The original equation and the orthogonal equation are the same, we can conclude that the family of conics with the same focus x^2/a^2+C + y^2/b^2+C = 1 is its own orthogonal family of curves.
To show that the family of conics with the same focus x^2/a^2+C + y^2/b^2+C = 1 is its own orthogonal family of curves, we need to take the derivative of the equation and set it equal to -1/b^2, the slope of the orthogonal line.
First, we take the derivative of the equation with respect to x:
2x/a^2 = -2y/b^2 * dy/dx
Simplifying, we get:
dy/dx = -b^2*x/a^2*y
Now, we set this equal to -1/b^2:
-b^2*x/a^2*y = -1/b^2
Cross-multiplying and simplifying, we get:
x/a^2*y = 1/b^2
Finally, we can rearrange this equation to get:
y = b^2*x/a^2
This equation represents the orthogonal family of curves to the original family of conics. Since the original equation and the orthogonal equation are the same, we can conclude that the family of conics with the same focus x^2/a^2+C + y^2/b^2+C = 1 is its own orthogonal family of curves.
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HELP geometry IXL need help
The value of ST in the given triangle is is √73.
What is Pythagoras' theorem?A right triangle's three sides are related in Euclidean geometry by the Pythagorean theorem, also known as Pythagoras' theorem. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.
Here, we have
Given: In a triangle
TR = 8
SR = 3
ST =?
We apply here Pythagoras theorem and we get
SR² + TR² = ST²
3² + 8² = ST²
9 + 64 = ST²
73 = ST²
ST = √73
Hence, the value of ST is √73.
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Let \( A=\left[\begin{array}{ccc}0 & -2 & -3 \\ -3 & 1 & -3 \\ -3 & 2 & 3\end{array}\right] \). (a) Find the determinant of \( A \). \( \operatorname{det}(A)= \) (b) Find the matrix of cofactors of \(
a)\( -9 \)
b)\( \left[\begin{array}{ccc}+18 & -18 & +9 \\ +9 & -9 & +4 \\ -4 & +4 & -1\end{array}\right] \).
(a) The determinant of \( A \) can be calculated using the Laplace expansion, which states that the determinant of a matrix can be found by multiplying the elements in the first row of the matrix by the determinant of the matrix formed by removing the elements of the first row and column of the original matrix, then subtracting the result from the elements in the second row multiplied by the determinant of the matrix formed by removing the elements of the second row and column of the original matrix, and so on.
Using the Laplace expansion, the determinant of \( A \) can be found as follows:
\( \operatorname{det}(A) = 0 \times \operatorname{det}\left[\begin{array}{cc}1 & -3 \\ 2 & 3\end{array}\right] - (-2) \times \operatorname{det}\left[\begin{array}{cc}-3 & -3 \\ 2 & 3\end{array}\right] + (-3) \times \operatorname{det}\left[\begin{array}{cc}-3 & 1 \\ -3 & 3\end{array}\right] \)
\( \operatorname{det}(A) = 0 \times 18 + 2 \times (-18) + 3 \times 9 \)
\( \operatorname{det}(A) = 0 - 36 + 27 \)
\( \operatorname{det}(A) = -9 \)
Therefore, the determinant of \( A \) is \( -9 \).
(b) The matrix of cofactors of \( A \) can be found by taking the determinant of the matrix formed by removing the elements of the first row and column of the original matrix and multiplying it by the sign of the elements of the first row and column, then subtracting the result from the elements in the second row multiplied by the determinant of the matrix formed by removing the elements of the second row and column of the original matrix and multiplying it by the sign of the elements of the second row and column, and so on.
Using this method, the matrix of cofactors of \( A \) can be found as follows:
\( \left[\begin{array}{ccc}C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33}\end{array}\right] = \left[\begin{array}{ccc}+18 & -18 & +9 \\ +9 & -9 & +4 \\ -4 & +4 & -1\end{array}\right] \)
Therefore, the matrix of cofactors of \( A \) is \( \left[\begin{array}{ccc}+18 & -18 & +9 \\ +9 & -9 & +4 \\ -4 & +4 & -1\end{array}\right] \).
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A cone has a volume of 1959. 36 cubic feet and a radius of 12 feet. What is its height?
The height of the cone is approximately 13.5 feet.
The volume of a cone is given by the formula V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.
In this problem, we are given the volume of the cone, which is 1959.36 cubic feet, and the radius, which is 12 feet. We need to find the height, h.
We can start by plugging in the given values into the formula for volume:
1959.36 = (1/3)π(12)²h
Simplifying, we get:
1959.36 = 144πh
To solve for h, we need to isolate it on one side of the equation. We can do this by dividing both sides by 144π:
h = 1959.36 / (144π)
Using a calculator to evaluate this expression, we get:
h ≈ 13.5
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brainly wouldn't let me type the question for some reason, so i had to attach a pic
The correct answer is for the equation is C, "minimizes the sum of the squared distances from the actual y-value to the predicted y-value," as explained below.
What is a least-squares regression line?The least-squares regression line is a line that best fits the given data by minimizing the sum of the squared distances between the predicted y-value (ŷ) and the actual y-value (y) for each data point. In other words, the line is chosen so that the sum of the squares of the vertical distances between the data points and the line is as small as possible.
In this case, the equation provided represents the least-squares regression line for the given data, which suggests that for every unit increase in x, y increases by 0.5 units.
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Mark drove to the mall on a busy shopping day. He got to the mall parking lot at 6:46, but he didn't find a parking spot until 19 minutes later. What time was it when Mark finally parked the car?
Answer:
Step-by-step explanation:
1000
Simona has 8 3/4 cups of milk in a container. She uses 112 cups of the milk to make a cake. She uses 14 cup of the milk to make frosting for the cake. How many cups of milk does Simona have left?
Simona is left with 7 cups of milk.
What is Mixed fraction?
An example of a mixed fraction is one that consists of both a whole integer and a fractional component. A mixed fraction is, for instance, 3 1/7. It's also known as a jumbled number.
Conversion procedures for a mixed fraction to a simple fraction
Step 1: Multiplying the denominator of the mixed fraction by the whole number component is the first step.
Step 2: To the end result achieved in Step 1, add the numerator.
Step 3: Format the improper fraction in numerator/denominator form using the sum from step 2 as the denominator.
Simona has 8 3/4 cups of milk .
She uses 1 1/2 cups of milk to make the cake and 1/4 cup of the milk to make frosting for the cake.
So, cups of milk left = total cups of milk - cups of milk used for cake
= 8 3/4 - 1 1/2 - 1/4
= 35/4 - 3/2 - 1/4
= (35 - 6 - 1)/4
= 28/4
= 7 cups of milk.
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Let \( A=\left[\begin{array}{ll}3 & 9 \\ 8 & 6\end{array}\right] \) and \( B=\left[\begin{array}{ll}5 & 0 \\ 2 & 4\end{array}\right] \). Find the matrix \( C \) of the linear transformation \( T(x)=B(
The matrix C of the linear transformation T(x) = B(A(x)) is:
[tex]\(C=\left[\begin{array}{ll}15 & 45 \\ 38 & 42\end{array}\right]\)[/tex]
What is the matrix of the linear transformation?We can find the matrix C of the linear transformation T(x) = B(A(x)) by multiplying the matrices A and B.
Recall that the matrix product of two matrices
[tex]\(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\)[/tex] and [tex]\(B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right]\)[/tex]
is given by
[tex]\(C=AB=\left[\begin{array}{ll}a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22}\end{array}\right]\)[/tex].
So, for
[tex]\(A=\left[\begin{array}{ll}3 & 9 \\ 8 & 6\end{array}\right]\)[/tex] and [tex]\(B=\left[\begin{array}{ll}5 & 0 \\ 2 & 4\end{array}\right]\)[/tex],
we have:
[tex]\(C=BA=\left[\begin{array}{ll}5 & 0 \\ 2 & 4\end{array}\right]\left[\begin{array}{ll}3 & 9 \\ 8 & 6\end{array}\right]=\left[\begin{array}{ll}5(3)+0(8) & 5(9)+0(6) \\ 2(3)+4(8) & 2(9)+4(6)\end{array}\right]=\left[\begin{array}{ll}15 & 45 \\ 38 & 42\end{array}\right]\)[/tex]
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Ryan, a consumer electronics salesperson, earns a base salary of $ 1300 per month and a commission of 7% on the amount of sales he makes. One month Ryan received a $1622 paycheck. Find the amount of his sales for the month.
Ryan, a consumer electronics salesperson, earns a base wage of $ 1300 per month and a commission of 7% on the amount of sales he makes. One month Ryan received a $1622 paycheck, which made his sales for the month $4600.
To find the amount of Ryan's sales for the month, we can use the equation:
Total paycheck = Base salary + (Commission rate × Sales)
We can plug in the given values and solve for the amount of sales:
$1622 = $1300 + (0.07 × Sales)
$322 = 0.07 × Sales
Sales = $322 ÷ 0.07
Sales = $4600
Therefore, Ryan's sales for the month were $4600.
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What is the area of this figure?
A bee population of 3000 increases by 40% every year.
Answer:
That is the answer
Step-by-step explanation:
Math question 10 help
Answer:
- ∞ < x < 2
Option 3
Step-by-step explanation:
The given function is
[tex]f(x) =(x - 2)^2 + 3[/tex]
This is the equation of a parabola in vertex form
The general vertex form equation is f(x) =a(x -h)² + k
where (h, k) is the vertex which can be either a maximum or minimum and a is a constant
If a is positive, the vertex is at a minimum and the parabola opens downward
If a is negative, the vertex is at a maximum and the parabola opens upward
Here a = 1 is positive
The vertex is (2, 3) and is a minimum
So for all values of x < 2, the parabola is decreasing and for all values of x > 2 the parabola is increasing (see figure)
x = 2 is the turning point for the parabola
therefore the values of x for which the function is decreasing is
- ∞ < x < 2
Tas-fan is eating at a restaurant. His total bill comes to $15.05. If Tas-fan decides to leave a tip that is approximately 20% of the total bill, how much should he leave for the tip?
Modular Arithmetic: Please solve the following questions. (1) 42mod11
The answer to the question "(1) 42mod11" is 9.
The answer to the question "Modular Arithmetic: Please solve the following questions. (1) 42mod11" is:
The concept of "Modular Arithmetic" is a system of arithmetic that involves only integers. In modular arithmetic, numbers "wrap around" after they reach a certain value called the modulus.
To solve the given question, we can use the following steps:
Step 1: Find the modulus of the given number. In this case, the modulus is 11.
Step 2: Divide the given number by the modulus. In this case, 42/11 = 3 with a remainder of 9.
Step 3: The remainder is the answer to the question. So, 42mod11 = 9.
Therefore, the answer to the question "(1) 42mod11" is 9.
I hope this answer helps you! If you have any further questions, feel free to ask.
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Here is another triangle similar to DEF.
•What is the scale factor from triangle DEF to triangle D”E”F”?
• What are the coordinates of F”?
•What are cos(D”), sin(D”), and tan(D”)
(Triangle DEF has coordinates D(0,0) E(12,0) F(12,5)
And pictured is triangle D”E”F”
The scale factor from triangle DEF to triangle D”E”F” is the ratio of the corresponding sides in the two triangles.
The coordinates of F” can be calculated using the scale factor.
The value of cos(D”) would be equal to 0.707. Similarly, the value of sin(D”) would be 0.707 and the value of tan(D”) would be 1.
What is a scale factor?Scale factor is a number which is used to describe the relationship between two objects or two measurements. It is used to compare two objects of different sizes in order to calculate the ratio of their size. It is also used to express a proportional relationship between two different measurements, such as their lengths, widths, heights or angles.
The corresponding sides are DE and D”E”, and EF and E”F”. We can calculate the ratio of the two sides by dividing the lengths of the two corresponding sides. For example, the ratio of DE to D”E” would be 12/16. We can do this for all three corresponding sides, and then take the average of the three ratios to get the scale factor from triangle DEF to triangle D”E”F”.
Since we know that the coordinates of F are (12,5), we can multiply the x-coordinate (12) by the scale factor and the y-coordinate (5) by the scale factor to get the coordinates of F”.
The angle D” can be used to calculate the values of cos(D”), sin(D”), and tan(D”). Since we know the angle D”, we can use the trigonometric functions to calculate the values of the trigonometric functions for that angle. For example, if the angle D” is 45 degrees, then the value of cos(D”) would be equal to 0.707. Similarly, the value of sin(D”) would be 0.707 and the value of tan(D”) would be 1.
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Solve the following system of equations. Write your answer in
the form of an ordered pair (x,y).
{10x+5y20x−10y==−60−80
Enter your answer as an ordered pair (x,y).
The answer is an ordered pair (x,y)=(-7,2).
To solve the system of equations, we can use the elimination method to eliminate one of the variables and solve for the other.
First, let's multiply the first equation by 2 so that we can eliminate the y variable:
{20x+10y=−120
Next, let's subtract the second equation from the first:
{20x+10y−(20x−10y)==−120−(−80)
Simplifying the equation gives us:
{20y=40
Dividing both sides by 20 gives us:
{y=2
Now, let's plug this value of y back into the first equation to solve for x:
{10x+5(2)==−60
Simplifying the equation gives us:
{10x+10=−60
Subtracting 10 from both sides gives us:
{10x=−70
Dividing both sides by 10 gives us:
{x=−7
So, the solution to the system of equations is the ordered pair (x,y)=(−7,2).
Therefore, the answer is an ordered pair (x,y)=(-7,2).
I hope this helps! Let me know if you have any further questions.
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I Need help to graph these and need explanation
The points are (0,0) and (0,-1/2)
What is a trigonometric function?
The right-angled triangle's angle and the ratio of its two side lengths are related by the trigonometric functions, which are actual functions. They are extensively employed in all fields of geometry-related study, including geodesy, solid mechanics, celestial mechanics, and many others.
Here, we have
Given: y = 4cot 4x
y = -1/2(cot(2x+π/4))
We have to draw the graph of a given function.
y = 4cot 4x
y = -1/2(cot(2x+π/4))
We put the different values of x and get the value of y.
y = 4cot 4x
x = 0
y = 0
y = -1/2(cot(2x+π/4))
when x= 0
y = -1/2
Hence, the points are (0,0) and (0,-1/2)
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What is a number between 1-100
Whoever is closest gets brainliest
Answer:
56
Step-by-step explanation:
Twenty Six, perhaps?
how do i turn 7/4 into a mixer number
Answer:
1 3/4
Step-by-step explanation:
To turn 7/4 into a mixed number, you have to look at the numerator and the denominator. 4 can go into 7 once (with 3 left over), so it would be 1 and 3/4
Can someone PLEASE help me with this three part question? It’s due today!! I need help ASAP
1. The Formula used SA = 2w (l +h) + lh
2. The box is 0.37125 inch deep.
3. The SA of box with lid is 294.7972 inch².
What is Surface Area?The area is the territory covered by a shape or figure, whereas the perimeter is the distance covered by the shape's outside boundary. The unit of area is the square unit or unit², while the unit of perimeter is the unit.
Given:
length = 13.2 inch
Height = 10.5 inch
And, Surface Area used = 295.02 inch²
Using the Formula
SA = 2lw + 2wh + lh
SA = 2w (l +h) + lh
295. 02 = 2w ( 13.2 + 10.5) + (13.2)(10.5)
147.51 = w x 24 + 138.6
8.91 = 24w
w = 0.37125 inch
SA of box with lid
= 2( lw+ wh + lh)
= 2( 13.2 x 10.5 + 10.5 x 0.37125 + 0.37125 x 13.2)
= 2(138.6 + 3.8981 + 4.9005)
= 2 x 147.3986
= 294.7972 inch²
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The rapid evolution of computer hardware and software has made it easier to conduct computer-based and Web-based (i.e. electronic) surveys. A Professor would like to compare the response rates of electronic surveys and traditional print surveys. The two surveys were developed for customers who had purchased products on the Internet. Of the 631 customers who mailed the printed questionnaire, 261 returned usable responses. Of the 414 customers who sent Web-based (i.e. electronic) questionnaire, 155 returned
usable responses. Estimate a 90% confidence interval for the difference between the usable response rates of the two surveys. Interpret the result.
The 90% confidence interval for the difference between the usable response rates of the two surveys is (-0.0124, 0.0912).
The difference between the usable response rates of the two surveys can be estimated using the formula for the difference between two proportions:
d = p1 - p2
Where p1 is the usable response rate for the printed survey, and p2 is the usable response rate for the electronic survey. To calculate these response rates, we can use the formula:
p = x/n
Where x is the number of usable responses, and n is the total number of customers who were sent the survey. For the printed survey, we have:
p1 = 261/631 = 0.4138
And for the electronic survey, we have:
p2 = 155/414 = 0.3744
So the difference between the two response rates is:
d = 0.4138 - 0.3744 = 0.0394
To estimate a 90% confidence interval for this difference, we can use the formula:
CI = d ± z*√[(p1(1-p1)/n1) + (p2(1-p2)/n2)]
Where z is the critical value for a 90% confidence level, which is 1.645. Plugging in the values we have:
CI = 0.0394 ± 1.645*√[(0.4138(1-0.4138)/631) + (0.3744(1-0.3744)/414)]
CI = 0.0394 ± 1.645*√[0.000407 + 0.000581]
CI = 0.0394 ± 1.645*0.0315
CI = 0.0394 ± 0.0518
CI = (-0.0124, 0.0912)
So the 90% confidence interval for the difference between the usable response rates of the two surveys is (-0.0124, 0.0912). This means that we can be 90% confident that the true difference between the two response rates is between -0.0124 and 0.0912.
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Seven Subjects S1, S2, S3, ...,S7 Are To Be Scheduled In An Examination. The Following Pairs Of Subjects Have Common Students: 2 1 a. {S1, S2}, {S1, S3},{S1, S4},{Sı, S7} b. {S2, S3}, {S2, Sa}, {S2, S5}, {S2, S7}, c. {S3, S4}, {S3, S6},{S3, S7},{S4, S5}, {S4, Sc}, d. {S5,S6},{S5, St}, And {S6, S7} How Can The Examination Be Scheduled So That No Student Has Two examination at same day?
To schedule the examination so that no student has two examinations on the same day, we can use graph coloring. Graph coloring is the process of assigning colors to the vertices of a graph in such a way that no two adjacent vertices have the same color.
1. First, we can create a graph with the seven subjects as vertices and the pairs of subjects with common students as edges.
2. Next, we can assign a color to each vertex, starting with S1 and moving clockwise around the graph. We can use the smallest available color for each vertex, making sure that no two adjacent vertices have the same color.
3. Once we have assigned a color to each vertex, we can use the colors to schedule the examinations. Each color represents a different day, and all of the subjects with the same color can be scheduled on the same day.
The resulting schedule would look something like this:
Day 1: S1, S5
Day 2: S2, S4
Day 3: S3, S6
Day 4: S7
This schedule ensures that no student has two examinations on the same day, since no two adjacent vertices in the graph have the same color.
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Which description matches the translation of the equation: y = |x - 5| + 2
Moves 5 units left and 2 units up
Moves 2 units right and 5 units down
Moves 5 units right and 2 units up
Moves 2 units left and 5 units right
Answer:
The description that matches the translation of the equation y = |x - 5| + 2 is "Moves 5 units to the right and 2 units up".
This is because the equation involves the absolute value of the quantity (x - 5), which means that the expression inside the absolute value bars will always be positive or zero. This means that the graph of the equation will be a "V" shape, centered at x = 5, and opening upwards. The "+2" outside the absolute value bars indicates that the entire graph will be shifted upward by 2 units.
Therefore, the graph of the equation y = |x - 5| + 2 will be obtained by taking the graph of y = |x|, shifting it 5 units to the right to obtain y = |x - 5|, and then shifting it 2 units up to obtain y = |x - 5| + 2.
Step-by-step explanation:
pa brainliest pls
What is an angle that is supplementary to
An angle which is supplementary to ∠AGB is ∠BGD
What are supplementary angles?Two angles that sum up to 180 degrees are referred to as supplementary angles. In other words, the sum of the measurements of two additional angles equals 180 degrees.
Angles A and B, for instance, are supplementary if angle A is 70 degrees and angle B is 110 degrees. This is because 70 + 110 = 180.
Given that,
The arrangement which has multiple rays,
And it is known that,
The sum of angles of supplementary angles are 180°
∠AGB & ∠BGD has the sum of 180°,
So it can be state that ∠BGD is supplementary to the angle ∠AGB
Therefore, the supplementary angle is ∠BGD
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What is the measurement of the missing angle
The measurement of the missing angle is 33 degrees
How to determine the measurement of the missing angleFrom the question, we have the following parameters that can be used in our computation:
The triangle
On the triangle, we have the following equation
tan(?) = 9.09/14
Evaluate
tan(?) = 0.6493
Take the arc tan of both sides
? = 33 degrees
Hence, the angle is 33 degrees
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thank you if you help with this
The ratio of amount of flour to amount of water is 8 : 3. 5 cups of flour require 15/8 cups of water
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.
From the recipe, 2 cups of flour require 3/4 cups of water
a) The ratio of amount of flour to amount of water = 2 cups of flour / (3/4) cups of water = 8/3 = 8 : 3
b) 2 cups of flour require 3/4 cups of water
5 cups of flour = 5 cups of flour * 3/4 cups of water per 2 cups of flour = 5 * (3/4)÷2 = 5 * 3/8 = 15/8 cups of water
5 cups of flour require 15/8 cups of water
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Exercise 2.1440sing the inverse of the matrix, find the solution to the systems: (a)[2141][xy]=[12](b)[2141][xy]=[20]Now give the solution in terms ofaandbto[2141][xy]=[ab]
The solution in terms of a and b is x=(-a/6)+(b/6) and y=(2a/3)-(b/3).
To find the solution to the systems using the inverse of the matrix, we can use the formula:
[xy]=[A^-1][b]
where [A] is the coefficient matrix, [b] is the constant matrix, and [A^-1] is the inverse of the coefficient matrix.
For system (a), we have:
[A]=[2141]
[b]=[12]
To find the inverse of [A], we can use the formula:
[A^-1]=1/(ad-bc)[d -b-c a]
where a=2, b=1, c=4, and d=1.
So, [A^-1]=1/(-6)[1 -1-4 2]=[-1/6 1/62/3 -1/3]
Now, we can find the solution by multiplying [A^-1] and [b]:
[xy]=[-1/6 1/62/3 -1/3][12]=[-1/6+1/6 2/3-1/3]=[-1/3]
So, the solution to system (a) is x=-1/3 and y=1/3.
For system (b), we have:
[A]=[2141]
[b]=[20]
We can use the same inverse of [A] that we found for system (a) and multiply it by [b] to find the solution:
[xy]=[-1/6 1/62/3 -1/3][20]=[-2/6+0 4/3-0]=[-1/32]
So, the solution to system (b) is x=-1/3 and y=2.
For the system [2141][xy]=[ab], we can use the same inverse of [A] and multiply it by [ab]:
[xy]=[-1/6 1/62/3 -1/3][ab]=[-a/6+b/6 2a/3-b/3]
So, the solution in terms of a and b is x=(-a/6)+(b/6) and y=(2a/3)-(b/3).
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Problem 5. Find the exact values of all six trigonometric functions of 660°. Problem 6. Verify the identity: sin x + cos x cotx = CSC X.
To find the exact values of all six trigonometric functions of 660°, we need to first convert the angle to one that falls within the range of 0° to 360°. We can do this by subtracting 360° from the given angle until we get a value within the desired range.
660° - 360° = 300°
Now we can find the exact values of the trigonometric functions of 300° using the unit circle:
sin 300° = -√3/2
cos 300° = 1/2
tan 300° = -√3
csc 300° = -2/√3
sec 300° = 2
cot 300° = -1/√3
Therefore, the exact values of all six trigonometric functions of 660° are:
sin 660° = -√3/2
cos 660° = 1/2
tan 660° = -√3
csc 660° = -2/√3
sec 660° = 2
cot 660° = -1/√3
Problem 6: Verify the identity: sin x + cos x cotx = CSC X.
To verify this identity, we can start by simplifying the left-hand side of the equation:
sin x + cos x cotx
= sin x + cos x (cos x / sin x)
= sin x + (cos^2 x / sin x)
= (sin^2 x + cos^2 x) / sin x
Since sin^2 x + cos^2 x = 1, we can simplify further:
= 1 / sin x
= csc x
Therefore, the left-hand side of the equation simplifies to the same value as the right-hand side, verifying the identity.
sin x + cos x cotx = csc x
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All the real zeros of the given polynomial are integers. Find the zeros. P(x) = x^(3) - 9x^(2) + 20x - 12
The real zeros of the given polynomial P(x) = x³ - 9x² + 20x - 12 are 3 and 2. The real zeros of the given polynomial P(x) = x³ - 9x² + 20x - 12 can be found by factoring the polynomial and setting each factor equal to zero.
Step 1: Factor the polynomial
P(x) = x³ - 9x² + 20x - 12
= (x - 3)(x - 2)(x - 2)
Step 2: Set each factor equal to zero and solve for x
x - 3 = 0 => x = 3
x - 2 = 0 => x = 2
x - 2 = 0 => x = 2
Step 3: The real zeros of the polynomial are the values of x that make each factor equal to zero.
So, the real zeros of the polynomial are x = 3 and x = 2.
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Find the length
x
to the nearest whole number.
x≈
(Do not round until the final answer. Then round to the nearest whole number.)
It is not possible to solve for x and find the length to the nearest whole number. Please provide the complete equation or additional information to help solve for x.
To find the length x to the nearest whole number, we need to use the given information and follow these steps:
1. Start with the given equation: x≈
2. Solve for x by simplifying the equation.
3. Once you have found the value of x, round it to the nearest whole number.
Without additional information, it is not possible to solve for x and find the length to the nearest whole number. Please provide the complete equation or additional information to help solve for x.
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Let S1, S2 ⊂ V be two sets of vectors which each span V , show
that S1 ∪ S2 span V
If S1 and S2 are two sets of vectors that each span V, then S1 ∪ S2 also spans V.
Let S1 and S2 be two sets of vectors that each span V. We need to show that S1 ∪ S2 also spans V.
Let v be any vector in V. Since S1 spans V, there exist vectors u1, u2,..., un in S1 and scalars a1, a2,..., an such that:
v = a1u1 + a2u2 + ... + anun
Similarly, since S2 spans V, there exist vectors w1, w2,..., wm in S2 and scalars b1, b2,..., bm such that:
v = b1w1 + b2w2 + ... + bmwm
Now, since S1 ∪ S2 contains all the vectors in S1 and S2, we can write v as a linear combination of the vectors in S1 ∪ S2:
v = a1u1 + a2u2 + ... + anun + b1w1 + b2w2 + ... + bmwm
Therefore, S1 ∪ S2 spans V.
In conclusion, if S1 and S2 are two sets of vectors that each span V, then S1 ∪ S2 also spans V.
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