The mass of the wire is found to be 40π√2 units.
How to find the mass?To calculate the mass of the wire which runs along the curve r ( t ) with the density function δ=5.
The general formula is,
Mass = [tex]\int_a^b \delta\left|r^{\prime}(t)\right| d t[/tex]
To find, we must differentiate this same given curve r ( t ) with respect to t to estimate |r'(t)|.
The given integration limits in this case are a = 0, b = 2π.
Now, as per the question;
The equation of the curve is given as;
r(t) = (4cost)i + (4sint)j + 4tk
Now, differentiate this same given curve r ( t ) with respect to t.
[tex]\begin{aligned}\left|r^{\prime}(t)\right| &=\sqrt{(-4 \sin t)^2+(4 \cos t)^2+4^2} \\&=\sqrt{16 \sin ^2 t+16 \cos ^2 t+16} \\&=\sqrt{16\left(\sin t^2+\cos ^2 t\right)+16}\end{aligned}[/tex]
Further simplifying;
[tex]\begin{aligned}&=\sqrt{16(1)+16} \\&=\sqrt{16+16} \\&=\sqrt{32} \\\left|r^{\prime}(t)\right| &=4 \sqrt{2}\end{aligned}[/tex]
Now, use integration to find the mass of the wire;
[tex]\begin{aligned}&=\int_a^b \delta\left|r^{\prime}(t)\right| d t \\&=\int_0^{2 \pi} 54 \sqrt{2} d t \\&=20 \sqrt{2} \int_0^{2 \pi} d t \\&=20 \sqrt{2}[t]_0^{2 \pi} \\&=20 \sqrt{2}[2 \pi-0] \\&=40 \pi \sqrt{2}\end{aligned}[/tex]
Therefore, the mass of the wire is estimated as 40π√2 units.
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The complete question is-
Find the mass of the wire that lies along the curve r and has density δ.
r(t) = (4cost)i + (4sint)j + 4tk, 0≤t≤2π; δ=5
If the polynomial function f(x)=(x+1)(x+3)9x-4) is rewritten in standard form, the degree would be ______ and the constant term would be _______.
.
If the polynomial function f(x)=(x+1)(x+3)9x-4) is rewritten in standard form, the degree would be 3 and the constant term would be 12 .
What is polynomial function?
In an equation such as the quadratic equation, cubic equation, etc., a polynomial function is a function that only uses non-negative integer powers or only positive integer exponents of a variable. For instance, the polynomial 2x+5 has an exponent of 1.A quadratic, cubic, quartic, and other functions involving only non-negative integer powers of x are examples of polynomial functions. We can define a polynomial's degree and provide a generic definition for it.f(x)=(x+1)(x+3)9x-4)
f(x ) = 3x³ + 36x² + 27x - 4x² - 16x - 12
f(x) = 3x³ + 32 x² + 11x - 12
in above function highest degree would be 3 and constant -12 .
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.
which set of points are collinear? d, s, y startfragment, d, s, y, p, t, y startfragment, p, t, y, y, d, h startfragment, y, d, h, h, t, s
The set of points (A) P, T, and Y are collinear.
What are collinear points?Collinear points are those that are on the same straight line or in the same plane. In Euclidean geometry, two or more points on a line that is close to or far from each other are said to be collinear.In general, three points A, B, and C are collinear if the sum of the lengths of any two line segments between AB, BC, and CA equals the length of the remaining line segment, that is, AB + BC = AC, AC +CB = AB, or BA + AC = BC.In response to the question -
Only one set of points - (P, T, Y) - can be seen to be on the same straight line. One of the options mentioned is - (P, T, Y).Therefore, the set of points (A) P, T, and Y are collinear.
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The correct question is given below:
Which set of points is collinear?
(A) P, T, Y
(B) D, S, Y
(C) H, T, S
(D) Y, D, H
What is the remainder when f(x) = x4 + 7x3 + 17x2 + 20x is divided by x2 + 4?
a
\large x^2+13x+7+\frac{-52}{x^2+4}
b
\large x^2+7x+\frac{13x-8}{x^2+4}
c
\large x^2+7x+13+\frac{5x-52}{x^2+4}
d
\large x^2+1+\frac{1}{x^2+4}
The reminder by dividing f(x)= x⁴ + 7x³ + 17x² + 20x is divided by x² + 4 is x² + 7x + (-13x² - 8x)/(x² + 4) so none of the given option is correct.
What is a number system?The number system is a way to represent or express numbers.
Any of the multiple sets of symbols and the guidelines for utilizing them to represent numbers are included in the Number System.
Given the dividend f(x)= x⁴ + 7x³ + 17x² + 20x and diviser x² + 4
The first time it will go to x²
Then,
Quotient = x²
Remainder = 7x³ - 13x²
The second time goes to 7x
Quotient = 7x
Remainder = -13x² - 28x
The third time it is not divisible so (-13x² - 8x)/(x² + 4) will be the reminder.
Together it is x² + 7x + (-13x² - 8x)/(x² + 4).
Hence "The reminder by dividing f(x)= x⁴ + 7x³ + 17x² + 20x is divided by x² + 4 is x² + 7x + (-13x² - 8x)/(x² + 4)".
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A recipe uses both carrots and onions. The recipe uses 4 times more carrots than onions. If the recipe uses a total of 25 ounces of chopped carrots and onions, how many ounces of onions does the recipe use?
The ounces of onions is 5 ounces.
How to calculate the value?Let the onions be represented by x.
The recipe uses 4 times more carrots than onions. The carrots will be 4 × x = 4x.
Therefore, the equation will be:
4x + x = 25
5x = 25
Divide by 5
x = 25/5
x = 5
Therefore, the ounces of onions is 5 ounces.
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Answer:
Step-by-step explanation:
The ounces of onions is 5 ounces.
The recipe uses 4 times more carrots than onions. The carrots will be 4 × x = 4x.
Therefore, the equation will be:
4x + x = 25
5x = 25
Divide by 5
x = 25/5
x = 5
Therefore, the ounces of onions are 5 ounces.
I NEED HELP QUICK
In the image above angle 11 is 93 degrees, what is angle 11?
Answer:
angle 11 is an obtuse angle
Step-by-step explanation:
93 < 90
⬆️ ⬆️
obtuse right
Let f(x)=2x^2-9x-5 and g(x)=x-5. what are (f•g)(x) and (f/g)(x)?
The values of the composite functions are (f.g)(x) = 2x^3 - 19x^2 + 40x + 25 and (f/g)(x) = 2x + 1
How to evaluate the composite functions?The functions are given as:
f(x) = 2x^2 - 9x - 5
g(x) = x - 5
The composite function (f.g)(x) is calculated as
(f.g)(x) = f(x) * g(x)
This gives
(f.g)(x) = (2x^2 - 9x - 5) * (x - 5)
Evaluate the product
(f.g)(x) = 2x^3 - 9x^2 - 5x - 10x^2 + 45x + 25
Evaluate the like terms
(f.g)(x) = 2x^3 - 19x^2 + 40x + 25
The composite function (f/g)(x) is calculated as
(f/g)(x) = f(x)/g(x)
This gives
(f/g)(x) = (2x^2 - 9x - 5)/(x - 5)
Factorize the numerator
(f/g)(x) = (2x + 1)(x - 5)/(x - 5)
Evaluate the quotient
(f/g)(x) = 2x + 1
Hence, the values of the composite functions are (f.g)(x) = 2x^3 - 19x^2 + 40x + 25 and (f/g)(x) = 2x + 1
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The ratio of a to b is constant, and
a = 9 when b = 6. What is the value of
b when a = 2?
Answer:
Step-by-step explanation:
a : b
9 : 6
2 : x
Cross multiply
6*2 = 9x
12 = 9x
12/9 = x
4/3 = x
What are the prime factors of 4325
Refer to the number line. find the coordinate of point c that is 78 of the distance from m to j.
The coordinate of point c is such that point c is 7/8 of m to j is [ 7/8(x₂ - x₁), 7/8(y₂ - y₁) ].
What are coordinates?A pair of numbers that employ the horizontal and vertical distinctions from the two reference axes to represent a point's placement on a coordinate plane. typically expressed by the x-value and y-value pairs (x,y).
Coordinates are always written in the form of small brackets the first term will be x and the second term will be y.
For example (5,3) here 5 will be for the x and 3 will be for the y.
Another example is (9,0) here 9 is for x and 0 is for y.
Let's say the coordinate of m → (x₁,y₁)
j → (x₂ ,y₂)
Now since point c is proportion with 7/8
So,
It will be 7/8 times of difference between of x and y coordinate of points m and j
So,
x = 7/8(x₂ - x₁) and y = 7/8(y₂ - y₁)
Hence "The coordinate of point c is such that point c is 7/8 of m to j is [ 7/8(x₂ - x₁), 7/8(y₂ - y₁) ]".
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One bridge design has supports that form an X in the center. Find the measure of each numbered angle.
The measure of angle is 52, 128 and 132
What are angles?A figure which is formed by two rays or lines that shares a common endpoint is called an angle.
Given:
one angle = 52
So,
<2 = 52 (vertically opposite angle)
<1 + <2 = 180 ( linear pair)
<1 + 52 = 180
<1 = 128
and, <1 = < 3 ( vertically opposite angle)
<3 = 132
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Adeli opened a savings account with an initial deposit of $2000 and will not make any additional deposits or withdrawals the account earns 5% interest compounded annually what is the total amount that adele will have in her account in the end of the 3 years
The total amount that Adele will have in her account at the end of the 3 years is $2,315.25.
Given that, principal=$2000, rate of interest=5% and time period=3 years.
How to calculate the compound interest?The compound interest formula is used to calculate compound interest, sometimes known as "interest on interest". The formula for compound interest is [tex]A = P(1 + \frac{r}{100})^{nt}[/tex], where P= principal balance, r= interest rate, n= number of times interest is compounded per time period and t= number of time periods.
Now, A=2000(1+5/100)³
=2000(1+0.05)³
=2000(1.05)³
=$2,315.25
Therefore, the total amount that Adele will have in her account at the end of the 3 years is $2,315.25.
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Please, help me. I have no idea.
Given g (x) = -x-4 , find g(-2)
Answer:
-2
Step-by-step explanation:
g(x) = -x-4
g(-2) = -(-2)-4
2 - 4
-2
Answer:
g(-2) = -2
Step-by-step explanation:
Hello!
To evaluate for -2, we can substitute -2 for x in the equation.
Evaluateg(x) = -x - 4g(-2) = -(-2) - 4g(-2) = 2 - 4g(-2) = -2g(-2) is -2.
7. On a recent math test, the boys averaged
81 and the girls averaged 90. If there are twice
as many boys in the class as girls, what was
the overall average for the class?
Bonnie made 7 pies. there were 6 slices in each apple pie and 7 slices in each peach pie. if bonnie made 44 slices of pie in total, how many of each pie did she make?
She make 5 apple pie and 2 peach pie.
In the given statement is:
Bonnie made 7 pies. there were 6 slices in each apple pie and 7 slices in each peach pie. if Bonnie made 44 slices of pie in total.
To find the how many of each pie did she make
Now, According to the question :
Let start the solution
Total number of slices made by Bonnie is 44
Let Number of apple pies be x
and, number of peach pies is (7 - x) ............(1)
Pieces of an apple pie is 6x
Pieces of peach pie = 7 (7 - x)
Calculation
6x + 7(7 - x) = 44
6x + 49 -7x = 44
-x = 44 - 49
-x = -5
x = 5
Put the value of x in equation (1)
7 - x = 7 - 5 = 2
She make 5 apple pie and 2 peach pie.
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I need help if someone can be so kind to give me the answer and explain how to do
Verify if f(x)= 5-3x/2 and g(x)= 5-2x/3 are inverses. Please show step by step.
Since f(g(x)) = g(f(x)) = x, hence the function f(x) and g(x) are inverses of each other.
Inverse of functionsIn order to determine if the function f(x) and g(x) are inverses of each other, the composite function f(g(x)) = g(f(x))
Given the function
f(x)= 5-3x/2 and
g(x)= 5-2x/3
f(g(x)) = f(5-2x/3)
Substitute
f(g(x)) = 5-3(5-2x)/3)/2
f(g(x)) = (5-5+2x)/2
f(g(x)) = 2x/2
f(g(x)) = x
Similarly
g(f(x)) = 5-2(5-3x/2)/3
g(f(x)) = 5-5+3x/3
g(f(x)) = 3x/3
g(f(x)) =x
Since f(g(x)) = g(f(x)) = x, hence the function f(x) and g(x) are inverses of each other.
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y(a-b)=c(y+a) what's the answer
Linear equation: a, b, c, and r are real numbers (and if a, b, and c are not all equal to 0)
then ax + by + cz = r is called a linear equation in three variables. (The
“three variables” are the x, the y, and the z.)
The numbers a, b, and c are called the coefficient of the equation. The
number r is called the constant of the equation
y(a-b)=c(y+a)
ya-yb=cy+ca
ya-yb-cy=ca
y(a-b-c)=ca
[tex]y=\frac{ca}{a-b-c}[/tex]
case 1:
if either C and a are zero the y is zero,
case 2:
if
[tex]c\neq 0 \\a\neq 0\\y\neq 0[/tex]
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Which equation best represents the graph shown below? Explain in detail how you arrived at your answer by stating each of the mathematical transformations necessary and to produce the graph
Answer step by step
A. Y = (x - 2)^2 - 3
B. Y = (x + 2)^2 - 3
C. Y = - ( x - 2 )^2 - 3
D. Y = 1/2 ( x + 2)^2 - 3
The quadratic equation is:
y = (x - 2)^2 - 3
So the correct option is A.
Which equation best represents the graph shown below?
The graph clearly is the graph of a quadratic equation, there we can see that:
The y-intercept is y = 1.
The vertex is (2, -3)
Remember that if we know that a quadratic equation has a vertex (h, k), then the equation can be written as:
y = a*(x - h)^2 + k
In this case, the vertex is (2, -3), replacing that we get:
y = a*(x - 2)^2 - 3
Now, we can find the value of a, the leading coefficient, by using the fact that the y-intercept is y = 1, evaluating in zero we should get:
1 = a*(0 - 2)^2 - 3
1 = a*4 - 3
1 + 3 = a*4
4/4 = a = 1
Then we conclude that the quadratic equation is:
y = (x - 2)^2 - 3
So the correct option is A.
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A rectangle with a total of 80 square feet has been broken into two rectangles with areas of 56
square feet and 24 square feet as shown. The common length of the two smaller rectangles is
a whole number that is greater than either width 1 or width 2, which are both whole numbers.
From this information, determine the length, width 1, and width 2. Explain how you found
your answers.
The length of the rectangle is 8 feet, width 1 is 7 feet and width 2 is 3.
A rectangle of area 80 square feet is broken and forms two square rectangles of area 56 square feet and 24 square feet, which has a common length.
Area of the rectangle = length × width
Now suppose the length is x feet, and let width 1 be a and width 2 be b.
Therefore we get that
rectangle first is of area = x a
rectangle second is of area = x b
Therefore original rectangle area = area of first rectangle + area of the second rectangle
80= x a + x b
80 = x( a + b)
As it is given that the length is common and is greater than both the width of the rectangle
56 = 8 × 7
24 = 8 × 3
So we will take x = 8 and a =7, b= 3
Now put these numbers in the equation to see if they fulfill the criteria
80 = 8( 7 + 3)
80 = 80
Therefore we get a length is 8, a width of 1 is 7, and a width of 2 is 3.
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6.
Part A: Annotate the expressions.
Expression C
20-(-3)
Expression D
-20-(-3)
Part B: How are the expressions similar?
Part C: Explain the steps you would take in simplifying
Expression D.
Answer:
expression d is your right answer
is 90.790170 rational or irrational
Answer:
irrational number
Step-by-step explanation:
because it has no repeating pattern
The number x = 90.790170 is rational number as it is a finite decimal and can be represented as a fraction.
Given data:
Let the number be represented as x.
Now, the value of x = 90.790170
Rational numbers can be positive, negative, or zero, and they can be written in the form a/b, where a is the numerator and b is the denominator, with b not equal to zero.
The number 90.790170 is a rational number. A rational number is any number that can be expressed as a fraction of two integers, where the denominator is not zero.
Since 90.790170 is a finite decimal (it doesn't repeat or go on forever), it can be represented as a fraction of two integers.
Hence, 90.790170 is rational.
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Factoring All Methods
Answer:
5(3n - 4)(3n - 10)
Step-by-step explanation:
15n² - 110n + 200 ← factor out 5 from each term
= 5(3n² - 22n + 40) ← factor the quadratic
consider the factors of the product of the coefficient of the n² term and the constant term which sum to give the coefficient of the n- term.
product = 3 × 40 = 120 and sum = - 22
the factors are - 12 and - 10
use these factors to split the n- term
3n² - 12n - 10n + 40 ( factor the first/second and third/fourth terms )
3n(n - 4) - 10(n - 4) ← factor out (n - 4) from each term
(n - 4)(3n - 10)
then
15n² - 110n + 200 = 5(n - 4)(3n - 10)
Answer:
Hello,
Step-by-step explanation:
[tex]15n^2-110n+200=15(n^2-\dfrac{22}{3}x +\dfrac{40}{3} )\\[/tex]
1. I am thinking about 2 numbers with sum=22/3 and product=40/3.
Those numbers are 4 and 10/3 since 4*10/3=40/3 and 4+10/3=22/3
[tex]15(n^2-\dfrac{22}{3}x +\dfrac{40}{3} )=15(n-\dfrac{10}{3} )(n-4)=5*(3n-10)(n-4)[/tex]
2.
[tex]15(n^2-\dfrac{22}{3}x +\dfrac{40}{3} )=15(n^2-2*\dfrac{11}{3} x+(\dfrac{11}{3})^2 -\dfrac{121}{9} +\dfrac{120}{9})\\\\=15((n-\dfrac{11}{3})^2 -\dfrac{1}{9})\\\\=15( n-\dfrac{11}{3} -\dfrac{1}{3})*(n-\dfrac{11}{3} +\dfrac{1}{3})\\=15(n-\dfrac{10}{3})(n-4)\\\\=5(3n-10)(n-4)\\[/tex]
3.
Using discriminant ...
In a school election there were four candidates: Tom, Rebecca, Olly and Wendy.
540 students voted in the election.
5% of the votes were for Tom
of the votes were for Rebecca
The ratio of the number of votes for Olly to the number of votes to Wendy was 1:2
How many votes were for Wendy?
5% of votes were for Tom: 27
2/9 of the votes were for Rebecca: 114
213 votes are for Wendy
Find the slope of each of the lines below (-4,-2),(6,1)
Scientist A dissolved 1.0 kilogram of salt in 3.0 liters of water. Scientist B dissolved 2.0 pounds of salt into 7.0 pints of water. Which scientist made a more concentrated salt solution? Use dimensional analysis to find the concentration of salt that each scientist made. Find the equivalents from your table of measures.
The scientist that made a more concentrated salt solution is scientist A.
How to compute the value?Since Scientist A dissolved 1.0 kilogram of salt in 3.0 liters of water. The concentration will be:
= 1/3 = 0.33
Scientist B dissolved 2.0 pounds of salt into 7.0 pints of water. The concentration will be:
= 2/7
= 0.285
Therefore, the scientist that made a more concentrated salt solution is scientist A.
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Type a digit that makes this statement true. 8,353,05_
The digits that make the given statement true are 3 or 7.
5,156,936 is divisible by 4.5,156,976 is divisible by 4.What are the rules of divisibility of 4?The basic rule for divisibility by 4 is that if the number formed by the last two digits of a number is divisible by 4, the original number is divisible by 4 as well; this is because 100 is divisible by 4, and so adding hundreds, thousands, and so on is simply adding another number that is divisible by 4.So, 5,156,9_6 to be divisible by 4:
If the last two digits are divisible by 4, the sum is divisible by 4.As a result, _6 must be divisible by 4.There are two options here: 3 (as in 36) or 7. (as in 76).Therefore, the digit that makes the given statement true is 3 or 7.
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The correct question is given below:
Type a digit that makes this statement true.
5,156,9_6 is divisible by 4.
The solution set of a linear system whose augmented matrix is [ 1 2 3 ] is the same as the solution set of =, if =[ 1 2 3 ].
x+2x+x=d is solution set of a linear system whose augmented matrix.
What does matrix mean?
A collection of numbers lined up in rows and columns to form a rectangular array is called a matrix. The elements, or entries, of the matrix are the integers. In addition to numerous mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics.First statement
[ 1 2 3 | d][x]
[ 1 2 3]x=d
x+2x+3x=d
Second statement
Ax=d
Given that A = [ 1 2 3]
[ 1 2 3]x=d
x+2x+3x=d
x+2x+x=d
Then, they are going to have the same solution
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solve the following quadratic equations by using square roots
(x-2)² = 0
(x-2)² =36
9a²-100 =0
Answer:
Step-by-step explanation:
no solution
Examine the distribution of the target variable. what is the proportion of individuals who purchased organic products?
The proportion of individuals who purchased organic products is 24.77%.
The target variable is the variable whose values are modeled and predicted by other variables. A predictor variable is a variable whose values will be used to predict the value of the target variable
The target variable (also called the dependent variable) used in the analysis for this tutorial is a categorical variable that differentiates clinicians in the study that indicated that they would be Highly Likely to Use Product X from those that had a Low Likelihood to Use the new product
Organic food is grown without the use of synthetic chemicals, such as human-made pesticides and fertilizers, and does not contain genetically modified organisms (GMOs). Organic foods include fresh produce, meats, and dairy products as well as processed foods such as crackers, drinks, and frozen meals.
According to the plot the distribution of target variable is as follows
0’s: 167181’s
5505
The proportion of individuals who purchased organic products are: 5505/22223
= 0.2477
=24.77%
That is around 24.77% people have purchased organic products.
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