
3) Angelo is correct.
What is a function?A function is a relation that maps elements from one set called the domain into another set called the range, such each element from the domain is mapped into only one of the elements of the range.
Thus, if an element of the domain is being mapped into two or more elements of the range, then the relation is not a function.
1) If you look at the relationship in the right, you can see that the input -60 is being mapped into two different outputs, this, it is not a function.
2) The domain is the set in the left:
D: {-60, 20, 70}
And the range is the set in the right side:
R: {-6, 3, 2, 5}
3) Angelo is correct, functions are relations, but not all relations are functions, only the relations that meet the condition written above.
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What equation represents Twenty five is one sixth of the number n
The equation that represents "Twenty five is one sixth of the number n" is:25 = (1/6) * n
The equation "25 = (1/6) * n" represents the relationship between the number 25 and an unknown number, n. It states that 25 is equal to one sixth of n. In other words, if we multiply the unknown number n by one sixth (or divide n by 6), we will obtain the value of 25. This equation can be used to solve for the value of n by multiplying 25 by 6, which would give us the unknown number. It provides a clear mathematical expression of the given relationship and allows us to find the value of n based on the given condition.The equation states that 25 is equal to one sixth of the number n. This means that if we take one sixth of the value of n, we will get 25 as the result. In other words, if we divide the value of n by 6, the quotient will be 25.
To understand it conceptually, imagine that we have a number n and we want to find one sixth of that number. By dividing the number by 6, we can determine how many equal parts make up the whole number, and one of those parts is equal to 25.
This equation is useful in various situations, such as solving problems involving proportions or determining unknown values when given a fraction of a number.
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(federal income taxes and piecewise functions mc) determine f(−2) for a piecewise function f of x in three pieces. the function is defined by part 1, which is x cubed for x less than negative 3, part 2, which is 2 times x squared minus 9 for negative 3 is less than or equal to x which is less than 4, and part 3 which is 5 times x plus 4, for x greater than or equal to 4. −1 −6 8 9
Answer:
9
Step-by-step explanation:
For the data in the table, does y vary directly with x? If it does, write an equation for the
direct variation.
x y
8 11
16 22
24 33
Answer:
Below in bold.
Step-by-step explanation:
The definition of direct variation is y = kx where k is a constant.
So, if y/x is a constant for the data, we have direct variation.
So, checking the rows in the given table:
y/x = 11/8
y/x = 22/16 = 11/8
y/x = 33/24 = 11/8
- we see that this is direct variation.
So, k = 11/8 and
The equation is
y = (11/8) x
or
y = 11x/8
or
y = 1.375x in decimal form.
Answer:
y = (11/8)x----------------------------------
Direct variation equation:
y = kx, where k- variation coefficientUse the pairs of coordinates and find the value of k, then compare:
11 = 8k ⇒ k = 11/822 = 16k ⇒ k = 22/16 = 11/833 = 24k ⇒ k = 33/24 = 11/8The value of k is same with all ordered pairs and the equation is:
y = (11/8)xhelp please struggling
Answer:
Step-by-step explanation:
There are just two triangles stuck together , here. Don't let that throw you off your game :DD
solve for a
then b
for a it's 180 = 2*36 +a ( b/c of the red marks , again)
180 = 72 +a
180-72 =a
108 = a
Now solve for b
180 = 2b + 42 (red marks tell us this formula works, again)
180-42 = 2b
138 =2b
138/2 =b
69 = b
Do you see , now, how I get those formulas???
Please complete the proof.
The line l and line m are parallel.
What are parallel lines?Parallel lines are those lines that are equidistant from each other and never meet, no matter how much they may be extended in either directions.
Given that, transversal t cuts line l and line m.
∠2≅∠7
Transversal t cuts line l and line m (Given)
∠2≅∠7 (Given)
∠2≅∠3 (Vertical angles theorem)
∠3≅∠7 (Converse of corresponding angles postulates)
l║m proved
Hence, the line l and line m are parallel.
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make this graph into y=mx+b form
Answer: y=-3x-5
Step-by-step explanation:
(0,-5) (-1,-2)
[tex]\displaystyle\\\frac{x-x_1}{x_2-x_1} =\frac{y-y_1}{y_2-y_1}[/tex]
x₁=0 x₂=-1 y₁=-5 y₂=-2
[tex]\displaystyle\\\frac{x-0}{-1-0}=\frac{y-(-5)}{-2-(-5)} \\\\\frac{x}{-1} =\frac{y+5}{-2+5} \\\\-x=\frac{y+5}{3}[/tex]
Multiply both parts of the equation by 3:
[tex]-3x=y+5\\\\-3x-5=y+5-5\\\\-3x-5=y[/tex]
[tex]Thus,\ y=-3x-5[/tex]
The census bureau gives this distribution for the number of a family household's related children under the age of 1818 in american households in 2016: number of children 00 11 22 33 44 proportion 0. 580. 58 0. 180. 18 0. 160. 16 0. 60. 06 0. 20. 02 in this table, 44 actually represents four or more (the census bureau does not provide separate information on households with 5,5, 6,6, or more children under the age of 18). 18). But for purposes of this exercise, assume that it means only households with exactly four children under the age of 18. 18. This is also the probability distribution for the number of children under 1818 in a randomly chosen household. The expected value of this distribution is the average number of children under 1818 in a household. What is this expected value? (use decimal notation. Give your answer as an exact number. )
The average number of children in a family under the age of 1818 is 0.76, which is the predicted value of this distribution.
The product of each potential result and its associated probability yields the anticipated value of a probability distribution. The probable outcomes in this scenario are 0, 1, 2, 3, & 4 minors, with the associated probability shown in the table.
We may add the products of each result and its probability to determine the estimated return of this distribution:
Expected value = (0)(0.58) + (1)(0.18) + (2)(0.16) + (3)(0.06) + (4)(0.02)
= 0 + 0.18 + 0.32 + 0.18 + 0.08
= 0.76
Therefore, 0.76 children below the age of 18 per household are the anticipated value of this distribution.
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In problem 3, the wind blows harder and the top of the
balloon is now 15 feet over from the stake. What is the
height of the balloon now? Draw a diagram to help you
solve the problem and explain your answer.
To solve this problem, we can use the Pythagorean theorem to find the height of the balloon. In particular, we can consider the right triangle formed by the stake, the top of the balloon, and the part of the string that is attached to the stake.
[asy]
unitsize(2 cm);
pair A, B, C, D, E;
A = (0,0);
B = (0,15);
C = extension(A,B,A+(1,0),B+(1,0));
D = extension(A,C,A+(0,1),C+(0,1));
E = extension(B,D,B+(0,-1),D+(0,-1));
draw(A--B--C--cycle);
draw(A--D--E--cycle);
draw(rightanglemark(A,D,C,2));
label("$A$", A, S);
label("$B$", B, N);
label("$C$", C, E);
label("$D$", D, E);
label("$E$", E, W);
[/asy]
Let $a$ be the distance from the stake to the top of the balloon, $b$ be the height of the balloon, and $c$ be the length of the string. We know that $a = 15$ feet and $c = 15$ feet, so we can use the Pythagorean theorem to find $b$:
$a^2 + b^2 = c^2$
$15^2 + b^2 = 15^2$
$225 + b^2 = 225$
$b^2 = 0$
$b = \sqrt{0} = 0$ feet
Therefore, the height of the balloon is 0 feet. This is because the balloon has been pulled up by the wind so that its top is directly above the stake, and therefore its height is 0 feet.
helppp please i dont know how to do this
The solution of the pairs of lines are as follows,
(1) Line 1 and line 2 are perpendicular to each other.
(2) Line 1 and line 3 are parallel to each other.
(3) Line 2 and line 3 are perpendicular to each other.
The slope of the line is a tangent angle made by line with horizontal. i.e. m =tanx where x in degrees.
Here,
Calculate the slope of each line,
LIne 1
3y = 2x + 5
y = 2/3x + 5/3
Compared with the standard equation of line y = mx + c,
Slope m = 2/3
Similarly.
The slope of line 2 = -3/2
The slope of line 3 = 2/3
Now. properties of pair of lines state that the slope of parallel lines is equal and the slope of perpendicular lines are negative reciprocal of each other,
So
Slope of line 1 = slope of line 3
But,
The slope of line 2 is the negative reciprocal of the slope of lines 1 and 3.
Thus, the solution of the pair of lines has been shown above.
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Given the piont and slope write the equation of the line 4,2 slope = 3
Answer:
I am not sure which form you want.
Points slope: y -2 = 3(x -4)
Slope intercept: y = 3x -10
Step-by-step explanation:
Point slope form for the line
y- y = m(x -x)
slope of 3 and the point (4,2) Use 4 for x and 2 for y
y -2 = 3(x -4)
Slope intercept form of a line
y = mx + b
2 = 3(4) + b
2 = 12 + b Subtract 12 from both sides
2 - 12 = 12 - 12 + b
-10 = b
y = mx + b
y = 3x -10
Anna is knitting hats to give as gifts. Each one will use 1.2 skeins, and she has 6 skeins of yarn available. With the yarn available, how many hats can Anna knit?
Answer:
5
Step-by-step explanation:
There are 6 total skeins, and 1.2 is needed to make one
So you would divide 6 by 1.2
6 / 1.2 = 5
And to confirm it is 5, you can now multiply 5 by 1.2
5 x 1.2 equals 6
Lin is selling boxes of cookies. Each box costs $3.75. Lin's goal is to earn more than $30 selling cookies.
If Lin sells b, boxes of cookies, write an inequality (using symbol < or >) that will be true whenever Lin makes her goal.
NEED ASAP!!
Answer: If Lin sells b boxes of cookies, she will earn 3.75b dollars. Therefore, the inequality that will be true whenever Lin makes her goal is:
3.75b > 30
This inequality can be read as "the total amount of money Lin earns selling cookies is greater than $30."
Step-by-step explanation:
a deck is shuffled well, what is the probability that the top card is ace of spades and the bottom is queen of hearts?
The probability that the top card is an Ace of Spades and the bottom card is a Queen of Hearts is extremely low, as there are 52 cards in a standard deck of cards and only two of those cards are an Ace of Spades and a Queen of Hearts.
Therefore, the probability of the top card being an Ace of Spades and the bottom card being a Queen of Hearts is 1/2652 or approximately 0.000038%.The probability of getting an Ace of Spades as the top card and a Queen of Hearts as the bottom card is 1/52 x 1/52 = 1/2652. This can be expressed as a percentage by multiplying the fraction by 100 which gives us 0.000038%. The probability that the top card is an Ace of Spades and the bottom card is a Queen of Hearts is extremely low, as there are 52 cards in a standard deck of cards and only two of those cards are an Ace of Spades and a Queen of Hearts.
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Autumn has a smart phone data plan that costs $50 per month that includes 9 GB of data, but will charge an extra $10 per GB over the included amount. How much would Autumn have to pay in a month where she used 3 GB over the limit? How much would Autumn have to pay in a month where she used went over by xx GB?
Answer:
1. $80 dollars for the month she used 3 GB over her data limit.
2. 10x + 50.
Explanation:
1. Autumn used 3 GB data more than her plan offered. $10 dollars a gigabyte of data times 3 gigabytes equals 30, plus 50 dollars for the base smartphone plan (which, off topic, sucks) equals 80 dollars for the month.
2. 10x + 50 means 10 dollars per X gigabytes, plus the base 50 dollars a month Autumn has to pay.
a rectangular room is 3 3 times as long as it is wide, and its perimeter is 56 56 meters. find the dimension of the room.
Consequently, the room has a 7 meter width. The room's length is
[tex]= 7 * 3 meters = 21 meters[/tex]
What is rectangle?
A rectangle in Euclidean plane geometry is a quadrilateral with four right angles. You might also describe it as follows: a quadrilateral that is equiangular, which indicates that all of its angles are equal. The parallelogram might also have a straight angle. Squares are rectangles with four equally sized sides. A quadrilateral of the shape of a rectangle has four 90-degree vertices and equal parallel sides. As a result, it is sometimes referred to as an equirectangular rectangle. Because its opposite sides are equal and parallel, a rectangle is also known as a parallelogram.
Given that the length of the rectangular space is three times its breadth
The chamber has a 56-meter perimeter.
Assume that the room's width is x.
As a result, the room is three times as long.
The room's perimeter is now equal to 2(3x + x).
[tex]Here, 2(3x + x) = 56\\ 2 * 4x = 56\\ 8x = 56\\ x = 7.[/tex]
Consequently, the room has a 7 meter width.
The room's length is
[tex]= 7 * 3 meters\\= 21 meters[/tex]
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The cost of 4 avocados and 5 tomatoes is $3.76. Six avocados and 3 tomatoes cost $4,02. Find the cost of each avocado and each tomatoe.
Answer:
It costs $0.49 for each avocado and $0.36 for each tomato.
Step-by-step explanation:
This is a system of equations problem.
Let x represent the avocados and y represent tomatoes:
4x + 5y = 3.76 <--- scenario 1
6x + 3y = 4.02 <---- scenario 2
If we graph these out on a graphing calculator, the lines will intersept at point (0.49,0.36)
The x coordinate represents how much it costs for each avacado and the y coordinate represents how much it costs for each tomato
Select the correct anwer. What i the value of thi expreion when c = -4 and d = 10?
A. 2
B. 9
C. 21
D. 41
Reet
The expression be 1/4(c³+ d²) then the value exists 9.
What is meant by PEMDAS order of operations?The order of operations is a rule that specifies the right procedure to follow when evaluating a mathematical equation. Using the acronym PEMDAS, we can memorize this order: parentheses, exponents, multiplication and division (left to right), addition and subtraction (from left to right).
The rules that specify the order in which we should perform several operations to solve an expression are known as the order of operations.
Parentheses, Exponents, Multiplication, Division (from left to right), Addition, and Subtraction are in the following order: PEMDAS (from left to right).
Let the given expression be 1/4(c³+ d²)
The value of c = -4 and d = 10
substitute the values in the above equation, we get
Follow the PEMDAS order of operations
Calculate within parentheses ((-4)³ + (10)²) = 36
1/4((-4)³ + (10)²)
simplifying the above equation, we get
= 1/4 × 36
= 9
Therefore, the correct answer is option B. 9.
The complete question is:
What is the value of this expression when c = -4 and d = 10?
1/4(c³+ d²)
A. 2
B. 9
C. 21
D. 41
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please help asap i have 2 days
Answer:
first option
Step-by-step explanation:
inequalities of the type | x | > a have solutions of the form
x < - a or x > a
given
| 7 + m | - 4 > 8 ( add 4 to both sides )
| 7 + m | > 12
then
7 + m < - 12 (subtract 7 from both sides )
m < - 19
OR
7 + m > 12 ( subtract 7 from both sides )
m > 5
solution is m > 5 or m < - 19
sketch the region enclosed by the given curves. y = 4/x, y = 16x, y = 1 16 x, x > 0
In order to plot the region enclosed by the given graphs we have to find the intersection points of all the 3 graphs taking 2 at a time
so finding intersection point of the y=4/x and y=16x
4/x=16x
=> [tex]x^{2}[/tex] = 1/4
=> x = 1/2 as x>0
and y=16x
=> y =8
now finding intersection point of y=4/x and y=1/16x
=> 4/x =x/16
=> [tex]x^{2}[/tex] = 64
=> x = 8 as x>0
and y= 4/x
=> y=1/2
now finding intersection point of y=16x and y=1/16x
the intersection point of both line is x=0
so y=16x
=> y=0
so we will get the intersection point of all the lines now after plotting all the points below we will get the graph as attached below.
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if we repeatedly and infinitely take random samples of size n from a population, what will the distribution of the sample means look like?
When n is large the distribution of the sample means will approach a normal distribution.
The central limit theorem (CLT) is a statistical theory that predicts that given a large enough sample size from a population with a certain level of variation, the mean of all the samples from that population will be close to the population mean.
To put it another way, the central limit theorem predicts with absolute certainty the form of the distribution of means when we take repeated samples from a certain population. In particular, the distribution of means derived from repeated sampling will tend toward normality as sample sizes increase.
In other words, when n is large, the distribution of the sample means will resemble a normal distribution if we continually pick independent, random samples of size n from any population.
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A water balloon is thrown upward from a height of 5 feet with an initial velocity of 35 feet per second. The quadratic function \large h\left(t\right)=-16t^2+35t+5 represents the height of the balloon, h, in feet t seconds after it is thrown. When does the water balloon reach the height of 20 feet? round your answer to the nearest thousandth.
Since, The equation of the height with the time is provided and the balloon is thrown upwards against gravity from 5 feet, The answer is 0.255 sec.
What do you mean by equation?A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
What is initial and final velocity?When gravity first exerts force on an object, its initial velocity defines how quickly the object moves. The final velocity, on the other hand, is a vector number that gauges a moving body's speed and direction after it has reached its maximum acceleration.
initial height = 5
final height =20
height to travel =20-5 =15
[tex]15 = 16t^{2}+35t+5\\16t^{2}+35t-10 =0[/tex]
solving we get, t = 0.255 or -2.44
since, -2.44 is not possible,
t = 0.255 seconds.
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Determine which answer in the solution set will make the equation true.
4f − 10 = 3f − 18
S: {−18, −8, 3, 8}
A. −8
B. 8
C. −18
D. 3
The answer in the solution set that makes 4f − 10 = 3f − 18 true is (a) -8
How to determine the answer in the solution set?From the question, we have the following parameters that can be used in our computation:
4f − 10 = 3f − 18
And we have the solution set to be
S: {−18, −8, 3, 8}
Subtract 3f from both sides of the equation
So, we have the following representation
f - 10 = - 18
Add 10 to both sides of the equation
This gives
f = -8
Hence, the solution is (a) -8
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