The ODE
[tex]M(x,y)\,\mathrm dx+N(x,y)\,\mathrm dy=0[/tex]
is exact if
[tex]\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}[/tex]
We have
[tex]M=-xy\sin x+2y\cos x\implies M_y=-x\sin x+2\cos x[/tex]
[tex]N=2x\cos x\implies N_x=2\cos x-2x\sin x[/tex]
so the ODE is indeed not exact.
Multiplying both sides of the ODE by [tex]\mu(x,y)=xy[/tex] gives
[tex]\mu M=-x^2y^2\sin x+2xy^2\cos x\implies(\mu M)_y=-2x^2y\sin x+4xy\cos x[/tex]
[tex]\mu N=2x^2y\cos x\implies(\mu N)_x=4xy\cos x-2x^2y\sin x[/tex]
so that [tex](\mu M)_y=(\mu N)_x[/tex], and the modified ODE is exact.
We're looking for a solution of the form
[tex]\Psi(x,y)=C[/tex]
so that by differentiation, we should have
[tex]\Psi_x\,\mathrm dx+\Psi_y\,\mathrm dy=0[/tex]
[tex]\implies\begin{cases}\Psi_x=\mu M\\\Psi_y=\mu N\end{cases}[/tex]
Integrating both sides of the second equation with respect to [tex]y[/tex] gives
[tex]\Psi_y=2x^2y\cos x\implies\Psi=x^2y^2\cos x+f(x)[/tex]
Differentiating both sides with respect to [tex]x[/tex] gives
[tex]\Psi_x=-x^2y^2\sin x+2xy^2\cos x=2xy^2\cos x-x^2y^2\sin x+\dfrac{\mathrm df}{\mathrm dx}[/tex]
[tex]\implies\dfrac{\mathrm df}{\mathrm dx}=0\implies f(x)=c[/tex]
for some constant [tex]c[/tex].
So the general solution to this ODE is
[tex]x^2y^2\cos x+c=C[/tex]
or simply
[tex]x^2y^2\cos x=C[/tex]
We are to verify and confirm if the given differential equations are exact or not. Then solve for the exact equation.
The first differential equation says:
[tex]\mathbf{(-xy \ sin x + 2y \ cos x) dx + 2(x \ cos x) dy = 0 }[/tex]
Recall that:
A differential equation that takes the form [tex]\mathbf{M(x,y)dt + N(x, y)dy = 0 }[/tex] will be exact if and only if:
[tex]\mathbf{\dfrac{\partial M }{\partial y} = \dfrac{\partial N }{\partial x}}[/tex]From equation (1), we can represent M and N as follows:
[tex]\mathbf{M = (-xy \ sin x + 2y \ cos x)}[/tex][tex]\mathbf{N = (2x \ cos x)}[/tex]Thus, taking the differential of M and N, we have:
[tex]\mathbf{ \dfrac{\partial M}{\partial y }= M_y = -x sin x + 2cos x}[/tex]
[tex]\mathbf{ \dfrac{\partial N}{\partial x }= N_x = 2 cos x + 2x sin x}[/tex]
From above, it is clear that:
[tex]\mathbf{\dfrac{\partial M }{\partial y} \neq \dfrac{\partial N }{\partial x}}[/tex]
∴
We can conclude that the equation is not exact.
Now, after multiplying the given differential equation in (1) by the integrating factor μ(x, y) = xy, we have:
[tex]\mathbf{ = \mathsf{(-x^2y^2 sin x + 2xy^2cos x ) dx +(2x^2ycos x ) dy = 0 --- (2)}}[/tex]Representing the equation into form M and N, then:
[tex]\mathbf{M = -x^22y^2 sin x +2xy^2 cos x}[/tex]
[tex]\mathbf{N = 2x^2y cos x}[/tex]
Taking the differential, we have:
[tex]\mathbf{\dfrac{\partial M}{\partial y }= M_y = -2x^2y sin x + 4xy cos x }[/tex]
[tex]\mathbf{\dfrac{\partial N}{\partial x} =N_x= 4xycos \ x -2x^2 y sin x}[/tex]
Here;
[tex]\mathbf{\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x} }[/tex]
Therefore, we can conclude that the second equation is exact.
Now, the solution of the second equation is as follows:
[tex]\int_{y } M dx + \int (not \ containing \ 'x') dy = C[/tex]
[tex]\rightarrow \int_{y } (-x^2y^2 sin(x) +2xy^2 cos (x) ) dx + \int(0)dy = C[/tex]
[tex]\rightarrow-y^2 \int x^2 sin(x) dx +2y ^2 \int x cos (x) dx = C[/tex] ---- (3)
Taking integrations by parts:
[tex]\int u v dx = u \int v dx - \int (\dfrac{du}{dx} \int v dx) dx[/tex]
∴
[tex]\int x^2 sin (x) dx = x^2 \int sin(x) dx - \int (\dfrac{d}{dx}(x^2) \int (sin \ (x)) dx) dx[/tex]
[tex]\to x^2 (-cos (x)) \ - \int 2x (-cos \ (x)) \ dx[/tex]
[tex]\to -x^2 (cos (x)) \ + \int 2x \ cos \ (x) \ dx[/tex] ----- replace this equation into (3)
∴
[tex]\rightarrow-y^2( -x^2 cos (x) \ + \int 2x \ cos \ (x) \ dx) +2y ^2 \int x cos (x) dx = C[/tex]
[tex]\mathbf{\rightarrow -x^2 y^2 cos (x) \ -2y ^2 \int x \ cos \ (x) \ dx +2y ^2 \int x cos (x) dx = C}[/tex]
[tex]\mathbf{x^2y^2 cos (x) = C\ \text{ where C is constant}}[/tex]
Therefore, from the explanation, we've can conclude that the first equation is not exact and the second equation is exact.
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In one town, 4242% of all voters are Democrats. If two voters are randomly selected for a survey, find the probability that they are both Democrats. Round to the nearest thousandth if necessary.
Answer:
hi
Step-by-step explanation:
PLEASE HELP!!! Solve -(6)^x-1+5=(2/3)^2-x by graphing. Round to the nearest tenth.
X = 1.8 <--- ANSWER
Answer:
x= 1.8
Step-by-step explanation:
We have been given the equation;
-(6)^(x-1)+5=(2/3)^(2-x)
We are required to determine the value of via graphing. To do this we can split up the right and the left hand sides of the equation to form the following two separate equations;
y = -(6)^(x-1)+5
y = (2/3)^(2-x)
We then proceed to graph the two equations on the same graph. The solution will be the point where the equations will intersect. Find the attachment below for the graph;
The value of x is 1.785. To the nearest tenth we have x = 1.8
When water flows across farm land, some soil is washed away, resulting in erosion. An experiment was conducted to investigate the effect of the rate of water flow (liters per second) on the amount of soil (kilograms) washed away. The data are given in the following table.Flow Rate: .31 .85 1.26 2.47 3.75Eroded soil: .82 1.95 2.18 3.01 6.07The association between flow rate and amount of eroded soil is:A. negativeB. impossible to determine because both variables are categoricalC. neither postive nor negativeD. positive
Answer:
D
Step-by-step explanation:
[tex]\left[\begin{array}{cc}Flow Rate&Eroded Soil\\0.31&0.82\\0.85&1.95\\1.26&2.18\\2.47&3.01\\3.75&6.07\end{array}\right][/tex]
As we can see, as flow rate increases, eroded soil also increases. So the association is positive.
The association between flow rate and amount of eroded soil is positive.
The association between flow rate and amount of eroded soil is:
D. positive
The data shows that as the water flow rate increases, the amount of eroded soil also increases. This positive relationship indicates that higher water flow leads to more soil erosion.
Mike wants to redesign a box. Currently, it’s length is 20 cm, it’s width is 30 cm and it’s height is 40cm. he wants to keep the volume and the length unchanged and increase the height by 25 percent. What will be the new width of the box?
Answer:
24 cm
Step-by-step explanation:
The product of height and width will remain the same, so the new width (w) will be given by ...
w·(new height) = (old width)·(old height)
w·(1.25·40 cm) = (30 cm)(40 cm)
w = (30 cm)/1.25 = 24 cm . . . . . . . divide by 1.25·40 cm and simplify
_____
This derives from the fact that volume and length are unchanged. The formula for the volume in terms of length, width, and height is ...
V = LWH
Then the product of W and H is the constant ...
V/L = WH . . . . . divide by L
For our purpose, we only need to know that V and L are unchanged, so the product WH is unchanged. We don't need to know their values.
Of course, increasing the height by 25% is equivalent to multiplying it by 1.25:
H + 25/100·H = H·(1 + 0.25) = 1.25H
Which of the following expressions is true
A. 2^4 * 2^3 = 2^12
B. 3^3 * 3^6 >3^8
C. 4^2 * 4^2 >4^4
D. 5^5 * 5^2 =5^10
3³ X 3⁶ > 3⁸ is the correct expression.
What is Equation?Equations are mathematical statements containing two algebraic expressions on both sides of an 'equal to (=)' sign.
Here, Let first equation is true,
then, LHS = 2⁴ X 2³
= 2⁴⁺³
= 2⁷
≠ RHS
Option A is false.
Again assume option B is true.
Now check, LHS = 3³ X 3⁶
= 3³⁺⁶
= 3⁹
>3⁸
LHS > RHS
Thus, 3³ X 3⁶ > 3⁸ is the correct expression.
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Solve the problem of exponential growth. In 1985 an antique automobile club had 23,000 members. Since then its membership has grown at an average rate of 5% per year. Assuming this trend continues, how many members will there be in 2020? Round to the nearest thousand.
Answer:
[tex]127,000\ members[/tex]
Step-by-step explanation:
In this problem we have an exponential function of the form
[tex]f(x)=a(b)^{x}[/tex]
where
a is the initial value
b is the base
The base is equal to
b=1+r
r is the average rate
In this problem we have
a=23,000 members
r=5%=5/100=0.05
b=1+0.05=1.05
substitute
[tex]f(x)=23,000(1.05)^{x}[/tex]
x ----> is the number of years since 1985
How many members will there be in 2020?
x=2020-1985=35 years
substitute in the function
[tex]f(x)=23,000(1.05)^{35}=126,868\ members[/tex]
Round to the nearest thousand
[tex]126,868=127,000\ members[/tex]
The club is expected to have about 126868 members in 2020.
[tex]\[N(t) = N_0 \times (1 + r)^t\][/tex]
In this problem:
[tex]- \( N_0 = 23,000 \)- \( r = 0.05 \) (5% growth rate)- \( t = 2020 - 1985 = 35 \)[/tex]
Substituting these values into the formula, we get:
[tex]\[N(35) = 23,000 \times (1 + 0.05)^{35}\][/tex]
[tex]\[N(35) = 23,000 \times (1.05)^{35}\][/tex]
[tex]\[(1.05)^{35} \approx 5.516\][/tex]
Now, multiply by the initial number of members:
[tex]\[N(35) = 23,000 \times 5.516 \approx 126,868\][/tex]
A function in which each y value has more than one corresponding x value is called a
A: nonlinear function
B: many to one function
C one to one function
D linear function
For each Y value to have more than one x value would be a linear function.
It would make a straight line.
The correct answer is B: many-to-one function. This is the type of function where one y value is associated with multiple x values, differing from linear, nonlinear, and one-to-one functions.
A function in which each y value has more than one corresponding x value is known as a many-to-one function. This type of function allows for multiple x values to be paired with a single y value. This should not be confused with nonlinear functions or one-to-one functions. Moreover, while linear function is often associated with straight lines, in algebra, a linear function may actually include more than one term, each having a single multiplicative parameter. For example, y = ax + bx² is linear in this algebraic sense because terms x and x² each have one parameter (a and b). However, a function like y = [tex]x^b[/tex], where b is an exponent rather than a multiplicative parameter, is an example of a nonlinear function as it cannot be accurately described using linear regression.
Mike correctly found the slope and y intercept of the line passing through the points(-5,-2)and (3,14)as follows.
Answer:
y= 2x+8
Step-by-step explanation:
The standard form of an equation of line in slope-intercept form is:
y= mx+b
Where m is the slope of the line and b is the y-intercept of the line.
To obtain the equation from the given information we need to put the values of m and b into the standard form of equation.
We are given
m = 2
and
b = 8
So putting the values of m and b
y = (2)x +8
y= 2x+8 ..
THE ANSWER IS :y=2x +8
just know its right.
For a school fundraiser Jeff sold large boxes of candy for three dollars each and small boxes of candy for two dollars each if you sold 37 boxes in all for total of $96 how many more large boxes then small boxes did he sell
A. 7
B. 8
C. 8
D. 12
Answer:
C
Step-by-step explanation:
Answer: The correct option is
(A) 7.
Step-by-step explanation: Given that for a school fundraiser, Jeff sold large boxes of candy for three dollars each and small boxes of candy for two dollars each.
Also, Jeff sold 37 boxes in all for total of $96.
We are to find the number of large boxes more than that of small boxes.
Let x and y represents the number of large boxes and small boxes respectively.
Then, according to the given information, we have
[tex]x+y=37~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\3x+2y=96~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]
Multiplying equation (i) by 2 and subtracting from equation (ii), we get
[tex](3x+2y)-(2x+2y)=96-37\times2\\\\\Rightarrow x=96-74\\\\\Rightarrow x=22.[/tex]
From equation (i), we get
[tex]22+y=37\\\\\Rightarrow y=37-22\\\\\Rightarrow y=15.[/tex]
We have,
[tex]x-y=22-15=7.[/tex]
Thus, there are 7 more large boxes than small boxes.
Option (A) is CORRECT.
At a coffee shop, the first 100 customers'
orders were as follows.
Small
Medium
Large
Hot
22
Cold
If we choose a customer at random, what
is the probability that his or her drink will
be cold?
[? ]%
Answer:
[tex]P=25\%[/tex]
Step-by-step explanation:
We have a sample of 100 clients.
To find the probability that a randomly selected customer chooses a cold drink, we must first count how many people in the sample chose cold drinks.
The table shows that cold drinks were
8 small, 12 medium and 5 large.
Then the number of cold drinks was:
[tex]8 + 12 + 5 = 25[/tex]
Now the probability of someone selecting a cold drink is:
[tex]P = \frac{25}{100}[/tex]
[tex]P = 0.25 = 25\%[/tex]
Answer:p=25%
For the acellus people
Step-by-step explanation:
In order to qualify for a role in a play, an actor must be taller than 64 inches but shorter than 68 inches. The inequality 64 < x < 68, where x represents height, can be used to represent the height range. Which is another way of writing the inequality? x > 64 and x < 68 x > 64 or x < 68 x < 64 and x < 68 x < 64 or x < 68
Answer:
Option A is correct.
Step-by-step explanation:
64 < x < 68
This inequality represent that the height x should be greater than 64 and less than 68.i.e.
x>64 and x<68
So, Option A is correct.
Answer:
x>64 and x<68
Step-by-step explanation:
Anyone who could help with this it would be amazingly appreciated.
The sum of angles in ΔABD is ...
2a + b + d = 180
The sum of angles in ΔAHG is ...
a + h + 90 = 180
Solving this second equation for a, we have ...
a = 180 -90 -h = 90 -h
Substituting this into the first equation, we have ...
2(90 -h) +b +d = 180
180 -2h +b +d = 180 . . . . . eliminate parentheses
b + d = 2h . . . . . . . . . . . . . add 2h-180
h = (1/2)(b+d) . . . . . . . . . . divide by 2
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 5 sin(x), y = 5 cos(x), 0 ≤ x ≤ π/4; about y = −1
Step-by-step explanation:
First we need to find which function is the outside radius and which is the inside radius (by which I mean which is farther and which is closer to the axis of rotation). We can do this by graphing, but we can also do this by evaluating each function at the end limits.
At x = 0:
y = 5 sin 0 = 0
y = 5 cos 0 = 5
At x = π/4:
y = 5 sin π/4 = 5√2 / 2
y = 5 cos π/4 = 5√2 / 2
So y = 5 cos x is the outside radius because it is farther away from y = -1, and y = 5 sin x is the inside radius because it is closer to y = -1.
Here's a graph:
desmos.com/calculator/5oaiobcpww
The volume of the rotation is:
V = π ∫ₐᵇ [(R−y)² − (r−y)²] dx
where a is the lower limit, b is the upper limit, R is the outside radius, r is the inside radius, and y is the axis of rotation.
Plugging in:
V = π ∫₀ᵖ [(5 cos x − -1)² − (5 sin x − -1)²] dx
V = π ∫₀ᵖ [(5 cos x + 1)² − (5 sin x + 1)²] dx
V = π ∫₀ᵖ [(25 cos² x + 10 cos x + 1) − (25 sin² x + 10 sin x + 1)] dx
V = π ∫₀ᵖ [25 cos² x + 10 cos x + 1 − 25 sin² x − 10 sin x − 1] dx
V = π ∫₀ᵖ [25 (cos² x − sin² x) + 10 cos x − 10 sin x] dx
V = π ∫₀ᵖ [25 cos (2x) + 10 cos x − 10 sin x] dx
V = π [25/2 sin (2x) + 10 sin x + 10 cos x] from 0 to π/4
V = π [25/2 sin (π/2) + 10 sin (π/4) + 10 cos (π/4)] − π [25/2 sin 0 + 10 sin 0 + 10 cos 0]
V = π [25/2 + 5√2 + 5√2] − 10π
V = π (5/2 + 10√2)
V ≈ 52.283
In this exercise it is necessary to calculate the volume of the rotating solid, in this way we have:
[tex]V= 52.3[/tex]
First we need to find which function is the outside radius and which is the inside radius (by which I mean which is farther and which is closer to the axis of rotation). We can do this by graphing, but we can also do this by evaluating each function at the end limits.
[tex]x=0\\y=5sin(x)\\y=5sin(0)=0\\y=5cos(0)= 5\\\\x=\pi/4\\y=5sin(\pi/4)= 5(\sqrt{2/2})\\y=5cos( \pi/4)= 5/\sqrt{2}[/tex]
So [tex]y = 5 cos x[/tex] is the outside radius because it is farther away from y = -1, and [tex]y = 5 sin x[/tex] is the inside radius because it is closer to y = -1. Here's a graph (first image). The volume of the rotation is:
[tex]V=\pi\int\limits^a_b {[(R-y)^2-(r-y)^2]} \, dx[/tex]
Where a is the lower limit, b is the upper limit, R is the outside radius, r is the inside radius, and y is the axis of rotation. Plugging in:
[tex]V=\pi\int\limits^a_b {[(R-y)^2-(r-y)^2]} \, dx\\= \pi\int\limits^p_0 {[(5 cos(x) + 1)^2-(5 sin(x) + 1)^2]} \, dx\\=\pi\int\limits^p_0 {[(25 cos^2 (x) + 10 cos (x) + 1)-(25 sin^2 (x) + 10 sin (x) + 1)]} \, dx\\=\pi\int\limits^p_0 {[(5 cos^2 (x) + 10 cos (x) + 1 -25 sin^2 (x)- 10 sin (x)- 1]} \, dx\\=\pi\int\limits^p_0 {[(25 (cos^2( x)- sin^2( x)) + 10 cos( x)- 10 sin( x)]} \, dx\\=\pi\int\limits^p_0 {[(25 cos (2x) + 10 cos (x)- 10 sin (x)]} \, dx\\[/tex]
[tex]=\pi[(25/2 sin (2x) + 10 sin x + 10 cos (x)]\\\\=\pi[(25/2 sin (\pi/2) + 10 sin (\pi/4) + 10 cos (\pi/4)] - \pi [25/2 sin( 0) + 10 sin( 0) + 10 cos(0)]\\\\=\pi[25/2+5\sqrt{2}+5/\sqrt{2}]-10\pi\\\\\V= 52.3[/tex]
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The expression is the result of applying the change of base formula to a logarithmic expression.
Which could be the original expression?
x= 8 is the correct answer
Answer:
x = 8
Step-by-step explanation:
Given equation is,
[tex]log_5(10x-1)=log_5(9x+7)[/tex]
We know that,
[tex]log_a(b)=log_a(c)\implies b = c[/tex]
[tex]\implies 10x -1 = 9x + 7[/tex]
Subtracting 9x on both sides,
[tex]x - 1 = 7[/tex]
Adding 1 on both sides,
[tex]x = 8[/tex]
Hence, the solution would be x = 8
Please help me simplified this
For this case we have that by definition, the perimeter of the quadrilateral shown is given by the sum of its sides:
Let "p" be the perimeter of the quadrilateral, then:
[tex]p = 7 + y + 7 + x\\p = 7 + 7 + x + y\\p = 14 + x + y[/tex]
So, the perimeter of the figure is: [tex]14 + x + y[/tex]
Answer:
[tex]p = 14 + x + y[/tex]
Answer:
perimeter = x + y + 14
Step-by-step explanation:
The perimeter is the sum of the side lengths:
perimeter = x + 7 + y + 7
The constants can be combined to give ...
perimeter = x + y + 14
___
An "equation" will have an equal sign somewhere. Since you want to find the perimeter, it makes sense for the equation to show you how to find the perimeter. (I have used "perimeter" to represent the perimeter. It is often represented using the letter P. Of course, you can choose any variable or other representation you like.)
What is the slope of a line that is perpendicular to the line y = 1? -1,0,1,or undefined are the choices.
ANSWER
undefined
EXPLANATION
We want to find the slope of a line that is perpendicular to the line y = 1
The line y=1 is parallel to the x-axis.
In other words, the line y=1 is a horizontal line.
The line perpendicular to y=1 is a vertical line.
The slope of a vertical line is undefined
Answer:
its an undefined slope
Step-by-step explanation:
the line y=−1 has slope 0 so any line perpendicular to it will have an undefined slope
The angle formed by the intersection of a secant and a tangent is equal to the sum of the measures of the intercepted arcs.
True
False
Answer: False
Step-by-step explanation:
The measure of the created angle is equal to the difference of the measure of the arcs.
The angle formed by the intersection of a secant and a tangent is equal to half the measure of the intercepted arc, not the sum of the measures of the intercepted arcs.
Explanation:The statement is False. The angle formed by the intersection of a secant and a tangent is half the measure of the intercepted arc, not equal to the sum of the measures of the intercepted arcs.
For example, let's consider a circle with a secant and a tangent line intersecting at a point. The intercepted arc is represented by the section of the circumference between the two points of intersection. The angle formed by the intersection of the secant and the tangent is equal to half the measure of this intercepted arc.
So, in conclusion, the angle formed by the intersection of a secant and a tangent is equal to half the measure of the intercepted arc.
x cubed - y cubed factor completely
ANSWER
[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} + xy + {y}^{2}][/tex]
EXPLANATION
We want to factor:
[tex] {x}^{3} - {y}^{3} [/tex]
completely.
Recall from binomial theorem that:
[tex]( {x - y)}^{3} = {x}^{3} - 3 {x}^{2} y + 3x {y}^{2} - {y}^{3} [/tex]
We make x³-y³ the subject to get:
[tex] {x}^{3} - {y}^{3} = ( {x - y)}^{3} + 3 {x}^{2} y - \:3x {y}^{2}[/tex]
We now factor the right hand side to get;
[tex]{x}^{3} - {y}^{3} = ( {x - y)}^{3} + 3 {x} y(x - y)[/tex]
We factor further to get,
[tex]{x}^{3} - {y}^{3} = (x - y)[( {x - y)}^{2} + 3 {x} y][/tex]
[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} - 2xy + {y}^{2} + 3 {x} y][/tex]
This finally simplifies to:
[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} + xy + {y}^{2}][/tex]
To factor the expression x³ - y³ completely, use the formula for factoring a difference of cubes: a³ - b³ = (a - b)(a² + ab + b²). Plug in x for a and y for b to get (x - y)(x² + xy + y²).
To factor the expression x³ - y³ completely, we can use the formula for factoring a difference of cubes. The formula is:
a³ - b³ = (a - b)(a² + ab + b²)
Using this formula, we can plug in x for a and y for b.
x³ - y³ = (x - y)(x² + xy + y²)
Therefore, the expression x³ - y³ can be factored completely as (x - y)(x² + xy + y²).
For each x and n, find the multiplicative inverse mod n of x. Your answer should be an integer s in the range 0 through n - 1. Check your solution by verifying that sx mod n = 1.(a) x = 52, n = 77(b) x = 77, n = 52(c) x = 53, n = 71(d) x = 71, n = 53
To find the multiplicative inverse of x modulo n, we need to find an integer s such that (x * s) mod n = 1. We can calculate the multiplicative inverses for the given values: (a) x = 52, n = 77: s = 65. (b) x = 77, n = 52: s = 33. (c) x = 53, n = 71: s = 51. (d) x = 71, n = 53: s = 9.
Explanation:To find the multiplicative inverse modulo n of x, we need to find an integer s such that (x * s) mod n = 1. Let's calculate the multiplicative inverses for the given values:
a) For x = 52 and n = 77:
x * s ≡ 1 (mod n)
52 * s ≡ 1 (mod 77)
s ≡ 65 (mod 77)
So, the multiplicative inverse of 52 modulo 77 is 65.
b) For x = 77 and n = 52:
77 * s ≡ 1 (mod 52)
s ≡ 33 (mod 52)
The multiplicative inverse of 77 modulo 52 is 33.
c) For x = 53 and n = 71:
53 * s ≡ 1 (mod 71)
s ≡ 51 (mod 71)
The multiplicative inverse of 53 modulo 71 is 51.
d) For x = 71 and n = 53:
71 * s ≡ 1 (mod 53)
s ≡ 9 (mod 53)
The multiplicative inverse of 71 modulo 53 is 9.
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etermine if the following statement is true or false. If there is clearance for 95% of males, there will certainly be clearance for all women in the bottom 5%. A. The statement is true because some women will have sitting knee heights that are outliers. B. The statement is false because the 95th percentile for men is greater than the 5th percentile for women. C. The statement is true because the 95th percentile for men is greater than the 5th percentile for women. D. The statement is false because some women will have sitting knee heights that are outliers.
Answer:
C. The statement is true because the 95th percentile for men is greater than the 5th percentile for women.
Step-by-step explanation:
We suspect relevant data is missing, including exactly what clearance the the statement is referring to. It would be nice to know where the various men's and women's percentiles lie.
We give the above answer on the basis of the assumption that the largest of men are always larger than the smallest of women.
Emerson is making a box without a top from a rectangular piece of cardboard, with dimensions 12in by 16in, by cutting out square corners with side length x in.
A) Write an equation for the volume V of the box in terms of x.
B) Use technology to estimate the value of x that gives the greatest volume. Round the value to the nearest tenth.
C) Assume Emerson used the value of d you found in Part (b) to make his box. What were the dimensions of Emerson’s box?
Answer:
Step-by-step explanation:
The volume of a rectangular box is width times length times height:
V = wlh
After the cardboard is folded, the width is 12 - 2x, the length is 16 - 2x, and the height is x.
So the volume is:
V = (12 - 2x) (16 - 2x) x
If we graph this, we get a wave: desmos.com/calculator/rsjosgzuxz
The wave is the highest at around x = 2.3 in.
If we set x = 2.3:
w = 12 - 2x = 7.4
l = 16 - 2x = 11.4
h = x = 2.3
What is the value place of the 5 in 956? Please answer ASAP!
Since the 5 is in the tens place, that means it represents 5 tens, or 50.
Answer:
the tens' place
Step-by-step explanation:
refer to the attachment
Use the order of operations to simplify the expression: 2^3 + 5 × 10 ÷ 2 – 3^3 + (11 – 8) Question 5 options: A)9 B)10 C)41 D)55 my answer when I worked it ended up being 3. Clearly, that’s not a choice
For this case we have that according to the order of PEMDAS algebraic operations, it is established that:
P: Any calculation is made inside the parentheses, making the most internal ones first.
E: Any exponential expressions are simplified.
MD: All the multiplications and divisions are made, from left to right, as they appear.
AS: All sums and subtractions are made, from left to right, as they appear.
So:
[tex]2 ^ 3[/tex] + 5 * 10 ÷ 2-[tex]3 ^ 3[/tex] + (11-8) =
[tex]2 ^ 3[/tex]+ 5 * 10 ÷ 2-[tex]3 ^ 3[/tex] + 3 =
8 + 5 * 10 ÷ 2-27 + 3 =
8 + 50 ÷ 2-27 + 3 =
8 + 25-27 + 3 =
33-27 + 3
6 + 3 =
9
Answer:
9
Describe the piece wise function below by evaluating the function for given values of the domain.
Answer:
y = -3 (x ≤ -2)y = 2x - 3 (-2 < x ≤ 2)y = 5 (2 < x)Step-by-step explanation:
The leftmost piece of the function is constant at -3.
The middle piece is a line with a slope of 2 that intersects the y-axis at -3, so has equation y = 2x-3. (The slope-intercept form of the equation of a line is y = slope·x + y-intercept. The slope is found by counting the number of vertical grid squares that correspond to each horizontal grid square.)
The rightmost piece is constant at +5.
___
We don't know what your choices are, so we can't tell you which to select.
The piecewise function is a horizontal line restricted between x = 0 and x = 20. To evaluate the function, substitute the given values into the function.
Explanation:The given piecewise function is:
f(x) =
a horizontal line for 0 ≤ x ≤ 20restricted to the portion between x = 0 and x = 20To evaluate the function for given values of the domain, you can substitute the given values of x into the function and calculate the corresponding y-values. For example, to find f(5), substitute x = 5 into the function and evaluate.
It is important to note that the question does not provide the equation or specific values of the function, so the evaluation of the function cannot be performed without additional information.
The function f(x) = x2 - 6x + 9 is shifted 5 units to the left to create g(x). What is g(x)?
Answer:
g(x) = x^2 + 4x + 4
Step-by-step explanation:
In translation of functions, adding a constant to the domain values (x) of a function will move the graph to the left, while subtracting from the input of the function will move the graph to the right.
Given the function;
f(x) = x2 - 6x + 9
a shift 5 units to the left implies that we shall be adding the constant 5 to the x values of the function;
g(x) = f(x+5)
g(x) = (x+5)^2 - 6(x+5) + 9
g(x) = x^2 + 10x + 25 - 6x -30 + 9
g(x) = x^2 + 4x + 4
What is the value of x?
Answer:
x = 3
Step-by-step explanation:
3/2.25 = 4/x Cross multiply
3x = 2.25 * 4 Combine the right
3x = 9 Divide by 3
3x/3 = 9/3
x = 3
A spinner is divided into four equal sections that are numbered 2, 3, 4, and 9. The spinner is spun twice. How many outcomes have a product less than 20 and contain at least one even number?
Answer with Step-by-step explanation:
On spinning the spinner twice,we have 16 different outcomes:.
We write the outcomes with their product:
Product
2 2 4
2 3 6
2 4 8
2 9 18
3 2 6
3 3 9
3 4 12
3 9 27
4 2 8
4 3 12
4 4 16
4 9 36
9 2 18
9 3 27
9 4 36
9 9 81
outcomes have a product less than 20 and contain at least one even number are in bold letters.
Hence, outcomes have a product less than 20 and contain at least one even number are:
10
There are 8 outcomes of spinning the spinner twice that result in a product less than 20 and contain at least one even number.
To determine how many outcomes of spinning a spinner twice result in a product less than 20 and contain at least one even number, we start by listing all possible outcomes.
The spinner is divided into four equal sections: 2, 3, 4, and 9. When spun twice, there are 16 outcomes in total:
(2,2)(2,3)(2,4)(2,9)(3,2)(3,3)(3,4)(3,9)(4,2)(4,3)(4,4)(4,9)(9,2)(9,3)(9,4)(9,9)We now filter these outcomes to find those with a product less than 20 and at least one even number:
(2,2) - Product: 4(2,3) - Product: 6(2,4) - Product: 8(3,2) - Product: 6(3,4) - Product: 12(4,2) - Product: 8(4,3) - Product: 12(4,4) - Product: 16The total number of outcomes that meet the criteria is 8.
A production manager tests 10 batteries and finds that their mean lifetime is 468 hours. She then designs a sales package for this type of battery. It states that consumers can expect the battery to last approximately 500 hours. This is an example of what phase of inferential statistics? A.Probability-based inference B.Data organization C.Data gathering
iits probalty like based analyse
Answer:
A.Probability-based inference
Step-by-step explanation:
Inferential statistics uses a random sample of data from a population to draw inferences about the population. One can make generalizations about a population.
The answer as per scenario is A.Probability-based inference.
The manager made an inference based on probability.
What is (f⋅g)(x)? f(x)=x^2−3x+2 g(x)=x^3−8 Enter your answer, in standard form, in the box. (f⋅g)(x)=
Answer:
[tex]f(g(x))=x^{6}-19x^{3}+90[/tex]
Step-by-step explanation:
We have been given the following equations;
f(x)=x^2−3x+2
g(x)=x^3−8
We are required to determine the composite function (f⋅g)(x). This simply means that we shall substitute the function g(x) in place of x in the function f(x);
[tex]f(g(x)) = g(x)^{2}-3g(x)+2\\\\f(g(x))=(x^{3}-8)^{2}-3(x^{3}-8)+2\\\\f(g(x))=x^{6}-16x^{3}+64-3x^{3}+24+2\\\\f(g(x))=x^{6}-19x^{3}+90[/tex]
Vance is designing a garden in the shape of an isosceles triangle. The base of the garden is 30 feet long. The function y = 15 tan θ models the height of the triangular garden. What is the height of the triangle when θ = 30 ° ? What is the height of the triangle when θ = 40 ° ? Vance is considering using either θ = 30 ° or θ = 40 ° for his garden. Compare the areas of the two possible gardens. Explain how you found the areas.
Answer:
Part 1) The height of the triangle when θ = 30° is equal to [tex]8.66\ ft[/tex]
Part 2) The height of the triangle when θ = 40° is equal to [tex]12.59\ ft[/tex]
Part 3) The area of triangle with θ = 30° is less than the area of triangle with θ = 40°
Step-by-step explanation:
Part 1) What is the height of the triangle when θ = 30 ° ?
we have
[tex]y=15tan(\theta)[/tex]
substitute the value of theta in the equation and find the height
[tex]y=15tan(30\°)=8.66\ ft[/tex]
Part 2) What is the height of the triangle when θ = 40 ° ?
we have
[tex]y=15tan(\theta)[/tex]
substitute the value of theta in the equation and find the height
[tex]y=15tan(40\°)=12.59\ ft[/tex]
Part 2) Vance is considering using either θ = 30 ° or θ = 40 ° for his garden
Compare the areas of the two possible gardens
step 1
Find the area when θ = 30 °
The height is [tex]8.66\ ft[/tex]
Remember that the area of a triangle is equal to the base multiplied by the height and divided by two
so
[tex]A=(1/2)(30)(8.66)=129.9\ ft^{2}[/tex]
step 2
Find the area when θ = 40°
The height is [tex]12.59\ ft[/tex]
Remember that the area of a triangle is equal to the base multiplied by the height and divided by two
so
[tex]A=(1/2)(30)(12.59)=188.85\ ft^{2}[/tex]
Compare the areas of the two possible gardens
The area of triangle with θ = 30° is less than the area of triangle with θ = 40°