The answer to the sum is: ∑ k⁴ (k ranges from 1 to 6)
Express the sum in sigma notation :
According to the question,
1 = lower limit of summation
k = the index of summation
1 + 16 + 81 + 256 + 625 + 1296 = 1² + 4² + 9² + 16² + 25² + 36²
= 1⁴ + 2⁴ + 3⁴ + 4⁴ + 5⁴ + 6⁴ ( i.e 16 = 4²=(2²)²)
= ∑ k⁴ (k ranges from 1 to 6)
So, The answer to the sum is - ∑ k⁴ (k ranges from 1 to 6).
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Devan ran 27 miles last week, which was three times as far as Hailey ran.
The equation that represents the number of miles Hailey ran is [tex]$3 \cdot h=27$[/tex],
[tex]$h=\frac{1}{3} \cdot 27$[/tex] and [tex]$h=3 \cdot 27$[/tex].
What is meant by expression?A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself. Any one of the following mathematical operations can be used. A sentence has the following structure: The formula is (Number/Variable, Math Operator, Number/Variable).
The addition, subtraction, multiplication, and division arithmetic operators are used to write a group of numbers together to form a numerical statement in mathematics. The expression of a number can take on various forms, including verbal form and numerical form.
Let the given information be "Last week, Devan ran 27 miles, three times Hailey's distance".
Let h be the number of miles Hailey ran.
then, we get
[tex]$3 \cdot h=27$[/tex],
[tex]$h=\frac{1}{3} \cdot 27$[/tex] and
[tex]$h=3 \cdot 27$[/tex]
Therefore, the correct answer is option
a) [tex]$3 \cdot h=27$[/tex]
b) [tex]$h=\frac{1}{3} \cdot 27$[/tex]
d) [tex]$h=3 \cdot 27$[/tex]
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For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 106. F(x, y) = (6x + 5y)i + (5x + 4yj
f(x, y) = 3x + 2y + C
To know if a vector field is conservative, we have to check if it is the gradient of a scalar function.
The gradient of the function f(x, y) is given by
∇f(x, y) = (∂f/∂x, ∂f/∂y)
If a vector field F(x, y) is the gradient of a scalar function f(x, y), then the following conditions must be satisfied:
F(x, y) = ∇f(x, y)
Whether a given vector field satisfies this condition can be checked by taking the partial derivative with respect to x and y.
∂F/∂x = ∂(6x + 5y)/∂x = 6
∂F/∂y = ∂(5x + 4y)/∂y = 4
The partial derivative of a given vector field is the constant, the gradient of the scalar function. So vector fields are conservative and we can find a potential function f(x, y) such that F(x, y) = ∇f(x, y).
To find the potential function, we can integrate the partial derivative of F(x, y) with respect to x and y.
f(x, y) = ∫∂F/∂xdx + ∫∂F/∂ydy
= ∫6dx + ∫4dy
= 3x + 2y + C
where C is the constant of integration. So the potential function for a given vector field is
f(x, y) = 3x + 2y + C
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The function g is defined by g(x)=x² +4.
Find g (5a).
g (5a) =
Answer:
25a² +
Step-by-step explanation:
5a is put in the equation replacing the previous variable and then the whole square follows and hence. g(5a) =25a²+
Meryl stood X meters from the base of a lighthouse. The angle from where she stood to the top of lighthouse was 60 degrees. If the lighthouse was 66.5 meters tall, how far away was Meryl standing from the base of the lighthouse? Round your answer to the nearest whole meter.
The required distance of Meryl from the base of the lighthouse is 38.39 meters.
What is angle ?An angle is the formed when two straight lines meet at one point, it is denoted by θ.
Given that,
The height of the lighthouse = 66.5 meters.
Also, the angle to the top of the lighthouse = 60°.
To find the distance X from the base of the lighthouse,
use tan θ = perpendicular / base
Substitute the values here,
tan θ = 66.5 / X
tan 60 = 66.5 / X
1.732 = 66.5 / X
X = 66.5 / 1.732
X = 38.39 meters.
The distance of Meryl from the base of the lighthouse is 38.39 meters.
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Please show your work: Select ALL the transformations that occur to the parent function f(x) to form g(x)=0.8f(x−2)−8
Answer:
Step-by-step explanation:
Vertical shrink by 0.8
horizontal shift by 2 to the right
vertical shift by 8 down
A number is rounded off to the nearest thousand. The rounded number is less than the original
number. Which of these can be the original number?
a) 4832
b) 5789
c) 6315
d) 7956
Answer:
c or d or b but probely d
Step-by-step explanation:
because
What correctly explains 55 percent of 64
Answer:
15
Step-by-step explanation:
Round 55 percent to 40 percent and 64 to 60. Think of 40 percent as . So, of 60 is 15.
How to write the fractions in its simplest form?
Step-by-step explanation:
---------------------
Please what’s the answer
The coefficient of the given term is 2208
What is the binomial formula?The binomial formula is used for binomial expansion and is given by: [tex](x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}[/tex]
Given here (2u-s)²⁴
The 22nd term will be given by ;
T₂₂ = ²⁴C₂₂× (2u)²× (-s)²²
= 552×4× (u)²× (s)²²
= 2208 (u)²(s)²²
Hence, the coefficient of the given term is 2208
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a line with a slope of 1/10 passes through the point (20,2). what is its equation in slope-intercept form?
-------------------------------------------------------------------------------------------------------------
Answer: [tex]y= \frac{1}{10}x[/tex]
-------------------------------------------------------------------------------------------------------------
Given: [tex]\textsf{Slope = 1/10 and goes through (20, 2)}[/tex]
Find: [tex]\textsf{Slope in slope-intercept form}[/tex]
Solution: First we need to plug into the point slope formula which is [tex](y - y_1) = m(x - x_1)[/tex]. Let us plug in the values first and then solve for the slope-intercept form.
Plug in the values
[tex](y - y_1) = m(x - x_1)[/tex][tex](y - 2) = \frac{1}{10}(x - 20)[/tex]Simplify
[tex]y - 2 = (\frac{1}{10} * x)+(\frac{1}{10}*-20)[/tex][tex]y - 2 = \frac{1}{10}x-2[/tex][tex]y - 2 + 2 = \frac{1}{10}x-2 + 2[/tex][tex]y= \frac{1}{10}x[/tex]Answer: Therefore, the final answer in slope-intercept form for the line with a slope of 1/10 that passes through the point (20, 2) is [tex]y= \frac{1}{10}x[/tex]
Find the sample mean, variance, and standard deviation. (For standard deviation, round your answer to two decimal places.)
−8, −4, −3, 0, 1, 2
sample mean:
sample variance:
sample standard deviation:
The sample mean is -2.0, the sample variance is 11.67, and the sample standard deviation is 3.42.
The sample mean is the average of a given set of numbers. To find the sample mean of the set of numbers {-8, -4, -3, 0, 1, 2}, you add up all of the numbers and divide by the number of numbers in the set, which is 6. So, the sample mean is -2.0. The sample variance measures the variability of the data, or how spread out the data is. To calculate the sample variance, you first subtract the mean from each number and square the result of each. Then, you add up all of the squares and divide by the number of numbers in the set. In this case, the sample variance is 11.67. The sample standard deviation is the square root of the sample variance and is a measure of how far on average each number is from the mean. To calculate the sample standard deviation, you simply take the square root of the sample variance, which is 3.42.
Sample mean: -2.0 = (-8 + -4 + -3 + 0 + 1 + 2) / 6
Sample variance: 11.67 = (64 + 16 + 9 + 0 + 1 + 4) / 6
Sample standard deviation: 3.42 = √11.67
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URGENT!! ILL GIVE BRAINLIEST!!!! AND 100 POINTS!!!
The main similary between a cone and a cylinder: They both have a circular base, area of the base is A = π · r². (Correct choice: B)
What similarity does exist between a cone and a cylinder?In this problem we must compare a cone with a similar cylinder, that is, two solids with similar key measures (same height and same base radius). The area of the base is described by the following formula:
A = π · r²
Where r is the radius of the circular base, in square units.
The volume formula for each solid is shown below:
Cone
V = (1 / 3) · A · h
Cylinder
V = A · h
Where h is the height of the solid, in units.
The main conclusions are described below:
The volume of the cone is a third of the volume of cylinder.Both solids have a circular base. The cone has one circular base and the cylinder has two parallel circular bases.Three cones fit a similar cylinder.Hence, the right choice for this question is B.
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2. From basis to differential equation In each of the following problems you are given a set of basis functions for the general solution to homogeneous, linear DE with constant coefficients. Work backwards to a determine the DE (in unknown function y y(z)) which they solve. 1. The number of basis functions corresponds to the order of the differential equation (and dimension of the solution space). 2. You can determine all characteristic equation roots and their multiplicities from the basis functions. 3. From the characteristic equation, you can write the DE. (b) (e2e e cos (2z), e r sin(2r))
On solving the provided question, the differential equations we obtained are as follows - [tex]\frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -y = 0[/tex] And [tex]\frac{d^3y}{dx^3} +\frac{dy}{dx} -10y = 0[/tex]
what is differential equations?
An equation comprising one or more functions and their derivatives is known as a differential equation in mathematics. A function's derivative expresses the rate of change of the function at a specific moment. It is mostly employed in the sciences of physics, engineering, and biology.
Since there are 3 basic function, the order of the differential equation is 3. Since the basic finction e^{x} is repeating thrice, the characteristic roots are 1,1,1,(i.e the characteristic root is 1 with multiplicity 3). Hence the characteristic equation is (D-1)^{3} = 0.
[tex]\frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -y = 0[/tex]
Since there are 3 basic function, the order of the differential equation is According to the basic finctions, the characteristic roots are [tex]2,-1[/tex]±[tex]2i.[/tex] Hence the characteristic equation is (D-2)(D^{2}+2D+5) = 0. which means D^{3}+D-10 = 0.
[tex]\frac{d^3y}{dx^3} +\frac{dy}{dx} -10y = 0[/tex]
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Find the first three powers, A, A2, and A3, of the transition matrix below. Find the probability that state 1 changes to state 2 after three repetitions of the experiment. 0.3 0.1 0.6 A0.5 0.2 0.3 Type an integer or decimal for each matrix element.)
The probability that state 1 changes to state 2 after three repetitions of the experiment is 0.052.
The first power of the transition matrix A is simply the matrix itself:
A1 = [ 0.3 0.1 0.6 ]
[ 0.5 0.2 0.3 ]
The second power of the transition matrix is obtained by multiplying A by itself:
A2 = A1 * A1 = [ 0.30.3+0.10.5 0.30.1+0.10.2 0.30.6+0.10.3 ]
[ 0.50.3+0.20.5 0.50.1+0.20.2 0.50.6+0.20.3 ]
= [ 0.19 0.06 0.33 ]
[ 0.29 0.08 0.33 ]
The third power of the transition matrix is obtained by multiplying A2 by A:
A3 = A2 * A = [ 0.190.3+0.060.5 0.190.1+0.060.2 0.190.6+0.060.3 ]
[ 0.290.3+0.080.5 0.290.1+0.080.2 0.290.6+0.080.3 ]
= [ 0.161 0.052 0.327 ]
[ 0.247 0.068 0.327 ]
To find the probability that state 1 changes to state 2 after three repetitions of the experiment, we need to look at the second element of the first row of A3. This element is 0.052, so the probability that state 1 changes to state 2 after three repetitions of the experiment is 0.052.
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Convert this rational number to its decimal form and round to the nearest thousandth.
2/9
The decimal form of the given rational number 2/9 is 0.2
A rational number is a number which is defined in p/q form where q is not equals to 0 and p and q are real numbers or integers.
A decimal number is a standard system to represent the integers and non-integer numbers.
Now convert the given rational number 2/9 into decimal form by dividing the numerator, that is, 2 with the denominator, that is 9.
After division we will get the decimal form of the given rational number
2/9 = 0.222222
When we round off it to nearest thousand it will be written as 0.2
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Read the following statement:
If the measure of an angle is more than 90º, then it is an obtuse angle.
Which of the following choices is a counterexample to the statement?
the measure of an angle that is 180º
the measure of an angle that is 150º
the measure of an angle that is 91º
Answer:
the measure of an angle that is [tex]180^{\circ}[/tex]
Step-by-step explanation:
This would be a straight angle, not an obtuse angle.
In Exercises 1, 2, 3, 4, 5, and 6, show that v is an eigenvector of A and find the corresponding eigenvalue. A= [ -1 16 0 ] V = [1 -2]
Av = λv, so we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.
A*v = [-1 16 0] * [1 -2] = [-1 -32 0]
λv = [-2 -4 0]
Since Av = λv, we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.
1. First, we multiply the matrix A with vector v to find the product Av.
2. A*v = [-1 16 0] * [1 -2] = [-1 -32 0]
3. Then, we multiply the eigenvalue λ with vector v which gives the result λv.
4. λv = [-2 -4 0]
5. We compare Av and λv. If they are equal, then v is an eigenvector of A and the corresponding eigenvalue is λ.
6. Av = λv, so we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.
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Solve the compound inequality and give the answer in interval notation.
The solution to the compound inequality in interval notation is [-7, 5)
How to determine the solution to the compound inequalityFrom the question, we have the following parameters that can be used in our computation:
3x - 10 < -5x + 30 AND 3(-7x - 7) + 6 ≤ - 6x + 132
Evaluate the like terms in the above expression
So, we have the following representation
8x - 10 < 30 AND 3(-7x + 7) + 6 ≤ - 6x + 132
Open the brackets
This gives
8x - 10 < 30 AND -21x + 21 + 6 ≤ - 6x + 132
Evaluate the like terms in the above expression
So, we have the following representation
8x < 40 AND -15x ≤ 105
Divide both sides by the coefficient of x
This gives
x < 5 AND x ≥ -7
Combine the inequalities
[-7, 5)
Hence, the solution is [-7, 5)
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Can someone please help me solve this question?
Answer:
Step-by-step explanation:
Consider again the electric circuit in Problem 21 of Section 7.6. This circuit is described by the system of differential equations ()=(-4,-)(0) a. Show that the eigenvalues are real and equal if L = 4R2C. b. Suppose that R=1.12, C = 1 F, and L = 4 H. Suppose also that I(0) = 1 A and V(0) = 2 V. Find I(t) and V(t).
The solution of a differential equation to the system is then given by I(t) = (3/4)e-4.48t + (-1/4)e4.48t and V(t) = (3/4)e-4.48t – (2/4)e4.48t.
a. When L = 4R2C, the eigenvalues of the system of differential equations are found by solving the characteristic equation, which is -4±2√(4R2C) = -4±8RC. Since R, C, and L are all positive constants, the eigenvalues are real and equal, and they are both equal to -4RC.
b. Setting R = 1.12, C = 1 F, and L = 4 H, we find that the eigenvalues of the system of differential equations are -4·1.12·1 = -4.48. The solution to the system is then given by I(t) = Ae-4.48t + Be4.48t and V(t) = -4RAe-4.48t + 4RB e4.48t.
Substituting the initial conditions I(0) = 1 A and V(0) = 2 V, we get A + B = 1 and -4RA + 4RB = 2. Solving for A and B, we find that A = 3/4 and B = -1/4.
The solution to the system is then given by I(t) = (3/4)e-4.48t + (-1/4)e4.48t and V(t) = (3/4)e-4.48t – (2/4)e4.48t.
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7. A customer ordered 20 kg of fertilizer that contains 15% nitrogen. To
fulfill the customer's order, how much of the stock 30% nitrogen
fertilizer must be mixed with the 10% nitrogen fertilizer?
Answer:
5 kg of 30% nitrogen an d 15 kg of 10% nitrogen.
Step-by-step explanation:
ILL GIVE BRAINLY AND 10 POINTS HURRY |
State the area of the given trapezoid.
Responses
A 40 square meters40 square meters
B 20 square meters20 square meters
C 26 square meters26 square meters
D 32 square meters
Answer:
Step-by-step explanation:
The answer is 22 the reason why is because 5 + 5 + 8 + 4= 22 so you just add all the numbers up!
If X and Y are independent standard normal random variables, determine the joint density function of
U = XV
=X/Y
Then use your result to show that has a Cauchy distribution.
The joint density function of u and v
If X and Y are independent standard normal random variables, joint density function of u and v is 1/(π+v²).
We have X and Y are independent standard normal random variables. It is given that X and Y are two independent standard normal random variables.
That is, The probability density function of standard normal random variable X and Y as shown below:
f(x) = 1/√2π exp{ -x²/2}
f(y) = 1/√2π exp{ -y²/2}
The joint density function of X and Y is shown as
f(x,y) = 1/2π exp{ -(x² + y²)/2}
It is given that, U = X , V = x/y
=> Y = X/V = U/V
The Jacobean transformation is shown below:
J = | ∂x/∂u ∂x/∂v |
| ∂y/∂u ∂y/∂v |
= det [ 1 0 ; 1/v -u/v²]
= 1 ×(-u/v²) - 0×(1/v)
= -u/v²
The joint density function of u and v is,
f(u,v) = fₓₓ(x,y)|J| =1/2π exp{ -(u² + (u/v)²} u/v²
find the value of f(x/y)
fᵥ (V) = ∫ f(u,v) du where limits u = -∞ to u= ∞
=∫{1/2π exp{ -(u²+ (u/v)²)} u/v²du , u = -∞ to u = ∞
= 2 ∫1/2π exp{ -(u² + (u/v)²)} u/v²du, u = 0 to u = ∞
(since, integrand is even function of u)
Using the change of variable,
u²₁ = u²(1+v²) we have
fᵥ (V) = ∫ 1/(π+v²) exp( -u₁²/2) u/v² du₁, u = 0 to u = ∞
= 1/(π+v²) ∫exp( -u₁²/2) d(u₁/v²) = 1/(π+v²)(1) =1/(π+v²)
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a school board has a plan to increase participation in the pta. currently only about 16 parents attend meetings. suppose the school board plan results in logistic growth of attendance. the school board believes their plan can eventually lead to an attendance level of 48 parents. in the absence of limiting factors the school board believes its plan can increase participation by 30% each month. let m denote the number of months since the participation plan was put in place, and let p be the number of parents attending pta meetings.
The required logistic model is
[tex]p=\frac{48}{1+2e^{0.11394t} }[/tex]
As given in the question,
a school board has plan to increase the participation in the pta
number of parents attending the meetings is 16
the school board believes that their would be rise in attendance level of
48 parents
the percentage of attendance rise is 30%
a logistic model is
[tex]N = \frac{k}{1+be^{-rt} }[/tex]
where
k is a constant
the constant b is founded by
[tex]b = \frac{k}{N(0)}-1[/tex]
r is the intrinsic exponential growth rate
if there is no limiting factors then logistic model will be
[tex]N=N(0) e^{rt}[/tex]
so the corresponding growth factor for the equation will be
[tex]a=e^t[/tex]
so the variable can be obtained by using
[tex]r=lna[/tex]
A) p is the number of parents attending pta meetings
P(0) = 16
then the limiting value is k
where k = 48
therefore,
the carrying capacity ( limiting value ) k is 48
B) the constant for the logistic model
[tex]b=\frac{k}{p(o)} -1[/tex]
where
the value of k is 48
the value of p(o) is 16
now
[tex]b = \frac{48}{16} -1\\[/tex]
[tex]b= \frac{48-16}{16}[/tex]
[tex]b = \frac{32}{16}[/tex]
b = 2
the constant for b in logistic model is
b = 2
C)
the r value to construct the logistic model because the number of participation increased by 30%
a = 1+0.30
a = 1.30
to find r
r = ln a
r = ln (1.30)
r = 0.11394
D)
now substitute the values k = 48 and b=2 and r = 0.11394 in the equation
[tex]p=\frac{k}{1+be^{-rt} }[/tex]
=> p = [tex]\frac{48}{1+2e^{0.11394t} }[/tex]
this the required logistic model which represents the number of parents attending the PTA after this month
=> p= [tex]\frac{48}{1+2e^{0.11394t} }[/tex]
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Note:- The complete question is,
A school board has a plan to increase participation in the pta. currently only about 16 parents attend meetings. suppose the school board plan results in logistic growth of attendance. the school board believes their plan can eventually lead to an attendance level of 48 parents. in the absence of limiting factors the school board believes its plan can increase participation by 30% each month. let m denote the number of months since the participation plan was put in place, and let p be the number of parents attending pta meetings.
(a) What is the carrying capacity K for a logistic model of P versus m ?
K=
(b) Find the constant b for a logistic model.
b=
(c) Find the r value for a logistic model. Round your answer to three decimal places.
r =
(d) Find a logistic model for P versus m.
P=
exploration of the clinical significance of the findings, as opposed to the statistical significance
While statistical significance indicates the reliability of the study results, clinical significance reflects its impact on clinical practice.
Can you have clinical significance without statistical significance?
Clinical researchers and practitioners should concentrate on clinically relevant improvements rather than statistical significance, which is the emphasis of most research. A research result may be clinically important but not statistically important, and vice versa.
In clinical research, statistically significant study results are frequently considered to be of clinical significance. Clinical significance emphasizes the study's influence on clinical practice, whereas statistical significance highlights the validity of the study's findings.
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Write the point-slope form of the equation of the line that passes through the points (-8, 2) and (1, -4). Include your work in your final answer. Type your answer in the box provided or use the upload option to submit your solution.
Slope: -2/3
Point-Slope Form: y - 2 = -2/3 ( x - (-8))
The equation of line in point slope form is y = -2/3 x - 10/3
Define Equation of line.
A line in geometry is a collection of points that can be stretched infinitely in two different ways.To put it another way, a line is created by continually extending the two end points in any direction. As a result, we can assert that a line lacks end points. However, because it just has length and no width, it is a one-dimensional geometric form. Based on the information provided, we can create multiple forms of equations for lines algebraically, such as point slope form, slope-intercept form, and two point form.Two Point Form Formula,y - y₁ = m (x - x₁)
Given points are,
(-8, 2) and (1, -4)
We know,
Slope formula,
m = (y₂ - y₁) / (x₂ - x₁)
Two Point Form Formula,
y - y₁ = m (x - x₁)
Let, (x₁, y₁) = (-8, 2)
and, (x₂, y₂) = (1, -4)
First find the slope,
m = (-4 - 2) / (1 - (-8))
m = -6 / 9
m = -2/3
Now, point slope equation is,
y - 2 = -2/3(x -(-8) )
y - 2 = -2/3 ( x + 8)
y - 2 = -2/3 x - 2/3 * 8
y - 2 = -2/3 x - 16/3
y = -2/3 x - 16/3 + 2
y = -2/3 x - ( 16/3 - 2 )
y = -2/3 x - ( (16 - 6) / 3)
y = -2/3 x - ( 10/3 )
y = -2/3 x - 10/3
Hence, the equation of line in point slope form is y = -2/3 x - 10/3
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Given the equation 3x-12=18. Which equation are equivalent to the given equation above
3x=30
3x=6
3(x-4)=18
-3x+12=-18
6x-24=34
The graph shows the number of meters Jenny runs relative to the number of minutes she runs what is the constant of proportionality round your answer to the nearest hundredth (6,100)
Describe or show how to solve for sine, cosine, and tangent values.
Answer:
SOH CAH TOA
The ratios of sine, cosine and tangent.
Step-by-step explanation:
Sine: Opposite / Hypotenuse
Cosine: Adjacent / Hypotenus
Tangent: Opposite / Adjacent
Let f be a scalar field and F a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field.
(a) curl f
(b) div F
(c) del (upside down traingle) F
(d) curl(curl F)
(e) (grad F)X(divF)
(f) grad(div F)
Please show your work when you answer, thank you!
On solving the provided question, Triple product - A.(BXC) = B.(CXA)=C.(AXB) and AX(BXC) = B(A.C) - C(A.B)
What is vector field?A vector field is a vector association with each point in a subset of space in the field of vector mathematics and physics. For example, a vector field in the plane can be represented as a series of arrows. Each arrow is associated with a point in the plane and each has a specific size and direction.
Triple product -
A.(BXC) = B.(CXA)=C.(AXB)
AX(BXC) = B(A.C) - C(A.B)
Product Rules-
∇(fg) = f(∇g) + g()∇f
∇A.B = AX (∇XB) + BX(∇XA) + (AX∇)B + (B.∇)A
∇(fA) = f(∇.A) + A(∇.f)
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