The ODE is exact because [tex]M_y=N_x[/tex]. Then
[tex]F_x=8x+8y\implies F(x,y)=4x^2+8xy+g(y)[/tex]
[tex]F_y=8x+g'(y)=8x+4y\implies g'(y)=4y\implies g(y)=2y^2+C[/tex]
So we have
[tex]F(x,y)=4x^2+8xy+2y^2=C[/tex]
Final answer:
The given differential equation is exact because the partial derivatives My and Nx are equal, both being 8. Integrating and finding the potential function F(x, y), we get F(x, y) = 4x² + 8xy + 2y².
Explanation:
The equation given is (8x + 8y)dx + (8x + 4y)dy = 0. To determine if this differential equation is exact, we will compare the partial derivatives of M with respect to y (My) and N with respect to x (Nx). In this case, we have M(x, y) = 8x + 8y and N(x, y) = 8x + 4y. Calculating the partial derivatives, we get My = 8 and Nx = 8. Since My equals Nx, the differential equation is exact, implying there is a function F(x, y) such that its differential dF = M dx + N dy.
To find F(x, y), we integrate M with respect to x, yielding F(x, y) = 4x² + 8xy + h(y), where h(y) is an arbitrary function of y. To determine h(y), we differentiate F(x, y) with respect to y and equate it to N: Fy = 8x + h'(y) = 8x + 4y. From this equation, it follows that h'(y) = 4y, so integrating with respect to y gives h(y) = 2y². Therefore, the potential function F(x, y) that makes the differential exact is F(x, y) = 4x² + 8xy + 2y².
M1Q1.) Which plot represents a stemplot of the data?
Answer:
answer 1
Step-by-step explanation:
Since there's only 1 value in the 90-99 groups it is either 1 or 3.
Furthermore, there are only 2 values starting with 10.
This only fits with answer 1
The answer is A.
A stem plot shows every number with the tens place and above being on the left side while the ones place is on the right side.
For example,
9 is in the tens place while 4 is in the ones place. So, this would look like:
9 | 4
There is only one number with a "9" in the tens place and there is only two numbers with "10" in the tens place. This only sastisfys the first option.
Best of Luck!
An exam worth 269 points contains 36 questions. Some questions are worth 7 points, and the others are worth 8 points. How many 7 point and 8 point questions are on the test?
Answer:
There are 19 questions worth 7 points.
There are 17 questions worth 8 points.
Step-by-step explanation:
Assign variables:
Let x = number of questions worth 7 points.
Let y = number of questions worth 8 points.
First we deal with the number of questions.
There are 36 questions, so our first equation is:
x + y = 36
Now we deal with the points.
x questions worth 7 points each are worth 7x points.
y questions worth 8 points each are worth 8y points.
The total worth of all the questions is 7x + 8y.
The total worth of the exam is 269, so our second equation is
7x + 8y = 269
We have a system of 2 equations in 2 variables.
x + y = 36
7x + 8y = 269
Let's use the substitution method to solve the system of equations.
Solve the first equation for x.
x + y = 36
x = 36 - y
Substitute 36 - y in for x in the second equation.
7x + 8y = 269
7(36 - y) + 8y = 269
Distribute the 7.
252 - 7y + 8y = 269
Combine like terms.
252 + y = 269
Subtract 252 from both sides.
y = 17
There are 17 questions worth 8 points.
x + y = 36
x + 17 = 36
x = 19
There are 19 questions worth 7 points.
Check:
What does 19 questions at 7 points each and 17 questions at 8 points each add to?
19 * 7 + 17 * 8 = 133 + 136 = 269
The points add to 269, so our answer is correct.
Find and identify the traces of the quadric surface x2 + y2 − z2 = 36 given the plane. x = k Find the trace. Incorrect: Your answer is incorrect. Identify the trace. circle ellipse hyperbola parabola y = k
Answer:
Hyperbola.
Step-by-step explanation:
We start from the function:
[tex]x^{2} +y^{2} -z^{2} =36[/tex]
We may get the traces if we cut the surface in planes x=k. This is equivalent to replace x for k:
[tex]k^{2}+y^{2} -z^{2}=36\\y^{2} -z^{2}=36-k^{2}[/tex]
The last equation is the equation of an hyperbola, which varies its characteristics depending on K (the plane cut).
When x=k or y=k is substituted into the equation of the quadric surface x² + y² - z² = 36, the resulting traces are hyperbolas. These traces represent intersections of planes parallel to coordinate planes with the quadric surface.
Explanation:The trace of a quadric surface can be found by substituting a constant into the equation for one of the variables. The given quadric surface equation is x² + y² - z² = 36. If we substitute x = k into the equation, we get k² + y² - z² = 36. This is an equation of a hyperbola.
Similarly, for the case when y = k, we substitute this into our original equation to get x² + k² - z² = 36. This also represents a hyperbola.
The traces given by these equations represent the intersection of planes parallel to the coordinate planes and the quadric surface. In these particular cases, the traces are hyperbolas.
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4. Without performing the divisions, determine whether the integers 176521221 and 149235678 are divisible by 9 or 11. - Obtain the following aralization . The int e nted
Final answer:
By using the divisibility rules for 9 and 11, we can conclude that the integer 176521221 is divisible by 9 but not by 11, and the integer 149235678 is divisible by 9 but not by 11.
Explanation:
To determine whether the integers 176521221 and 149235678 are divisible by 9 or 11 without performing the division, we can use divisibility rules:
Divisibility by 9: Add up all the digits in the number. If the sum is divisible by 9, then the original number is also divisible by 9.
Divisibility by 11: Take the alternating sum of the digits. If the result is divisible by 11 (including 0), the original number is divisible by 11.
Divisibility of 176521221 by 9
1+7+6+5+2+1+2+2+1 = 27; Since 27 is divisible by 9, 176521221 is also divisible by 9.
Divisibility of 176521221 by 11
(1-7+6-5+2-1+2-2+1) = -3; Since -3 is not divisible by 11, 176521221 is not divisible by 11.
Divisibility of 149235678 by 9
1+4+9+2+3+5+6+7+8 = 45; Since 45 is divisible by 9, 149235678 is also divisible by 9.
Divisibility of 149235678 by 11
(1-4+9-2+3-5+6-7+8) = 9; Since 9 is not divisible by 11, 149235678 is not divisible by 11.
(From Textbook) A company makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat. If chicken costs 10 cents per ounce and grain costs 1 cent per ounce, how many ounces of each should the company use in each bag of dog food in order to minimize cost?
Answer:
The company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.
Step-by-step explanation:
Let x be the number of ounces of chicken
let y be the number of ounces of grains
We are given that Chicken has 10 grams of protein and 5 grams of fat per ounce.
Chicken has protein in 1 ounce = 10
Chicken has protein in x ounces = 10x
Chicken has fat in 1 ounce = 5
Chicken has fat in x ounces = 5x
We are also given that grain has 2 grams of protein and 2 grams of fat per ounce
Grain has protein in 1 ounce = 2
Grain has protein in y ounces = 2y
Grain has fat in 1 ounce = 2
Grain has fat in y ounces =2y
Now we are given that . A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat.
So, equation becomes:
[tex]10x+2y\geq 200[/tex] ---A
[tex]5x+2y\geq 150[/tex] ---B
Since x and y represent the ounces .
So, [tex]x\geq 0[/tex] ---C and [tex]y\geq 0[/tex] ----D
Plot A ,B ,C and D on the graph
Refer the attached graph so , the corner points are (0,100),(10,50) and (30,0)
We are also given that chicken costs 10 cents per ounce and grain costs 1 cent per ounce
So, Cost function becomes : [tex]c=10x+y[/tex]
At(0,100)
[tex]c=10x+y[/tex]
[tex]c=10(0)+100[/tex]
[tex]c=100[/tex]
At(10,50)
[tex]c=10x+y[/tex]
[tex]c=10(10)+50[/tex]
[tex]c=150[/tex]
At(30,0)
[tex]c=10x+y[/tex]
[tex]c=10(30)+0[/tex]
[tex]c=300[/tex]
Minimum cost is 100 cents at (0,100)
So, the company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.
help please quickly!!
Answer:
m∠XQZ = 72°
Step-by-step explanation:
We presume the "if ..." condition is intended to mean that ray QY lies between rays QX and QZ. Then by angle addition, you get
∠XQY + ∠YQZ = ∠XQZ
(4a +8) + (28) = (8a)
36 = 4a . . . . . . . . . . . . . subtract 4a
72 = 8a = ∠XQZ . . . . multiply by 2
A rectangle has a length 10 m less than twice its width. When 2 m are added to the width, the resulting figure is a square with an area of 196 m squared. Find the dimensions of the original rectangle.
Answer:
The length of the original rectangle is 14 meters and the width of the original rectangle is 12 meters
Step-by-step explanation:
Let
l ----> the length of the original rectangle
w ----> the width of the original rectangle
we know that
[tex]l=2w-10[/tex] ----> equation A
[tex]196=(w+2)^{2}[/tex] ----> area of a square
Solve for w
square root both sides
[tex](w+2)=(+/-)14[/tex]
[tex]w=14=(+/-)14-2[/tex]
[tex]w=14=14-2=12\ m[/tex]
[tex]w=14=-14-2=16\ m[/tex] -----> this solution not make sense
so
[tex]w=12\ m[/tex]
Find the value of L
[tex]l=2(12)-10=14\ m[/tex]
therefore
The length of the original rectangle is 14 meters and the width is 12 meters
Final answer:
The dimensions of the original rectangle are 12 meters (width) and 14 meters (length).
Explanation:
To find the dimensions of the original rectangle, let's first set up an equation.
Let the width of the rectangle be x meters.
The length of the rectangle is given as 10 m less than twice its width, so the length is (2x - 10) meters.
When 2 m are added to the width, the resulting figure is a square with an area of 196 m2.
The side length of a square with area A is √A, so the side length of the square is √196 = 14 meters.
Since adding 2 m to the width creates a square with side length 14 meters, we have:
x + 2 = 14
Subtracting 2 from both sides gives:
x = 12
Therefore, the dimensions of the original rectangle are 12 meters (width) and
(2*12 - 10)
= 14 meters (length).
use logarithmic differentiation to find dy/dx
y=(lnx)^x
Try this suggested solution.
All tangents to the circle are congruent and form a square. The perimeter of square ACEG is 24 cm. What is the length of line segment BC?
Answer:
3 cm
Step-by-step explanation:
The perimeter of the square is 24 cm, so the side length is 6 cm.
I assume B is the point between A and C where the tangent line intersects the circle. If so, B is the midpoint of AC, so it is half the length. Therefore, BC = 3 cm.
The length of line segment BC is 3 cm.
What is a Square ?A square is a polygon with four sides , all the sides of the square are equal.
It is given that the tangents of the circle are forming a square ,
The perimeter of the square is 24 cm.
the length of line segment BC = ?
The perimeter of the square is 4a
where a is the side of the square.
Substituting the values
24 = 4 * a
a = 6 cm
B is the mid point of the tangent length and therefore
BC = 6 /2 = 3 cm
Therefore , the length of line segment BC is 3 cm.
The missing image is attached with the answer.
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A study to determine the sensitivity and specificity of a new test for macular degeneration is conducted on 2430 people. Macular degeneration occurs at a rate of 16.72%. Your sample has the same prevalence of macular degeneration. You find that 377 people with macular degeneration tested positive with the new test. You also have a total of 561 positive test results in your study. CALCULATE THE SPECIFICITY of this test.
Question 2 options:
A)
83.29%
B)
98.45%
C)
92.86%
D)
67.20%
E)
90.91%
A 4:1 scale drawing of a bearing is shown on an A-size print. Using a ruler, you measure the inside diameter of the part on the paper and you get 1.50 inches. What is the actual part size in inches? A. 6.0 B. 1.5 C.4.0 D 375
Answer:
A. 6.0
Step-by-step explanation:
We have been given that the the inside diameter of the part on the paper is 1.50 inches.
We have been given that scale is 4:1 for actual size to drawing side.
[tex]\text{Scale}=\frac{\text{Actual size}}{\text{Map size}}=\frac{4}{1}[/tex]
Upon substituting 1.50 in our given proportion, we will get:
[tex]\frac{\text{Actual size}}{1.50\text{ inches}}=\frac{4}{1}[/tex]
[tex]\frac{\text{Actual size}}{1.50\text{inches}}\times 1.50\text{ inches}=\frac{4}{1}\times 1.50\text{ inches}[/tex]
[tex]\text{Actual size}=4\times 1.50\text{ inches}[/tex]
[tex]\text{Actual size}=6.0\text{ inches}[/tex]
Therefore, the actual size is 6.0 inches and option A is the correct choice.
Jerry King is a server in a restaurant that pays a salary of $43 per day. He also averages tips of 18% of his total gross food orders. Last week he worked 4 days and had total food orders of $2,312.5. What was his total gross pay for the week? Round intermediate calculations and final answer to the nearest cent 416.25
Answer:
His total gross pay for the week is $588.25.
Step-by-step explanation:
Consider the provided information.
Jarry has a salary of $43 per day and he worked for 4 days.
Thus, the salary of 4 days is:
4 × $43= $172
The average tips is 18% of his total gross food orders.
Last week he had total food orders of $2,312.5. Therefore, the total tips he received is:
[tex]2312.5 \times(\frac{18}{100})[/tex]
[tex]2312.5 \times(0.18)[/tex]
[tex]416.25[/tex]
Total tips he received is $416.25.
His total gross pay for the week = The salary of 4 days + Total tips he received.
His total gross pay for the week = $172 + $416.25
His total gross pay for the week = $588.25
Hence, his total gross pay for the week is $588.25.
Final answer:
Jerry King's total gross pay for the week, combining his salary and the tips he made on his food orders, is $588.25, after calculating an 18% tip on his total orders and adding his daily salary for the 4 days he worked.
Explanation:
The question asks us to calculate the total gross pay for Jerry King, a server who earns both a salary and tips based on his food orders. First, we need to calculate Jerry's earnings from tips. He averages an 18% tip on his total gross food orders. Since his total food orders for the week were $2,312.5, we calculate the tips as follows:
Tips = (Total Food Orders) × (Tip Percentage)
= $2,312.5 × 18%
= $2,312.5 × 0.18
= $416.25 (rounded to the nearest cent)
Next, we calculate Jerry's salary for the week. Jerry works 4 days per week at a salary of $43 per day:
Salary = (Daily Salary) × (Number of Days Worked)
= $43 × 4
= $172
Finally, to find Jerry's total gross pay for the week, we add the salary and the tips:
Total Gross Pay = Salary + Tips
= $172 + $416.25
= $588.25 (rounded to the nearest cent)
Therefore, Jerry King's total gross pay for the week is $588.25.
Michelle wants to order cookies. It is $6 for 4 pcs, how much would it cost for 20 pcs?
6 divided by 4 is 1.5.
1.5 times 20 is 30
it would cost michelle $30 to buy 20 pieces of cookies
Answer:
$30
Step-by-step explanation:
4pcs = $6
20pcs = x
Cross multiplying, we have
x = (20pcs * $6)/4pcs
= $120/4 = $30
A regression analysis can be used to determine a. if a linear relationship exists between two categorical variables. b. a cause-and-effect relationship between two variables. c. if the difference in the population means may be zero. d. if a linear relationship exists between two quantitative variables.
Answer: Option (d) is correct.
Step-by-step explanation:
Regression analysis can be used to determine if there is any linear relationship exists between the two quantitative variables.
There are one dependent variable and one or more than one independent variable in a single regression equation.
After running a simple regression, we get to know the relationship between the dependent variable and explanatory variable.
Various statistical software are used for running regression like STATA, E- Views, SPSS, etc.
Regression analysis is used primarily to determine if a linear relationship exists between two quantitative variables. While it can identify relationships between variables, it does not imply causation. It does not establish a relationship between two categorical variables or if the difference in the population means may be zero.
Explanation:Regression analysis is a statistical methodology often used in mathematics and related fields. Primarily, it is used to determine if a linear relationship exists between two quantitative variables.
For example, you could use regression analysis to see if there is a linear relationship between the age of a car (quantitative variable) and the distance it can travel on a tank of gas (another quantitative variable).
It's important to note however that while regression analysis can identify a correlation or relationship between variables, it does not necessarily imply causation, or a cause-and-effect relationship. There could be other related variables causing the effect.
For your options, regression analysis can't be used to determine if a linear relationship exists between two categorical variables or if the difference in the population means may be zero.
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USE INDUCTIVE REASONING TO PREDICT THE NEXT LINE IN THE PATTERN. 11 X 14 = 154 11 X 15 = 165 11 X 16= 176 THE NEXT LINE IS__X__=
Answer:
11 * 17 = 187.
Step-by-step explanation:
11 * 14 = 154
11 * 15 = 165
11 * 16 = 176
So we have the series 154, 165, 176 which has a common difference of 11.
so the next line is 11*17 = 176 + 11 = 187.
A dart is thrown upward with an initial velocity of 66 ft/s at an angle of elevation of 54°. Consider the position of the dart at any time t. Neglect air resistance. (Assume t is in seconds.) Find parametric equations that model the problem situation.
Final answer:
The parametric equations that model the problem situation in this case are x(t) = v0x * t, [tex]y(t) = v0y * t - (1/2) * g * t^2,[/tex], and vy(t) = v0y - g * t.
Explanation:
To find parametric equations that model the problem situation, we need to consider the horizontal and vertical components of the motion separately.
Horizontal Component:
The horizontal velocity remains constant throughout the motion. Therefore, the horizontal position can be given by the equation:
x(t) = v0x * t
where x(t) is the horizontal position at time t and v0x is the initial horizontal velocity.
Vertical Component:
The vertical position depends on the initial velocity, acceleration due to gravity, and time. We can use the following equations:
[tex]y(t) = v0y * t - (1/2) * g * t^2,[/tex]
vy(t) = v0y - g * t
where y(t) is the vertical position at time t, v0y is the initial vertical velocity, g is the acceleration due to gravity (approximately 32 ft/s2), and vy(t) is the vertical velocity at time t.
Determine whether S is a basis for P3. S = {4t - 12,5 +t3,5 +3t,-3t2 +2/3
Answer:
Yes , S is a basis for [tex]P_3[/tex].
Step-by-step explanation:
Given
S=[tex]\left\{4t-12,5+t^3,5+3t,-3t^2+\frac{2}{3}\righ\}[/tex].
We can make a matrix
Let A=[tex]\begin{bmatrix}-12&4&0&0\\5&0&0&1\\5&3&0&0\\\frac{2}{3}&0&-3&0\end{bmatrix}[/tex]
All rows and columns are linearly indepedent and S span [tex]P_3[/tex].Hence, S is a basis of [tex]P_3[/tex]
Linearly independent means any row or any column should not combination of any rows or columns.
Because a subset of V with n elements is a basis if and only if it is linearly independent.
Basis:- If B is a subset of a vector space V over a field F .B is basis of V if satisfied the following conditions:
1.The elements of B are linearly independent.
2.Every element of vector V spanned by the elements of B.
Identify the radius and center.
x^2 + y^2 - 8x - 6y + 16 = 0
The answer is:
Center: (4,3)
Radius: 3 units.
Why?To solve the problem, using the given formula of a circle, we need to find its standard equation form which is equal to:
[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]
Where,
"h" and "k"are the coordinates of the center of the circle and "r" is its radius.
So, we need to complete the square for both variable "x" and "y".
The given equation is:
[tex]x^2+y^2-8x-6y+16=0[/tex]
So, solving we have:
[tex]x^2-8x+y^2-6y=-16[/tex]
[tex](x^2-8x+(\frac{8}{2})^{2} )+(y^2-6y+(\frac{6}{2})^{2})=-16+(\frac{8}{2})^{2}+(\frac{6}{2})^{2}\\\\(x^2-8x+16)+(y^2-6y+9)=-16+16+9\\\\(x^2-4)+(y^2-3)=9[/tex]
Now, we have that:
[tex]h=4\\k=3\\r=\sqrt{9}=3[/tex]
So,
Center: (4,3)
Radius: 3 units.
Have a nice day!
Note: I have attached a picture for better understanding.
By completing the square on the given equation, the center of the circle is found to be (4, 3) and the radius is 3.
This process will transform the equation into a standard form of a circle equation, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius of the circle.
To start, we can rewrite the original equation by adding and subtracting the necessary constants inside the square terms:
x^2 - 8x can become (x - 4)^2 by adding and then subtracting 16 inside the equation.y^2 - 6y can become (y - 3)^2 by adding and then subtracting 9 inside the equation.Therefore, after completing the square, we get (x - 4)^2 + (y - 3)^2 = 1. Hence, the center of the circle is (4, 3) and the radius is 3.
Find the following: F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k Exercise Find the curl and the divergence of the vector field.
[tex]\vec F(x,y,z)=e^{xy}\sin z\,\vec\jmath+y\tan^{-1}\dfrac xz\,\vec k[/tex]
Divergence is easier to compute:
[tex]\mathrm{div}\vec F=\dfrac{\partial(e^{xy}\sin z)}{\partial y}+\dfrac{\partial\left(y\tan^{-1}\frac xz\right)}{\partial z}[/tex]
[tex]\mathrm{div}\vec F=xe^{xy}\sin z-\dfrac{xy}{x^2+z^2}[/tex]
Curl is a bit more tedious. Denote by [tex]D_t[/tex] the differential operator, namely the derivative with respect to the variable [tex]t[/tex]. Then
[tex]\mathrm{curl}\vec F=\begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\D_x&D_y&D_z\\0&e^{xy}\sin z&y\tan^{-1}\frac xz\end{vmatrix}[/tex]
[tex]\mathrm{curl}\vec F=\left(D_y\left[y\tan^{-1}\dfrac xz\right]-D_z\left[e^{xy}\sin z\right]\right)\,\vec\imath-D_x\left[y\tan^{-1}\dfrac xz\right]\,\vec\jmath+D_x\left[e^{xy}\sin z}\right]\,\vec k[/tex]
[tex]\mathrm{curl}\vec F=\left(\tan^{-1}\dfrac xz-e^{xy}\cos z\right)\,\vec\imath-\dfrac{yz}{x^2+z^2}\,\vec\jmath+ye^{xy}\sin z\,\vec k[/tex]
To find the curl and divergence of a given vector field, you first identify the vector's components. The curl is calculated using a determinant and the divergence is obtained by computing a dot product of the gradient operator with the vector field.
Explanation:In order to find the curl and the divergence of the vector field F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k, we first need to identify its components. The components are as follows: e^(xy) sin z, and y tan^−1(x/z).
The Curl of a vector field F in three dimensions is typically denoted as ∇ × F or curl F, where '∇' is the del operator. In Cartesian coordinates, this can be calculated using a determinant that involves the unit vectors î, ĵ, and k, the gradient operator, and the components of F.
The Divergence of a vector field F in three dimensions, typically denoted as ∇ . F or div F, is obtained by computing a dot product of the gradient operator with the vector field. This can also be calculated using Cartesian coordinates.
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Given the differential Equation y'+2y=2e^x ;solve this equation using the integration factor; solve for y to get the general solution.
[tex]y'+2y=2e^x\Longrightarrow y'=2e^x-2y[/tex]
If [tex]f'(x)=g(x)[/tex] then [tex]y=\int{g(x)dx}[/tex]
So we extract,
[tex]y=\int{2e^x-2x}dx[/tex]
Which becomes,
[tex]y=2e^x-x^2+C[/tex]
Hope this helps.
r3t40
A retiree receives $5120 a year interest from $40,000 placed in two bonds, one paying 14% and the other 12%. How much is invested in each bond? 3)
To find out how much is invested in each bond, set up a system of equations and solve for the values of x and y.
Explanation:To find out how much is invested in each bond, we can set up a system of equations.
Let x be the amount invested in the bond paying 14% interest, and y be the amount invested in the bond paying 12% interest.
We know that the retiree receives $5120 in interest each year. So, we have the equation:
x(0.14) + y(0.12) = 5120
Since the total amount invested is $40,000, we can also write the equation:
x + y = 40000
We can now solve this system of equations using any method, such as substitution or elimination, to find the values of x and y.
Find the inverse of the matrices if they Exist. Use the algorithm introduced in the section. [1 0 -2 -3 1 4 2 -3 4]
Answer:
\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right]
Step-by-step explanation:
Given is a matrix 3x3 as
[tex]\left[\begin{array}{ccc}1&0&2\\-3&1&4\\2&-3&4\end{array}\right][/tex]
|A| =2 hence inverse exists.
Cofactors are 16 20 7
6 8 3
2 2 1
Hence inverse =
[tex]\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right][/tex]
Solve one of the following two non-homogeneous differential equations using whatever technique your prefer. Put an "X" through the equation you would not like me to grade. If you do not technique you prefer. Put an "X" through the equation you would not like me to grade. If you do not put an "X" through one of the equations, I will grade whichever problem I prefer to grade. a) y" - 4y + 4y = 6xe^2x b) y" + 9y = 5 cos x - 7 sin x
Answer:
a.[tex]y(x)=c_1e^{2x}+c-2xe^{2x}+x^3e^{2x}[/tex]
b[tex]y(x)=c_1cos 3x+c_2 sin 3x-5 cos x+ 7sin x[/tex]
Step-by-step explanation:
1.[tex]y''-4y'+4y=6x e^{2x}[/tex]
Auxillary equation
[tex]D^2-4D+ 4=0[/tex]
[tex](D-2)(D-2)=0[/tex]
D=2,2
Then complementary solution =[tex] C_1e^{2x}+C_2xe^{2x}[/tex]
Particular solution [tex]=\frac{6 xe^{2x}}{(D-2)^2}[/tex]
D is replace by D+2 then we get
P.I=[tex]\frac{6xe^{2x}}{0}[/tex]
P.I=[tex]\frac{e^{ax}}{D+a} \cdot .V[/tex]
where V is a function of x
P.I=[tex]\frac}x^3e^{2x}[/tex]
By integrating two times
Hence, the general solution
[tex]y(x)=c_1e^{2x}+c-2xe^{2x}+x^3e^{2x}[/tex]
b.y''+9y=5 cos x-7 sin x
Auxillary equation
[tex]D^2+9=0[/tex]
D=[tex]\pm 3i[/tex]
[tex]C.F=c_1 cos 3x+ c_2sin 3x[/tex]
P.I=[tex]\frac{5 cos x-7 sin x}{D^2+9}[/tex]
P.I=[tex]\frac{sin ax}{D^2+bD +C}[/tex]
Then D square is replace by -a square
[tex] D^2 [/tex] is replace by - then we get
P.I=-5 cos x+7 sin x
The general solution
[tex]y(x)=c_1cos 3x+c_2 sin 3x-5 cos x+ 7sin x[/tex]
A national random sample of 654 women aged 20 – 29 years was taken, and each woman's body mass index (BMI) was measured. The sample data had mean BMI ????¯=26.8 and standard deviation ????=7.42 . What is the 95% ???? confidence interval (????,????) for the mean BMI of all young women?
Answer:
The required interval is (26.23 , 27.37)
Step-by-step explanation:
The mean is = 26.8
The standard deviation is = 7.42
n = 654
At 95% confidence interval, the z score is 1.96
To find the desired interval we will calculate as:
[tex]26.8+1.96(\frac{7.42}{\sqrt{654} } )[/tex]
And [tex]26.8-1.96(\frac{7.42}{\sqrt{654} } )[/tex]
[tex]26.8+0.57[/tex] and [tex]26.8-0.57[/tex]
= 27.37 and 26.23
So, the required interval is (26.23 , 27.37)
The 95% confidence interval for the mean BMI of all young women, given a sample of 654 women with a mean BMI of 26.8 and a standard deviation of 7.42 is approximately (26.23, 27.37). This is calculated by finding the standard error and then using that to find the interval around the sample mean which covers 95% of the probable values for the population mean.
Explanation:The question is asking you to calculate the 95% confidence interval for the mean BMI of all young women, given a sample size of 654 women with a mean BMI of 26.8 and a standard deviation of 7.42.
In order to do this, we first need to calculate the standard error, which is the standard deviation divided by the square root of the number of observations. This gives us 7.42 divided by the square root of 654, or about 0.29032.
The 95% confidence interval is calculated by taking the mean and adding and subtracting the standard error multiplied by the relevant z-score for a 95% confidence interval, which is 1.96 (from the standard normal distribution table). So we take 26.8 plus and minus 1.96 times 0.29032, giving us a 95% confidence interval of approximately (26.23, 27.37).
So, we can say with 95% confidence that the mean BMI for all young women is between 26.23 and 27.37.
Learn more about Confidence Interval here:https://brainly.com/question/34700241
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It is given that y is directly proportional to x.
What is the constant of variation?
Answer:
The constant of variation is -4
Step-by-step explanation:
If y is directly proprtional to x, then we write:
[tex]y \propto \: x[/tex]
If we introduce the constant of proportionality or variation k, then we obtain:
[tex]y = kx[/tex]
When x=1, y=-4 (We can choose any ordered pair from the table)
[tex] - 4 = k(1)[/tex]
[tex]k = - 4[/tex]
Therefore the constant of variation is -4.
Answer:
-4
Step-by-step explanation:
PLEASE HELP
Write equations for the vertical and horizontal lines passing through the point (-6, -5)
Answer:
see below
Step-by-step explanation:
A horizontal line has the same y value and is a constant y value
y = -5
A vertical line has the same x value and is a constant x value
x = -6
A vertical line is a line where all of the [tex]x[/tex] values are the same. In this case, [tex]\boxed{x=-6}[/tex], so that is the equation of the line.
A horizontal line is a line where all of the [tex]y[/tex] values are the same. Here, [tex]\boxed{y=-5}[/tex], so that is the second line.
Please help me with this
Answer:
Yes, the triangles are congruent by Hypotenuse angle congruence
Step-by-step explanation:
we know that
The Hypotenuse-Angle Congruence states that If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the triangles are congruent.
In this problem
The hypotenuse and an acute angle of the right triangle in the left are congruent to the hypotenuse and corresponding acute angle of the right triangle in the right
therefore
The triangles are congruent by Hypotenuse angle congruence
Write an equation of the circle with center (9, -3) and radius 8.
Answer:
(x-9)^2 + (y+3)^2 = 8^2
or
(x-9)^2 + (y+3)^2 = 64
Step-by-step explanation:
An equation for a circle can be written as
(x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r is the radius
(x-9)^2 + (y- -3)^2 = 8^2
(x-9)^2 + (y+3)^2 = 8^2
or
(x-9)^2 + (y+3)^2 = 64
A decade-old study found that the proportion of high school seniors who felt that "getting rich" was an important personal goal was 72% . Suppose that we have reason to believe that this proportion has changed, and we wish to carry out a hypothesis test to see if our belief can be supported. State the null hypothesis and the alternative hypothesis that we would use for this test.
Answer: [tex]H_0:p\leq0.72[/tex]
[tex]H_a:p\neq0.72[/tex]
Step-by-step explanation:
Given : A decade-old study found that the proportion of high school seniors who felt that "getting rich" was an important personal goal was 72% .
Let 'p' be the new proportion of high school seniors who felt that "getting rich" was an important personal goal .
Claim : [tex]p\neq0.72[/tex]
We know that the null hypothesis has equal sign.
Therefore , the null hypothesis for the given situation will be opposite to the given claim will be :-
[tex]H_0:p=0.72[/tex]
And the alternative hypothesis must be :-
[tex]H_a:p\neq0.72[/tex]
Hence, the null hypothesis and the alternative hypothesis that we would use for this test :
[tex]H_0:p\leq0.72[/tex]
[tex]H_a:p\neq0.72[/tex]
Final answer:
To test the belief that the proportion of high school seniors valuing wealth has changed, we would use H0: p = 0.72 as the null hypothesis and Ha: p
Explanation:
To conduct a hypothesis test regarding the proportion of high school seniors who believe that "getting rich" is an important goal, we state the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis is the statement that the proportion remains the same, while the alternative hypothesis states that the proportion has changed.
For a proportion of high school seniors who believe that getting rich is an important goal, if the decade-old study showed a proportion of 72%, we would have:
H0: p = 0.72
Ha: p ≠ 0.72
In the hypothetical scenario with a disease prevalence of 9.5% in the general population and finding 7% in a local survey, we would determine the null and alternative hypotheses regarding whether the local proportion is less than the national average as follows:
H0: p ≥ 0.095
Ha: p < 0.095
When conducting a hypothesis test, the null hypothesis typically represents no change or no effect, while the alternative hypothesis represents a change, difference, or effect that we are trying to detect.
Solve the initial value problem (explicit solution). y y' - cot t = 0 y(pi/2) = -1
This ODE is separable:
[tex]yy'-\cot t=0\implies y\,\mathrm dy=\cot t\,\mathrm dt[/tex]
Integrate both sides to get
[tex]\dfrac12y^2=\ln|\sin t|+C[/tex]
Given that [tex]y\left(\frac\pi2\right)=-1[/tex], we get
[tex]\dfrac12(-1)^2=\ln|\sin\dfrac\pi2\right|+C\implies C=\dfrac12[/tex]
Then
[tex]\dfrac12y^2=\ln|\sin t|+\dfrac12[/tex]
[tex]y^2=2\ln|\sin t|+1[/tex]
[tex]y^2=\ln\sin^2t+1[/tex]
[tex]\implies\boxed{y(t)=\pm\sqrt{\ln\sin^2t+1}[/tex]