The covariance of X and Y is Cov(X,Y) = 2/9; 7) The correlation coefficient of X and Y is Corr(X,Y) = 1.
Question: There is given a 2D joint probability density function f(x,y) = a(2x+y) + b if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. Find: 1) Coefficient a; 2) Marginal p.d.f. of X, marginal p.d.f. of Y; 3) E(X), E(Y), E (XY); 4) Var(X), Var(Y); 5) 0(X), (Y); 6) Cov(X,Y); 7) Corr(X,Y).
Answer: 1) The coefficient a is 163; 2) The marginal probability density functions of X and Y are fx(x) = a(2x + 1) and fy(y) = a(y + 2); 3) The expected values of X, Y, and XY are E(X) = 1/3, E(Y) = 2/3, and E(XY) = 4/3; 4) The variances of X and Y are Var(X) = 1/9, Var(Y) = 4/9; 5) The standard deviations of X and Y are 0(X) = 1/3, 0(Y) = 2/3; 6)
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Do this photos plisssssssss
the first question answer is 15/17
3x+4y=-30 2x-5y=72 find y
nvm sorry wrong answer
Answer:-12
Step-by-step explanation:
To solve for y, we can use the second equation to isolate x and substitute the result into the first equation:
2x - 5y = 72
2x = 5y + 72
x = (5y + 72)/2
Now we can substitute this expression for x into the first equation:
3x + 4y = -30
3((5y + 72)/2) + 4y = -30
Simplifying the left side:
(15/2)y + 108 + 4y = -30
Combining like terms:
(23/2)y = -138
Dividing both sides by (23/2):
y = -138 * 2/23
Simplifying:
y = -12
Therefore, the value of y that satisfies both equations is -12.
For each correspondence, (a) write the domain, (b) write the
range, and (c) determine whether the correspondence is a
function.
21.{(-3,3), (-2.5), (0,9) (4,-10)}
The domain of the correspondence is {-3, -2.5, 0, 4} and the range is {3, 9, -10}. This correspondence is not a function because it is not a one-to-one correspondence; for example, both -3 and 4 are mapped to the same value of 3.
The question is asking for the domain, range, and whether the correspondence is a function for the given set of ordered pairs: {(-3,3), (-2.5), (0,9), (4,-10)}.
The domain of a correspondence is the set of all possible input values. In this case, the domain is the set of x-values from the ordered pairs: {-3, -2.5, 0, 4}.
The range of a correspondence is the set of all possible output values. In this case, the range is the set of y-values from the ordered pairs: {3, -5, 9, -10}.
A correspondence is a function if each input (x-value) is associated with only one output (y-value). To determine if the given correspondence is a function, we need to check if any x-value is repeated with a different y-value.
In this case, there are no repeated x-values, so each x-value is associated with only one y-value. Therefore, the correspondence is a function.
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Write a sinusoidal function with amplitude 4, period 3???? and
vertical shift down 8 units.
A sinusoidal function is a type of periodic function that can be written in the form y = a sin(bx + c) + d, where a is the amplitude, b is the frequency, c is the phase shift, and d is the vertical shift.
Given the parameters in the question, we can plug them into the general form of a sinusoidal function to find the specific function.
Amplitude = 4
Period = 3π
Vertical shift = -8
The period of a sinusoidal function is related to the frequency by the equation period = (2π)/b. Therefore, we can solve for b to find the frequency:
3π = (2π)/b
b = (2π)/(3π)
b = 2/3
Plugging in the given values for amplitude, frequency, and vertical shift, we get the sinusoidal function:
y = 4 sin((2/3)x) - 8
This is the sinusoidal function that satisfies the given conditions. It has an amplitude of 4, a period of 3π, and a vertical shift down 8 units.
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The sides of a triangle have lengths 2, 6, and 7. What kind of triangle is it?
Answer:
Scalene triangle
Step-by-step explanation:
To determine the type of triangle, we need to compare the lengths of the sides to each other.
If all three sides have the same length, it's an equilateral triangle.
If two sides have the same length and the third is different, it's an isosceles triangle.
If all three sides have different lengths, it's a scalene triangle.
Using the lengths given, we see that all three sides have different lengths, so this is a scalene triangle.
Answer:
an obtuse triangle.
Step-by-step explanation:
To determine the type of triangle with sides of lengths 2, 6, and 7, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we can check whether this condition holds for the three sides of lengths 2, 6, and 7:
2 + 6 > 7 (True)
6 + 7 > 2 (True)
2 + 7 > 6 (True)
Since all three inequalities are true, the given lengths of 2, 6, and 7 form a valid triangle.
To determine the type of triangle, we can use the lengths of the sides and the Pythagorean theorem or trigonometric functions. For this triangle, we can use the Pythagorean theorem to find that the longest side (7) is opposite the largest angle, which is greater than 90 degrees. Therefore, this triangle is an obtuse triangle.
A manufacturer of refrigerators must ship at least 100 refrigerators to its two West coast warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25 refrigerators already, while warehouse B has 20 on hand. It costs $12 to ship a refrigerator to warehouse A and $10 to ship one to warehouse B. Union rules require that at least 300 workers be hired. Shipping a refrigerator t warehouse a requires 4 workers, while shipping a refrigerator to warehouse b requires 2 workers. How many refrigerators should be shipped to each warehouse to minimize costs? what is the minimum cost? How many refrigerators should be shipped to each warehouse to minimize cost? What is the minimum cost?
The manufacturer should ship 0 refrigerators to warehouse A and 80 refrigerators to warehouse B to minimize costs. The minimum cost is $800.
The objective of the problem is to minimize the cost of shipping the refrigerators to the two warehouses while also meeting the requirements of shipping at least 100 refrigerators and hiring at least 300 workers. This can be formulated as a linear programming problem with the following decision variables:
x1 = the number of refrigerators shipped to warehouse A
x2 = the number of refrigerators shipped to warehouse B
The objective function is the total cost of shipping the refrigerators:
minimize 12x1 + 10x2
The constraints are the requirements of shipping at least 100 refrigerators, hiring at least 300 workers, and the maximum capacity of each warehouse:
x1 + x2 >= 100
4x1 + 2x2 >= 300
x1 <= 75 (100 - 25)
x2 <= 80 (100 - 20)
Using the graphical method, we can plot the constraints and find the feasible region. The optimal solution will be at one of the corner points of the feasible region.
After plotting the constraints and finding the feasible region, we can evaluate the objective function at each of the corner points to find the minimum cost. The corner points and their corresponding costs are:
(75, 0) -> 12(75) + 10(0) = $900
(0, 80) -> 12(0) + 10(80) = $800
(50, 100) -> 12(50) + 10(100) = $1700
(60, 90) -> 12(60) + 10(90) = $1740
The minimum cost is $800, which occurs when 0 refrigerators are shipped to warehouse A and 80 refrigerators are shipped to warehouse B.
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What is the simplifed expression when(17−8c)is subtracted from (−c+5). Show work.
The simplified expression when(17−8c)is subtracted from (−c+5) is 12 - 7c.
What are algebraic expressions?When operations like addition, subtraction, multiplication, division, etc. are performed on variables and constants, we obtain algebraic expressions as the mathematical statement. Let's say James and Natalie were tinkering with matchsticks when they had the idea to arrange the sticks into numerical patterns. James used four matchsticks to make the numeral 4 in the air. Adding three additional matchsticks allowed Natalie to create a pattern with two 4s. They understood that they may continue to add three matchsticks each round to make a total of four.
The two expressions are:
17 - 8c and - c + 5
The subtraction of the two expressions are:
17 - 8c - (-c + 5)
17 - 8c + c - 5 = 12 - 7c
Hence, the simplified expression when(17−8c)is subtracted from (−c+5) is 12 - 7c.
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whats the first step tp solve n/4=-12
If y varies inversely as a, and x = 24 when y = 1. Find y when x = 3.
[tex]\qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}} ~\hspace{6em} \stackrel{\textit{constant of variation}}{y=\cfrac{\stackrel{\downarrow }{k}}{x}~\hfill } \\\\ \textit{\underline{x} varies inversely with }\underline{z^5} ~\hspace{5.5em} \stackrel{\textit{constant of variation}}{x=\cfrac{\stackrel{\downarrow }{k}}{z^5}~\hfill } \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{"y" varies inversely with "x"}}{y = \cfrac{k}{x}}\hspace{5em}\textit{we also know that} \begin{cases} x=24\\ y=1 \end{cases} \\\\\\ 1=\cfrac{k}{24}\implies 24=k\hspace{9em}\boxed{y=\cfrac{24}{x}} \\\\\\ \textit{when x = 3, what's "y"?}\qquad y=\cfrac{24}{3}\implies y=8[/tex]
Evaluate the expression without using a calculator.
sin(2arccos(1/3))
Exact Form of the expression is mathematically given as
[tex]\sin \left(2 \arccos \left(\frac{1}{3}\right)\right)[/tex]
What is an expression?Generally, An expression is a combination of variables, numbers, operators, and/or functions that represent a value.
[tex]\sin \left(2 \arccos \left (\frac{1}{3}\right)\right)[/tex]
Evaluate[tex]$\arccos \left(\frac{1}{3}\right)$.[/tex]
sin (2 *1.23095941)
Multiply 2 by 1.23095941.
sin (2.46191883)
The result can be shown in multiple forms.
Exact Form:
[tex]\sin \left(2 \arccos \left(\frac{1}{3}\right)\right)[/tex]
Decimal Form:
0.62853936
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What is ordered pair is a solution of the equation 6x+3y=15
Answer:
Step-by-step explanation:
Sorry, i can't heip you
PLEASE HELP PLSSS
a sequence of transformations that maps DOG to D'O'G' is a (answer choice) followed by a (answer choice)
answer choices:
1. rotation of about 180 degrees of the origin
2.rotation of about 90 degrees of the origin
3. translation 5 units right
4. translation 5 units left
The sequence of transformations that maps DOG to D'O'G' is a rotation of about 90 degrees of the origin, followed by a translation of 5 units right.
What is rotation?Rotation is a physical phenomenon where an object spins around an axis. It is a type of circular motion where an object moves in a circular path around a central point.
To accomplish this transformation, first, the point representing the letter D is rotated 90 degrees clockwise around the origin so that it is now pointing in the same direction as the letter O. After that, the point representing the letter D is translated 5 units to the right, resulting in the final transformation of DOG to D'O'G'.
Thus, the sequence of transformations that maps DOG to D'O'G' is a rotation of about 90 degrees of the origin, followed by a translation of 5 units right.
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HELP LOL!
Given ∠ BAD ≅ ∠BED, what can you conclude?
Responses
A Δ ABC is isosceles
B BD is the perpendicular bisector of AC
C AE ≅ DE
D ∠ ABD ≅ ∠ BDE
E Δ ABE is isosceles
Answer:
E. ∆ABE is isosceles
Step-by-step explanation:
Given ∠BAD ≅ ∠BED in ∆ABE you want to know what can be concluded.
Isosceles triangleAn isosceles triangle has sides of equal measure opposite angles of equal measure. The base angles A and E in triangle ABE are given as congruent, so we can conclude ∆ABE is isosceles.
Find three consecutive odd integers such that the product of the second and third minus the product of the first and second is 34. (If there is no solution, enter NO SOLUTION.)
There is no solution to this problem.
To find the three consecutive odd integers, we need to use algebra. Let's first represent the three consecutive odd integers as x, x+2, and x+4. Then we can set up an equation based on the given information:
(x+2)(x+4) - (x)(x+2) = 34
Simplifying the equation gives:
x^2 + 6x + 8 - x^2 - 2x = 34
Combining like terms gives:
4x + 8 = 34
Subtracting 8 from both sides gives:
4x = 26
Dividing by 4 gives:
x = 6.5
However, since we are looking for odd integers, there is no solution to this problem. Therefore, the answer is NO SOLUTION.
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Given sin A = and that angle A is in Quadrant I, find the exact sec A in simplest radical form using a rational denominator
For 50pt
Answer: Read the step by step explanation
Step-by-step explanation:
Since sin A = , we can use the Pythagorean identity to find cos A:
cos A = √(1 - sin²A) = √(1 - ( )²) = √(1 - ) = √()
Since A is in Quadrant I, both sin A and cos A are positive. Therefore, we can use the definition of secant to find sec A:
sec A = 1/cos A = 1/√() = √()/
To simplify the expression, we can rationalize the denominator by multiplying both the numerator and denominator by √():
sec A = (√()/)(√()/√()) = √()/ =
Therefore, the exact secant of A in simplest radical form using a rational denominator is .
if you help me I will make another question with 300 points
a.) Use the table below to calculate the average percent change in population in California from 2000-2009.
b.) If California's population in 2009 was 37,000,000 and the population trend were to continue, what would the population be in the year 2015?
The exponential growth equation is x ( t ) = 37,000,000 ( 1 + 1.3567% )⁶ , where x ( t ) is the population in the year 2015 = 40,115,896
What is exponential growth factor?The exponential growth or decay formula is given by
x ( t ) = x₀ × ( 1 + r )ⁿ
x ( t ) is the value at time t
x₀ is the initial value at time t = 0.
r is the growth rate when r>0 or decay rate when r<0, in percent
t is the time in discrete intervals and selected time units'
Given data ,
Let the exponential growth equation be represented as x ( t )
Now , the average percentage change from the year 2000 - 2009 is calculated by
The total percentage = ( 1.97+1.71+1.65+1.42+1.22+1.02+1.07+1.22+0.93 )
The total number of years = 9 years
So , the average percentage change = total percentage / number of years
On simplifying , we get
The average percentage change = 12.21 / 9 = 1.35667 %
b)
The population in 2009 was 37,000,000
So , the population growth rate r = 1.35667 %
And , the population in the years 2015 is given by
The number of years n = 6 years
x ( t ) = 37,000,000 ( 1 + 1.3567% )⁶
On simplifying , we get
x ( t ) = 37,000,000 ( 1.013567 )⁶
x ( t ) = 40,115,896
Hence , the population in the year 2015 is 40,115,896
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Identify the highlighted part of circle O shown below.
The highlighted part of circle shown below is known as segment.
What is the segment of a circle?A segment of a circle is a region of the circle that is bounded by a chord and the arc that it intersects. More specifically, a segment is the region between a chord and a minor or major arc of a circle.
The chord is the straight line that connects two points on the circumference of the circle, and the arc is the curved part of the circumference that lies between these two points. A segment is named according to its chord, for example, the segment determined by the chord AB is referred to as segment AB. The area of a segment of a circle can be calculated using the formula A = (1/2)r^2(θ-sinθ), where r is the radius of the circle, and θ is the central angle of the segment in radians.
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The compound shape below is formed from rectangle ABDE and right-angled triangle BCD. What is the area of this shape? Give your answer in cm² and give any decimal answers to 1 d.p. A 4cm B 9 cm E 4 cm D 15 cm C Not drawn accurately
The area of the shape formed from rectangle ABDE is 78 cm².
How to find area of a rectangle?To find the area of the compound shape, we need to find the area of rectangle ABDE and triangle BCD separately and then add them.
The rectangle ABDE has a length of 15 cm (DE) and a width of 4 cm (AB). Therefore, its area is:
Area of rectangle = length x width = 15 cm x 4 cm = 60 cm²
The triangle BCD has a base of 4 cm (BC) and a height of 9 cm (BD). Therefore, its area is:
Area of triangle = 1/2 x base x height = 1/2 x 4 cm x 9 cm = 18 cm²
Now, add the areas of the rectangle and the triangle to find the total area of the compound shape:
Total area = Area of rectangle + Area of triangle = 60 cm² + 18 cm² = 78 cm²
Therefore, the area of the compound shape is 78 cm².
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probability please help!!!!!!
The probability of choosing the lists of students from the class is given by combinations C = 2400 ways
What are Combinations?The number of ways of selecting r objects from n unlike objects is given by combinations
ⁿCₓ = n! / ( ( n - x )! x! )
where
n = total number of objects
x = number of choosing objects from the set
Given data ,
Let the total ways of choosing the students from the class be C
Now , the total number of boys = 12 boys
The total number of girls = 10 girls
And , the lists are in the order
A = { boy , girl , boy }
B = { girl , boy , girl }
The total number of ways C = A + B
From the combination of selecting boys and girls , we get
The total number of selecting A = 12 x 10 x 11 = 1320 ways
The total number of selecting B = 10 x 12 x 9 = 1080 ways
So , the total number of selecting the lists C = 2400 ways
Hence , the probability of choosing the lists of students from the class is given by C = 2400 ways
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5+5 x 5 is whattt EASY MATH 20 points
Answer:50
Step-by-step explanation:5+5=10 and 5 groups of 10 is 50
Answer:30
Step-by-step explanation:
5x5=25
25+5
30
100 POINTS IF YOU CAN HELP ME!!
Use the equation below to calculate the total cost of a payday loan for #6-8.
\Total Cost = Loan amount + finance charge ( 1 + Number of rollovers or renewals). Keep in mind, this does not include the initial loan cost, just the interest.
#6. Toby takes out a payday loan to pay for a medical bill that costs $450. The payday lender charges him $50 every two weeks. Toby has to roll over the loan 4 times before he can finally pay it off. What was the total cost of his loan?
7. Tabitha takes out a payday loan to pay for her son's emergency room bill that costs $300. The payday lender charges her $40 every two weeks. She has to roll over the loan 6 times before she can finally pay it off. What is the total cost of her loan?
8. Sariah takes out a payday loan to get a new set of tires in order to pass inspection. The bill is $500. She needs her car to get to and from work but does not have enough money in her savings to cover the bill so she takes out a payday loan. The payday loan will cost her $45 every two weeks. She has to roll over 7 times. What was the total cost of her loan?
Step-by-step explanation:
Toby's loan amount is $450 and the finance charge is $50 every two weeks. He rolls over the loan 4 times.
Total cost = $450 + ($50 x (1 + 4))
Total cost = $450 + ($50 x 5)
Total cost = $450 + $250
Total cost = $700
Therefore, the total cost of Toby's payday loan is $700.
Tabitha's loan amount is $300 and the finance charge is $40 every two weeks. She rolls over the loan 6 times.
Total cost = $300 + ($40 x (1 + 6))
Total cost = $300 + ($40 x 7)
Total cost = $300 + $280
Total cost = $580
Therefore, the total cost of Tabitha's payday loan is $580.
Sariah's loan amount is $500 and the finance charge is $45 every two weeks. She rolls over the loan 7 times.
Total cost = $500 + ($45 x (1 + 7))
Total cost = $500 + ($45 x 8)
Total cost = $500 + $360
Total cost = $860
Therefore, the total cost of Sariah's payday loan is $860
Answer:
#6.
Loan amount = $450
Finance charge = $50 every two weeks
Number of rollovers or renewals = 4
Total Cost = 450 + (50 x (1 + 4))
Total Cost = 450 + 250
Total Cost = $700
The total cost of Toby's loan is $700.
#7.
Loan amount = $300
Finance charge = $40 every two weeks
Number of rollovers or renewals = 6
Total Cost = 300 + (40 x (1 + 6))
Total Cost = 300 + 280
Total Cost = $580
The total cost of Tabitha's loan is $580.
#8.
Loan amount = $500
Finance charge = $45 every two weeks
Number of rollovers or renewals = 7
Total Cost = 500 + (45 x (1 + 7))
Total Cost = 500 + 360
Total Cost = $860
The total cost of Sariah's loan is $860.
please help me i have been sruggling for ages now
Answer:
13:00
Make a straight line from the vertex of the current Yuri line to 16:15
Step-by-step explanation:
Winter 2023 (online ) operties of Exponents Simplify. Write the answer using positive exponents only. (2x^(6)y^(4))^(4)
The answer using positive exponents only is 16x^(24)y^(16).
To simplify the expression (2x^(6)y^(4))^(4), we need to use the properties of exponents. Specifically, we need to use the property that states (a^(m))^n = a^(m*n). This means that when we raise a power to another power, we multiply the exponents.
Using this property, we can simplify the expression as follows:
(2x^(6)y^(4))^(4) = 2^(4) * x^(6*4) * y^(4*4) = 16 * x^(24) * y^(16)
So, the simplified expression is 16x^(24)y^(16).
Therefore, the answer using positive exponents only is 16x^(24)y^(16).
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Please help meeeeeeeeee
The answer of the given question based on the Equation is given the solution to the given equation is n = 21/4.
What is Quadratic equation?A quadratic equation is type of equation in an algebra that involves variable (usually represented by x) that raised to the second power (i.e., squared). The general form of the quadratic equation are given,
[tex]ax^2+bx+c = 0[/tex]
The highest power of variable is 2, hence , name "quadratic".
To solve this equation, we can start by simplifying each of the terms on both sides of the equation:
3n + √(n² + 8) + 3√(n² + 8) - √(n² + 8) = 8
Combining like terms, we get:
3n + 3√(n² + 8) = 8
Subtracting 3√(n² + 8) from both sides, we get:
3n = 8 - 3√(n² + 8)
Dividing both sides by 3, we get:
n = (8/3) - (√(n² + 8)/3)
Now, we can rearrange this equation to isolate the radical term on one side of the equation:
√(n² + 8)/3 = (8/3) - n
by Squaring both the sides of equation, we get:
n² + 8 = 64/9 - (16n/3) + n²
Simplifying this equation, we get:
16n/3 = 56/9
Dividing both sides by 16/3, we get:
n = 21/4
Therefore, the solution to the given equation is n = 21/4.
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O RATIONAL EXPRESSIONS Adding rational expressions with common der Subtract. (z)/(z^(2)+4z-12)-(2)/(z^(2)+4z-12) Simplify your answer as much as possible.
The solution to the given rational expressions is (z - 2)/(z^(2)+4z-12).
To subtract rational expressions with common denominators, we simply subtract the numerators and keep the same denominator.
In this case, our common denominator is (z^(2)+4z-12), so we can subtract/add the numerators:
So, we will subtract the numerators, (z) - (2), and keep the same denominator:
(z)/(z^(2)+4z-12)-(2)/(z^(2)+4z-12) = (z - 2)/(z^(2)+4z-12)
Now, we can simplify the numerator by combining like terms:
(z - 2)/(z^(2)+4z-12) = (z - 2)/(z^(2)+4z-12)
Since the numerator and denominator cannot be simplified any further, this is our final answer:
(z - 2)/(z^(2)+4z-12)
Therefore, the solution to the subtraction of the given rational expressions is (z - 2)/(z^(2)+4z-12).
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1. Classify each triangle by its side lengths and angle measurements. Circle the correct names.
Classify Using
Side Lengths
m
b.
Equilateral Isosceles
Math Day Or
Scalene
Classify Using
Angle Measurements
Acute Right Obtuse. Angel mea
There are three types of triangle according to sides and angles each.
What is a triangle?The triangle is a type of polygon having three sides, angles and vertices.
Classification of triangles :-
Based on sides :-
1) Scalene triangle :- A triangle with all different measurements of sides and angles are called a Scalene triangle.
2) Isosceles triangle :- A triangle with two of the sides and angles are congruent called an Isosceles triangle.
3) Equilateral triangles :- A triangle with all the same measurements of sides and angles is called an Equilateral triangle.
Based on angles :-
1) Acute angled triangle :- A triangle with all the angles less than 90°, are called acute angled triangle.
2) Obtuse angled triangle :- A triangle with all the angles greater than 90° and less than 180°, are called Obtuse angled triangle.
3) Right angled triangle :- A triangle with one angle equals 90° is called a Right angled triangle.
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a line segment is drawn between (2,8) and (5,4). find it’s gradient.
the slope goes by several names
• average rate of change
• rate of change
• deltaY over deltaX
• Δy over Δx
• rise over run
• gradient
• constant of proportionality
however, is the same cat wearing different costumes.
[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{8})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{4}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{8}}}{\underset{\textit{\large run}} {\underset{x_2}{5}-\underset{x_1}{2}}} \implies \cfrac{ -4 }{ 3 } \implies - \cfrac{4 }{ 3 }[/tex]
So I’ve been stuck on this for a very long time it’s so annoying please hellpppp me
A follow-up question will appear once you have correctly answered the ques Since p(-1)=0, write p(x) in the form p(x)=(x+1)q(x).
The polynomial p(x) in the form p(x) = (x+1)q(x), where q(x) is a polynomial of degree n-1.
Algebraic expressions called polynomials include coefficients and variables. Indeterminates are another name for variables. For polynomial expressions, we can do mathematical operations like addition, subtraction, multiplication, and positive integer exponents but not division by variables.
x2+x-12 is an illustration of a polynomial with a single variable. There are three terms in this example: x2, x, and -12.
Since we know that p(-1)=0, we can use this information to substitute -1 into p(x), giving us:
p(-1) = 0
We can then rearrange this equation to get:
0 = p(-1) = (x+1)q(x)
This gives us the form of p(x) that we are looking for:
p(x) = (x+1)q(x)
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Your boss hands you a memo with a summary of the monthly data. The number of imports is shown as f(x), and the number of exports is shown as g(x). Use the data in the table below, representing both functions, to explain to your boss the solution to the system of equations and what that solution represents. Use complete sentences.
Month f(x) = No. of imports g(x) = No. of exports
January (1) 2 3
February (2) 4 4
March (3) 6 5
April (4) 8 6
Answer:
Based on the data in the table, we can set up a system of equations to represent the relationship between the number of imports (f(x)) and the number of exports (g(x)):
2x + 3y = z (Equation 1)
4x + 4y = z (Equation 2)
6x + 5y = z (Equation 3)
8x + 6y = z (Equation 4)
In this system, x represents the month number (January = 1, February = 2, etc.), y represents the number of exports, and z represents the total trade (exports plus imports). Each equation represents the data for a specific month.
To solve the system, we can use any method of solving systems of linear equations, such as substitution or elimination. However, we can also observe a pattern in the coefficients of the variables:
2 3 5 8
4 4 6 8
We can see that the coefficients of the x-term (the month number) increase by 2 each time, and the coefficients of the y-term (the number of exports) increase by 1 each time. This suggests that the equation for the nth month (where n is an integer between 1 and 4) can be expressed as:
2n + (n+2)y = z
We can test this by plugging in the values of n and y from any of the given months and verifying that the resulting value of z matches the actual value for that month. For example, if we use the data for March (n=3, y=5), we get:
2(3) + (3+2)(5) = z
6 + 25 = 31
And we can see that the actual value of z for March is indeed 31, which confirms that our equation works.
Using this equation, we can find the solution to the system for any given month by plugging in the appropriate values for n and y. For example, to find the total trade (z) for May (which would correspond to n=5), we can plug in n=5 and solve for z:
2(5) + (5+2)y = z
10 + 7y = z
To find the number of exports for May, we can plug in n=5 and solve for y:
2(5) + (5+2)y = z
10 + 7y = z
10 + 7y = 10 + 2y + 3
5y = -3
y = -3/5
However, since exports cannot be negative, this solution is not valid. This suggests that the data given in the table only applies to the months of January through April, and we cannot use this equation to find the solution for any month beyond April.
In summary, the system of equations represents the relationship between the number of imports and exports for each month from January to April. By observing the pattern in the coefficients of the equations, we can derive a general equation that can be used to find the total trade for any month between January and April. However, we cannot use this equation to find the solution for any month beyond April, as there is not enough data to determine the number of imports and exports for those months.