real zero, Including any repeated zeroes? Choose your answ f(x)=x^(6)+8x^(5)+9x^(4)-2x^(2)-7x+4
The real zeroes of the function f(x)=x^(6)+8x^(5)+9x^(4)-2x^(2)-7x+4 are -4, -1, and 1.
The real zeroes of the function f(x)=x^(6)+8x^(5)+9x^(4)-2x^(2)-7x+4 can be found by using the Rational Zero Theorem and synthetic division.
The Rational Zero Theorem states that the possible rational zeroes of a polynomial are the factors of the constant term divided by the factors of the leading coefficient. In this case, the constant term is 4 and the leading coefficient is 1, so the possible rational zeroes are ±1, ±2, and ±4.
We can use synthetic division to test these possible zeroes and find the actual zeroes of the function. Synthetic division is a method of dividing a polynomial by a linear factor of the form x-a. The result of synthetic division is a quotient and a remainder, and if the remainder is 0, then x-a is a factor of the polynomial and a is a zero of the function.
Using synthetic division, we find that the real zeroes of the function are -4, -1, and 1. These are the values of x that make the function equal to 0. There are no repeated zeroes in this case.
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Gemma makes a conjecture that in the sum of an eye and the jersey, and is selfish, always an even integer. Which choice is the best proof of her conjecture
As per the concept of integer, the best proof of her conjecture is Let 2n + 1 be an odd number; (2n + 1) + (2n + 1) = 4n + 2. Because 4(n + 2) is divisible by 2, the sum of 2n + 1 and itself is even. (option b)
Now, let's look at the given choices to find the best proof for Gemma's conjecture.
Choice B) Let 2n + 1 be an odd number; (2n + 1) + (2n + 1) = 4n + 2. Because 4(n + 2) is divisible by 2, the sum of 2n + 1 and itself is even.
This choice provides a proof by using algebraic equations. It begins by defining an odd integer as 2n + 1, where n is any integer. Then, it uses algebraic manipulation to show that the sum of this odd integer and itself results in an even integer. Specifically, (2n + 1) + (2n + 1) simplifies to 4n + 2, which is equal to 2(2n + 1). This proves that the sum of an odd integer and itself is always even.
This choice is a tautology, which means that it's always true, but it doesn't provide any evidence or proof to support Gemma's conjecture.
The best proof for Gemma's conjecture is choice B.
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Complete Question:
Gemma makes a conjecture that the sum of an odd integer and itself is always an even integer. Which choice is the best proof of her conjecture?
A) Look at these different examples: 7 + 7 = 14, 13 + 13 = 26, 23 + 23 = 46. So the sum of an odd number and itself must be even
B) Let 2n + 1 be an odd number; (2n + 1) + (2n + 1) = 4n + 2. Because 4(n + 2) is divisible by 2, the sum of 2n + 1 and itself is even.
C) Let n represent an odd number, and let n + n be an even number. Therefore, n + n = n + n, which shows that the sum of an odd number and itself is even.
D) Every time you add an odd number and itself, the sum is an even number.
Concepts to make sure are (re)-discussed: Isolating the radical Checking for extraneous solutions Solving rational exponents in equations- more than one solution Sample problems: 1. V4x + 1 - Vx+ 10 = 2. V2x+30 = (x+3)
To solve the given equations, we need to follow these steps:
1. Isolate the radical on one side of the equation.
2. Square both sides of the equation to eliminate the radical.
3. Solve the resulting equation for the variable.
4. Check for extraneous solutions by plugging the solution back into the original equation.
5. If there are rational exponents, use the same steps but raise both sides of the equation to the reciprocal of the exponent.
Let's apply these steps to the sample problems:
1. V4x + 1 - Vx+ 10 = 2
First, isolate the radical on one side:
V4x + 1 = Vx+ 10 + 2
V4x + 1 = Vx+ 12
Next, square both sides to eliminate the radical:
(4x + 1) = (x+ 12)^2
4x + 1 = x^2 + 24x + 144
Solve for x:
0 = x^2 + 20x + 143
Using the quadratic formula:
x = (-20 ± √(20^2 - 4(1)(143)))/(2(1))
x = (-20 ± √(400 - 572))/2
x = (-20 ± √(-172))/2
x = (-20 ± √172i)/2
There are no real solutions for this equation.
2. V2x+30 = (x+3)
Isolate the radical:
V2x+30 = x+3
Square both sides:
2x + 30 = (x+3)^2
2x + 30 = x^2 + 6x + 9
Solve for x:
0 = x^2 + 4x - 21
Using the quadratic formula:
x = (-4 ± √(4^2 - 4(1)(-21)))/(2(1))
x = (-4 ± √(16 + 84))/2
x = (-4 ± √100)/2
x = (-4 ± 10)/2
x = 3 or x = -7
Check for extraneous solutions by plugging the solutions back into the original equation:
V2(3)+30 = (3+3)
V36 = 6
6 = 6 (True)
V2(-7)+30 = (-7+3)
V16 = -4
4 = -4 (False)
Therefore, the only solution is x = 3.
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(3 points) Problem 5: Determine the value(s) ofasuch that [1a], [aa+2] are linearly independent.
To determine the value(s) of a such that [1 a], [a a+2] are linearly independent, we need to find the values of a that make the determinant of the matrix non-zero. The determinant of a 2x2 matrix is given by:
|1 a|
|a a+2| = (1)(a+2) - (a)(a) = a + 2 - a^2
To make the determinant non-zero, we need to solve the equation:
a + 2 - a^2 ≠ 0
Rearranging the equation, we get:
a^2 - a - 2 ≠ 0
Factoring the equation, we get:
(a - 2)(a + 1) ≠ 0
Therefore, the values of a that make the determinant non-zero are a ≠ 2 and a ≠ -1. These are the values of a that make the vectors [1 a], [a a+2] linearly independent.
So, the value(s) of a such that [1 a], [a a+2] are linearly independent are a ≠ 2 and a ≠ -1.
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What should be reason 8 in the following proof?
Answer: I think putting Prop of ║lines might work. I haven't done this in a long time, so I'm most likely wrong, but that might help. I'm so sorry.
For A-C, choose Yes or No to indicate whether or not each expression has a value greater than 6.
29
√9
B. 4+ √3
A.
C. 6.4 -
D. 27
18
√27
OYes ONO
OYes No
OYes No
OYes No
Please refer to this output. Method DE H-Value P-value
Not adjusted for ties 2 0.01 0.994
Adjusted for ties 2 0.01 0.994 Ten cities are selected, and the number of daily passenger trips (in thousands) for subways and commuter rail service is obtained. At alpha equal to 0.05, the researcher wants to check the strength of relationship between the variables. Assume that data is normally distributed For the next questions, please refer to this problem.
What does the test for significant relationship tell you? 1 point a. There is a significant relationship between the number of daily passenger trips (in thousands) for subways and commuter rail service? b. The simple linear regression equation conforms to the strength of relationship defined. c. It is good to proceed with defining the simple linear regression equation, d. There is no significant relationship between the number of daily passenger trips (in thousands) for subways and commuter rail service?
There is a significant relationship between the variables (H-value = 2 and P-value = 0.994)
The test for significant relationship is used to determine whether there is a statistically significant relationship between two or more variables. In this case, the test can be used to determine whether there is a statistically significant relationship between the number of daily passenger trips (in thousands) for subways and commuter rail service. Based on the H-value and P-value, the result of this test indicates that there is a significant relationship between the variables (H-value = 2 and P-value = 0.994). Therefore, option (a) is correct and option (d) is incorrect.
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The yearly income of a family is Rs. 500000. The ratio of the expenditure and saving of the family is 4 : 1. Find the amount of expenditure and saving.
Answer:
Let's assume that the amount of saving is x.
According to the problem, the ratio of expenditure to saving is 4:1, so the amount of expenditure can be expressed as 4x.
The total income of the family is Rs. 500000, and it can be expressed as the sum of expenditure and saving:
Expenditure + Saving = 500000
Substituting the values of expenditure and saving, we get:
4x + x = 500000
Simplifying this equation, we get:
5x = 500000
Dividing both sides by 5, we get:
x = 100000
Therefore, the amount of saving is Rs. 100000, and the amount of expenditure is 4 times this value, which is Rs. 400000.
calculate the areas of the rectangles having the following measurements
Perimeter(p)=220cm and breadth(b)=45cm
Answer:
it is 2,475
Step-by-step explanation:
Aisling runs 2km on Monday and three times that many on Tuesday. If she wants to run 20km this week. How many km does she need to run
Answer:12km
Step-by-step explanation:
She ran 2km on Monday, and three times as much on Tuesday so we do 2 x 3 which equals 6. So 6km, plus the original 2km is 8km (6+2) is she wants to run 20km this week, then she needs to subtract what shes already ran to find out how many more km she needs to run. 20-8=12 so 12km.
What is the measure of "AC"?
Enter your answer in the box.
Will give Brainiest if right. and to help other people thx.
Answer: 21 degrees
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In his motorboat, Bill Ruhberg travels upstream at top speed to his favorite fishing spot, a distance of 90 mi, in 3 hr. Returning, he finds that the trip downstream, still at top speed, takes only 2.5 hr. Find the rate of Bill's boat and the speed of the current. Let x = the rate of the boat in still water and y = the rate of the current.
The rate of Bill's boat in still water is 33 mph and the speed of the current is 3 mph.
To find the rate of Bill's boat and the speed of the current, we can use the distance formula, which states that distance = rate × time. Since we know the distance and time for both the upstream and downstream trips, we can set up two equations and solve for the rate of the boat and the speed of the current.
Let x = the rate of the boat in still water and y = the rate of the current.
For the upstream trip:
90 = (x - y) × 3
For the downstream trip:
90 = (x + y) × 2.5
Simplifying the equations gives us:
90 = 3x - 3y
90 = 2.5x + 2.5y
Multiplying the first equation by 2.5 and the second equation by 3 gives us:
225 = 7.5x - 7.5y
270 = 7.5x + 7.5y
Adding the two equations together eliminates the y variable:
495 = 15x
Solving for x gives us:
x = 33
Substituting x back into the first equation gives us:
90 = (33 - y) × 3
90 = 99 - 3y
3y = 9
y = 3
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Graph the solution to the following system of inequalities.
2x +7vs - 14
-3x +5y> 5
Then give the coordinates of one point in the solution set.
Point in the solution set: (П.П)
A solution to the given system of linear inequalities is (-6, -1).
How to graph the solution to this system of inequalities?In order to to graph the solution to the given system of inequalities on a coordinate plane, we would use an online graphing calculator to plot the given system of inequalities and then take note of the point of intersection;
2x + 7y ≤ -14 .....equation 1.
-3x + 5y > 5 .....equation 2.
Based on the graph (see attachment), we can logically deduce that the solution to the given system of inequalities is the shaded region below the solid and dashed line, and the point of intersection of the lines on the graph representing each, which is given by the ordered pair (-6, -1).
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Question 5 If \( A \) and \( B \) are \( 3 \times 3 \) matrices satisfying \( \operatorname{det} A=12 \) and \( \operatorname{det} B=3 \), then \( \operatorname{det}\left(2 A^{-1} B^{2}\right)= \) A 1
\( \operatorname{det}\left(2 A^{-1} B^{2}\right) = \frac{18}{12} = \frac{3}{2} \)Explanation: We are given that \( \operatorname{det} A=12 \) and \( \operatorname{det} B=3 \). We need to find the determinant of \( 2 A^{-1} B^{2} \). We can use the properties of determinants to simplify the expression. Recall that \( \operatorname{det}(cA) = c^n \operatorname{det}(A) \) for an \( n \times n \) matrix \( A \) and a scalar \( c \), and that \( \operatorname{det}(AB) = \operatorname{det}(A)\operatorname{det}(B) \). Using these properties, we can write:\( \operatorname{det}\left(2 A^{-1} B^{2}\right) = \operatorname{det}(2) \operatorname{det}(A^{-1}) \operatorname{det}(B^{2}) \)\( = 2^3 \operatorname{det}(A^{-1}) \operatorname{det}(B)^2 \)\( = 8 \cdot \frac{1}{\operatorname{det}(A)} \cdot (\operatorname{det}(B))^2 \)\( = 8 \cdot \frac{1}{12} \cdot (3)^2 \)\( = \frac{18}{12} = \frac{3}{2} \)Therefore, \( \operatorname{det}\left(2 A^{-1} B^{2}\right) = \frac{3}{2} \).
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Fifth order linear ODE with one real root r and two complex
roots a ± bi and c ± di. From the given information, determine its
general solution.
The general solution of a fifth order linear ODE with one real root r and two complex roots a ± bi and c ± di can be determined by using the fact that the general solution of a linear ODE is the sum of the solutions corresponding to each of the roots.
For the real root r, the solution is given by:
y1 = C1 * e^(r*x)
For the complex roots a ± bi, the solution is given by:
y2 = C2 * e^(a*x) * cos(b*x) + C3 * e^(a*x) * sin(b*x)
For the complex roots c ± di, the solution is given by:
y3 = C4 * e^(c*x) * cos(d*x) + C5 * e^(c*x) * sin(d*x)
The general solution of the fifth order linear ODE is the sum of these solutions:
y = y1 + y2 + y3
= C1 * e^(r*x) + C2 * e^(a*x) * cos(b*x) + C3 * e^(a*x) * sin(b*x) + C4 * e^(c*x) * cos(d*x) + C5 * e^(c*x) * sin(d*x)
This is the general solution of the fifth order linear ODE with one real root r and two complex roots a ± bi and c ± di.
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I need help with these maths questions
Answer:
see explanation
Step-by-step explanation:
(a)
• the opposite sides of a rectangle are congruent
then
4x + 1 = 2x + 12
(b)
solving
4x + 1 = 2x + 12 ( subtract 2x from both sides )
2x + 1 = 12 ( subtract 1 from both sides )
2x = 11 ( divide both sides by 2 )
x = 5.5
(c)
the perimeter (P) is the sum of the 4 sides of the rectangle , that is
P = 4x + 1 + x + 2x + 12 + x ( collect like terms )
= 8x + 13 ( substitute x = 5.5 )
= 8(5.5) + 13
= 44 + 13
= 57
Answer:
a) Because two opposite sides of the rectangle have the same sizes
B)
[tex]x = \frac{11}{2} [/tex]
c)
[tex]57[/tex]
Step-by-step explanation:
b)
[tex]4x + 1 = 2x + 12 \\ 4x - 2x = 12 - 1 \\ 2x = 11 \\ \frac{2x}{2} = \frac{11}{2} \\ x = \frac{11}{2} [/tex]
c) Perimeter
[tex]p = 2(x + y) \\ 2( \frac{11}{2} + 23) \\ 2( \frac{57}{2} ) \\ 57[/tex]
23 comes from
[tex]2x + 12 \\ x = \frac{11}{2} \\ 2( \frac{11}{2} ) + 12 \\ 11 + 12 \\ \\ 23[/tex]
Find the area of the trapezoid. Leave your answer in simplest radical form.
a) 156 ft^2
b) 78 ft^2
c) 78 √ 2 ft^2
d) 13 ft^2
The area of the trapezoid in simplest radical form is 78 feet².
Given a trapezoid.
Length of the bases are 10 feet and 16 feet.
We have to find the height.
Consider the smaller right triangle formed by the height of the trapezoid.
Triangle base length or one leg = 16 - 10 = 6 feet
Since one of the angle is 45°, the other angle in the right triangle is also 45°.
Since it is isosceles, opposite sides for 45° angles are same.
Other leg = 6 feet, which is the height.
Area of the trapezoid = 1/2 (a + b)h, where a and b are bases and h is the height.
A = 1/2 (10 + 16) 6
= 78 feet²
Hence the correct option is b.
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Si al salario por semana se le rebaja el 9. 5% para ccss y el salario es de 50. 000 ¿cuánto me pagan?
If the weekly salary is reduced by 9.5% for CCSS and the salary is 50000, then they would pay $45250.
The percent reduction in the weekly salary is = 9.5% = 0.095,
So, the amount of the reduction is;
⇒ reduction = 9.5% of 50,000
⇒ reduction = 0.095 x 50,000
⇒ reduction = 4750
So, the reduction in the weekly salary is 4750,
Now, we subtract the reduction from the original salary to get the new salary after the reduction,
⇒ new salary = 50000 - 4750
⇒ new salary = 45250
Therefore, the new salary after the 9.5% reduction for CCSS is $45250.
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Find the volume of the composite figure below.
Show work.
Answer:
1278in^3
Step-by-step explanation:
21×8=168
168×6=1008in^3
5×9=45
45×6=270in^3
1008+270=1278in^3
How to find the Zeros, Multiplicity, and Effect?
f(x)=-8x^(3)-20x^(2)
The Zeros of the equation would be x = -5/2.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
A mathematical equation is a statement with two equal sides and an equal sign in between. An equation is, for instance, 4 + 6 = 10. Both 4 + 6 and 10 can be seen on the left and right sides of the equal sign, respectively.
We are given the equation as;
[tex]f(x)=-8x^3-20x^2[/tex]
We can factor;
4x^2 ( 2x+5)
Using the zero product property
2x = 0
2x + 5 = 0
x = -5/2
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3
The drama club at Hawthorne Middle School is selling tickets to their spring musical. Student tickets
cost $10 and adult tickets cost $15. Last week, they sold 120 tickets for the Sunday matinee show. If those
ticket sales totaled to $1,400, how many adult tickets were sold? How many student tickets were sold?
40 adult tickets were sold and 80 student tickets were sold.
What is an equation?
An equation is a mathematical statement that asserts that two expressions are equal. It typically consists of variables, constants, and mathematical operations.
Let's use algebra to solve this problem.
Let's call the number of student tickets sold "s" and the number of adult tickets sold "a".
From the problem, we know two things:
The total number of tickets sold was 120:
s + a = 120
The total amount of money made from ticket sales was $1,400:
10s + 15a = 1400
Now we have two equations with two variables, so we can solve for "a" and "s".
Let's start by solving for "s" in the first equation:
s + a = 120
s = 120 - a
Now we can substitute this expression for "s" into the second equation:
10s + 15a = 1400
10(120 - a) + 15a = 1400
1200 - 10a + 15a = 1400
5a = 200
a = 40
So 40 adult tickets were sold.
Now we can substitute this value of "a" into the first equation to find "s":
s + a = 120
s + 40 = 120
s = 80
So 80 student tickets were sold.
Therefore, 40 adult tickets were sold and 80 student tickets were sold.
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College Algebua - „,1 Modellng with Quadratles Mahemet Koothall Thrun - Cine caact atswens, the and the fanctian ta ibcin your animen. - Show all necinary calnulations - Wrice par answis in ceptreste itriansich. reresest? 2. What a theytipenret' What dors it repericnt? Hha happris? ia at 10 irst be isurn wa fer be isle ta catis thin forchile College Algebra - 3.1 Modeling with QuadraticsMahomes Football Throw At a Kansas City Chiefs football practice, Patrick Mahomes is practicing his throws. He stands at one end of the football field and throws the ball. The ball's height, (in feet) as a function of horizontal distance,x(in feet), from Mahomes can be described by the following function:f(x)=−1201(x−88)2+151058- Give exact answers. Use only the function to obtain your answers. - Show all necessary calculations. - Write your answers in complete sentences. 1. What is the positivex-intercept? What does it represent? 2. What is they-intercept? What does it represent? 3. What is the maximum height that the foothall reaches?. What is the horizontal distance from Mahomes when this happens? 4. Travis Kelce is standing along the path of the football 160 feet from Patrick Mahomes. Travis can catch a ball that is at 10 feet or lower. Will he be able to catch this football? 5. What is the horizontal distance of the football from Patrick Mahomes when it first reaches a height of 40 feet?
1. The positive x-intercept of the function f(x)= -1201(x-88)^2 + 151058 is 88 feet.
2. The y-intercept of the function f(x) = -1201(x-88)^2 + 151058 is 151058 feet.
3. The maximum height that the football reaches is 151058 feet, and the horizontal distance from Patrick Mahomes when this happens is 88 feet.
4. Yes, Travis Kelce will be able to catch the football, since it reaches a height of 10 feet or lower at the horizontal distance of 160 feet from Patrick Mahomes.
5. The horizontal distance of the football from Patrick Mahomes when it first reaches a height of 40 feet is 176 feet.
1. The positive x-intercept represents the horizontal distance from Patrick Mahomes at which the football is initially released.
2. The y-intercept of the function represents the height of the football when it is released from Patrick Mahomes.
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Keegan was asked to graph (R90•D2)(Triangle ABC). Explain Keegan’s error.
Keegan's error is that he did not provide enough information to fully determine the result of the given composition of transformations, namely (R90•D2)(Triangle ABC).
To be able to graph the image of Triangle ABC under the composition of R90 (a 90-degree counterclockwise rotation) and D2 (a dilation with center the origin and scale factor 2), we need to know the center of rotation for the rotation R90.
The reason for this is that the composition of a rotation and a dilation is not commutative, meaning that the order of the transformations matters. Specifically, the result of the composition depends on whether the dilation is applied before or after the rotation. If the dilation is applied before the rotation, then the center of dilation becomes the center of rotation for the rotation. If the rotation is applied before the dilation, then the center of dilation is not affected by the rotation.
Therefore, without knowing the center of rotation for the rotation R90, we cannot determine the exact result of the composition (R90•D2)(Triangle ABC), and thus we cannot graph it accurately.
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Leon asks his classmates how many mystery books and how many adventure books they own. He states that because the mean number of mystery books, 5, is less than the mean number of adventure books, 10, there is less variability in the number of mysstery books. Do you agree?
To make any conclusions about the variability of the two datasets.
What is variable?a variable is a symbol or letter that represents a value that can change or vary. Variables are used to express mathematical relationships and equations, and they can take on different values depending on the context.
by question.
No, we cannot conclude that there is less variability in the number of mystery books based solely on the mean number of mystery books being less than the mean number of adventure books. The variability of the data is captured by the standard deviation or variance, and we would need this information to make any conclusion about the variability of the two types of books. It's possible that there is less variability in the number of mystery books,
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Fluffy, Spot, and Shampy have a combined age in dog years of 82. Spot is 14 years younger than Fluffy. Shampy is 6 years older than Fluffy. What is Fluffy's age, f, in dog years?
The age of fluffy is 30 in dog years. The solution has been obtained by using linear equation.
What is a linear equation?One is the degree of the linear equation. The absence of variables in linear equations with exponents greater than one is obvious. The graph's equation results in a straight line.
We are given that combined age of 3 dogs in dog years is 82.
Let Fluffy's age be 'f'.
Age of Spot = (f - 14)
Age of Shampy = (f + 6)
From this, we get
f + (f - 14) + (f + 6) = 82
On solving this, we get
⇒f + f - 14 + f + 6 = 82
⇒3f - 8 = 82
⇒3f = 90
⇒f = 30
Hence, the age of fluffy is 30 in dog years.
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How do I draw a parabola when given the directix and focus
The distance between any point on the parabola and the directrix is equal to the distance between that point and the focus. This property is what defines a parabola.
How to draw a parabola when given the directix and focus?To draw a parabola given the directrix and focus, follow these steps:
Draw the directrix as a straight line.Mark the focus point on the opposite side of the directrix from the vertex.Find the midpoint between the focus and the directrix, which is also the vertex of the parabola.,Draw a perpendicular line from the vertex to the directrix. This is the axis of symmetry of the parabola.Measure the distance between the focus and the vertex. This distance is called the focal length and is denoted by "p".From the vertex, mark a point "p" units above and below the vertex along the axis of symmetry. These points are called the "endpoints of the latus rectum."Draw lines through each endpoint of the latus rectum perpendicular to the axis of symmetry, extending to intersect the directrix. These two lines will be parallel to each other and equidistant from the axis of symmetry.Finally, draw a smooth curve through the focus point that passes through each endpoint of the latus rectum. This curve is the parabola.Learn more about parabola here: https://brainly.com/question/25651698
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Big ideas 7.5 (question)
1) A perpendicular bisector in kite ABCD is; BD
2) An isosceles triangle in kite ABCD is; ΔABC
3) A right triangle in kite ABCD is; ΔABM
What is a Kite?In geometry, a kite is defined as a quadrilateral with reflection symmetry across a diagonal.
here, we have,
1) A perpendicular bisector is defined as a line segment which bisects another line segment at 90 degrees.
Looking at the diagram, Line BD bisects Line AC and as such BD is the perpendicular bisector.
2) An Isosceles triangle is defined as a a triangle in which two sides have the same length.
In this case, in Triangle ABC, AB and BC have the same length and as such Triangle ABC is the Isosceles Triangle.
3) A right triangle is defined as a triangle, that has one of its interior angles equal to 90 degrees or any one angle is a right angle.
In this case if Mis the intersection of BD and AC, then the right angle triangle is ABM
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3)(9pts)Use Cramer's rule to find the values of the variablesx,y, andz( 3 points each). Show the determinants involved clearly and expand the numerator determinants as follows to get any credit: For variablex, expand along the first column; for variabley, expand along the second column; for variablez, expand along the third column. You will receive 0 points if you expand these determinants any other way, even if your answer is correct.−3x+0y−6z8x−2y+3z2x−y−4z=11=17=3
The values of the variables x, y, and z obtained by using Cramer's rule are 83/105, -277/105, and -99/105, respectively.
To find the values of the variables x, y, and z using Cramer's rule, we need to find the determinant of the coefficient matrix and the determinants of the matrices obtained by replacing the first, second, and third columns of the coefficient matrix with the constant terms.
The coefficient matrix is:
|−3 0 −6|
| 8 −2 3|
| 2 −1 −4|
The determinant of the coefficient matrix is:
|−3 0 −6| = (−3)(−2)(−4) − (0)(3)(2) − (−6)(8)(−1) − (−3)(−1)(3) − (0)(−4)(8) − (−6)(−2)(2)
= −24 − 0 − 48 − 9 − 0 − 24
= −105
The matrix obtained by replacing the first column with the constant terms is:
|11 0 −6|
|17 −2 3|
| 3 −1 −4|
The determinant of this matrix is:
|11 0 −6| = (11)(−2)(−4) − (0)(3)(3) − (−6)(17)(−1) − (11)(−1)(3) − (0)(−4)(17) − (−6)(−2)(3)
= 88 − 0 − 102 − 33 − 0 − 36
= −83
The matrix obtained by replacing the second column with the constant terms is:
|−3 11 −6|
| 8 17 3|
| 2 3 −4|
The determinant of this matrix is:
|−3 11 −6| = (−3)(17)(−4) − (11)(3)(2) − (−6)(8)(3) − (−3)(3)(3) − (11)(−4)(2) − (−6)(17)(2)
= 204 − 66 − 144 − 9 + 88 + 204
= 277
The matrix obtained by replacing the third column with the constant terms is:
|−3 0 11|
| 8 −2 17|
| 2 −1 3|
The determinant of this matrix is:
|−3 0 11| = (−3)(−2)(3) − (0)(17)(2) − (11)(8)(−1) − (−3)(−1)(17) − (0)(3)(8) − (11)(−2)(2)
= 18 − 0 + 88 − 51 − 0 + 44
= 99
Now we can use Cramer's rule to find the values of the variables x, y, and z:
x = (−83)/(-105) = 83/105
y = (277)/(-105) = -277/105
z = (99)/(-105) = -99/105
83/105, -277/105, and -99/105 are the values of x,y and z respectively.
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Failures have occurred at the following cumulative test times (Type II testing): 28, 146, 258, 426, 521, 1027, 1273 hours.
A) Fit the AMSAA growth model and estimate the MTTF at the conclusion of the test cycle.
B) On the basis of (A), how many more hours of test time will be necessary to achieve and MTTF of 3000 hours?
C) Assume that growth testing has resulted in achieving the desired MTTF of 3,000 hours. How many items must now be placed on test to obtain another 10 failures with 5,000 hours of test time available? (Assume a constant failure rate)
D) If the testing in (c) continues for only 600 more hours, what is the expected number
of failures?
AMSAA growth model
The MTTF at the conclusion of the test cycle is 232.59 hours. 22423.54 more hours of test time will be necessary to achieve and MTTF of 3000 hours. 52.15 items must now be placed on test to obtain another 10 failures with 5,000 hours of test time available. If the testing continues for only 600 more hours, 1.75 is the expected number of failures.
A) To fit the AMSAA growth model, we first need to calculate the cumulative number of failures (C) and the logarithm of the test time (lnT).
| Test Time (T) | C | lnT |
|---------------|---|-----|
| 28 | 1 | 3.33|
| 146 | 2 | 4.98|
| 258 | 3 | 5.55|
| 426 | 4 | 6.05|
| 521 | 5 | 6.26|
| 1027 | 6 | 6.93|
| 1273 | 7 | 7.15|
Next, we can use linear regression to estimate the parameters of the AMSAA model, β and η:
C = βlnT - η
Using linear regression, we get β = 1.11 and η = 3.82.
To estimate the MTTF at the conclusion of the test cycle, we can use the formula:
MTTF = (T/C)^(1/β)
Plugging in the values for T (1273), C (7), and β (1.11), we get:
MTTF = (1273/7)^(1/1.11) = 232.59 hours
B) To achieve an MTTF of 3000 hours, we need to solve for T in the equation: 3000 = (T/C)^(1/β)
Plugging in the values for C (7), β (1.11), and rearranging the equation, we get:
T = 3000^(β) * C = 3000^(1.11) * 7 = 23696.54 hours
Since we have already tested for 1273 hours, we need an additional 23696.54 - 1273 = 22423.54 hours of test time to achieve an MTTF of 3000 hours.
C) To obtain another 10 failures with 5000 hours of test time available, we can use the formula:
C = βlnT - η
Plugging in the values for C (10), β (1.11), η (3.82), and T (5000), and rearranging the equation, we get:
10 = 1.11ln(5000) - 3.82
ln(5000) = (10 + 3.82)/1.11 = 12.47
5000 = e^(12.47) = 260753.13
Therefore, we need to place 260753.13/5000 = 52.15 items on test to obtain another 10 failures with 5000 hours of test time available.
D) If the testing in (c) continues for only 600 more hours, we can use the formula:
C = βlnT - η
Plugging in the values for β (1.11), η (3.82), and T (600), and rearranging the equation, we get:
C = 1.11ln(600) - 3.82 = 1.75
Therefore, the expected number of failures in 600 more hours of testing is 1.75.
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Curt and Melanie are mixing blue and yellow paint to make seafoam green paint.use the percent equation to find how much yellow paint they should use
The percentage of yellow paint that has to be used is given as 0.45
How to find the amount of yellow paint that is to be used hereThe amount of seafoam paint is given as 1.5 quartz
The percentage of yellow paint in the seafoam paint is 30 percent
Hence the amount that would be in it that would be made of yellow paint is given as
1.5 x 30%
1.5 x 0.30
= 0.45
Hence we would conclude by saying that the amount of yellow paint that has to be used in order to make the paint should be 0.45
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Curt and Melanie are mixing blue and yellow paint to make seafoam green paint. 1.5 quarts of seafoam paint is made of 70% blue, 30% yellow. Use the percent equation to find how much yellow paint they should use.