a) To find k, we need to integrate f(x, y) over its entire domain and set it equal to 1 since f(x, y) is a valid probability density function. Therefore,
Integral from 0 to 3-x Integral from 0 to 3-x of k dy dx = 1
Integrating with respect to y first, we get
Integral from 0 to 3-x of k(3-x) dy dx = 1
Solving for k, we get
k = 1/[(3/2)^2] = 4/9
b) P(X + Y ≤ 1) can be found by integrating f(x, y) over the region where X + Y ≤ 1. Since f(x, y) is 0 for x + y > 3, this integral can be split into two parts:
Integral from 0 to 1 Integral from 0 to x of f(x, y) dy dx + Integral from 1 to 3 Integral from 0 to 1-x of f(x, y) dy dx
Evaluating this integral, we get
P(X + Y ≤ 1) = Integral from 0 to 1 Integral from 0 to x of (4/9) dy dx + Integral from 1 to 3 Integral from 0 to 1-x of 0 dy dx
= Integral from 0 to 1 x(4/9) dx
= 2/9
c) P(X^2 + Y^2 ≤ 1) represents the area of the circle centered at the origin with radius 1. Since f(x, y) is 0 outside the region where x + y < 3, this probability can be found by integrating f(x, y) over the circle of radius 1. Converting to polar coordinates, we get
Integral from 0 to 2π Integral from 0 to 1 of r f(r cosθ, r sinθ) dr dθ
= Integral from 0 to π/4 Integral from 0 to 1 of (4/9) r dr dθ + Integral from π/4 to π/2 Integral from 0 to 3-√(2) of (4/9) r dr dθ
= Integral from 0 to π/4 (2/9) dθ + Integral from π/4 to π/2 (3-√(2))2/9 dθ
= (π/18) + [(6-2√(2))/27]
= (2π-12+4√2)/54
d) P(Y > X) can be found by integrating f(x, y) over the region where Y > X. Since f(x, y) is 0 for y > 3 - x, this integral can be split into two parts:
Integral from 0 to 3/2 Integral from x to 3-x of f(x, y) dy dx + Integral from 3/2 to 3 Integral from 3-x to 0 of f(x, y) dy dx
Evaluating this integral, we get
P(Y > X) = Integral from 0 to 3/2 Integral from x to 3-x of (4/9) dy dx + Integral from 3/2 to 3 Integral from 3-x to 0 of 0 dy dx
= Integral from 0 to 3/2 (8/9)x dx
= 1/3
e) X and Y are not independent since the probability of Y > X is not equal to the product of
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Select the correct answer from each drop-down menu.
As x approaches positive infinity, m(x) approaches positive infinity.
As x approaches negative infinity, m(x) approaches negative infinity.
What is meant by function?An expression, rule, or law in mathematics that establishes a connection between one variable (the independent variable) and another variable (the dependent variable).
A graph is considered to be a function if any vertical line formed can cross it at most one point. The graph is not a function if it contains any locations where a vertical line can pass through it twice or more.
If a function produces the same outcome regardless of how the input is represented without changing its value, it is said to be well defined.
A function is a rule that is applied to a number, known as an input, in order to produce a result, known as an output.
when [tex]$x^n$[/tex] ( n is odd number and n > 0), a > 0 and c is any constant,
[tex]$$x \rightarrow \infty, a x^n+c \rightarrow+\infty[/tex]
[tex]$x \rightarrow-\infty, a x^n+c \rightarrow-\infty$[/tex]
The complete question is;
The function f(x) = x^3 has been transformed, resulting in function m.
m(x) = 1\3x^3 + 6
As x approaches positive infinity, m(x) approaches___
As x approaches negative infinity, m(x) approaches___
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What does 2x² mean?
A quadratic equation is a second-degree equation, meaning it contains at least one squared term.
what is quadratic equation ?A quadratic polynomial in a single variable is x ax2+bx+c=0, which is a quadratic equation. a 0. The fact that this polynomial is of second order guarantees that it has at least one solution according to the Fundamental Theorem of Algebra. Solutions could be straightforward or difficult. A quadratic equation is one that has four variables. This suggests that at least one word in it has to be squared. One of the common formulas for solving quadratic equations is "ax2 + bx + c = 0." where the undefined variable "X" is represented by the numerical coefficients or constants a, b, and c.
given
2x2-7x = 0.
As we already know, a quadratic equation is a second-degree equation, meaning it contains at least one squared term.
Regarding comparing the quadratic equation in standard form with the equation 2x27x=0
ax ²+bx+c=0:
We get a=2, b=7, and c=0.
The equation 2x27x=0 is therefore a quadratic equation.
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solve this for me. and get a brainlist.
Answer:
below
Step-by-step explanation:
Isosceles triangle have TWO equal legs....so
Leg + leg + base = perimeter
(4x-4) + ( 4x-4) + x = 37
9x -8 = 37
9x = 45
x = 5
the rate of change of a linear function is 4 what could be the two points on the line of this function
For the rate of change of 4 the two points will be A(2,8) and B(4,16).
What is coordinate geometry?A coordinate plane is a 2D plane that is formed by the intersection of two perpendicular lines known as the x-axis and y-axis. A coordinate system in geometry is a method for determining the positions of the points by using one or more numbers or coordinates.
Given that the rate of change of the linear function is 4. The coordinates of the two points will be calculated as,
Rate of change = ( y₂ - y₁ ) / ( x₂ - x₁ )
Rate of change = ( 16 - 8 ) / ( 4 - 2 ) = 8 / 2 = 4
Compare the coordinates,
( x₁, y₁ ) = ( 2, 8 )
( x₂, y₂ ) = ( 4, 16 )
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Find the perimeter of the polygon if
5cm, 4cm, and 3cm
P=
The perimeter of the polygon is found as the 42 cm.
Define the term perimeter of the polygon?The perimeter of an enclosed plane figure is referred to in geometry. The perimeter of an enclosed figure is the sum of the lengths of its outermost sides. It is the sum of the lengths of a polygon's sides. All sides add up to the perimeter.For the shown polygon:
Points A, B, C and D are tangents on the circle.
Thus, these tangent will subtend equal length from the points.
So,
AB = 3 + 4 = 7 cm
BC = 4 + 5 = 9 cm
As, ∠B ≅ ∠D, sides must also be equal.
CD = 5 + 4 = 9 cm
DA = 4 + 3 = 7 cm
Perimeter = AB + BC + BC + BC
= 7 + 9 + 9 + 7
Perimeter = 42 cm
Thus, the perimeter of the polygon is found as the 42 cm.
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How would the equation f (x) = log (x - 2) + 8 be shifted with respect to the parent function
Answer:
right 2 up 8 units
Step-by-step explanation:
the (x-2) is basically the opposite of what u think so it goes right
and +8 is normal so its up 8
Why is it called substitution method?
It is called the substitution method because we substitute the value of one variable with another one and solve the equation.
By changing the value of another variable in an equation, the variable can be eliminated using this method. Thus, the elimination by substitution method is the name given to this technique.
As we are aware, the substitution approach entails solving the equation to determine the value of the variable, which is then substituted in the other equation. The elimination approach, in contrast, involves removing variables from the equation such that the equation's system can be reduced to being the function of a single variable.
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A current flows at 5 mph. A boat can travel 20 mi downstream in the same amount of time it can go 12 mi upstream. What is the speed of the boat in still water?
If a current flows at 5 mph. A boat can travel 20 mi downstream in the same amount of time it can go 12 mi upstream. The speed of the boat in still water is 20mph.
How to find the speed?Boat speed x mph
Current speed 5 mph
Against current x - 5 mph
With current x + 5 mph
Distance against current 12
Distance with current 20
Time = same equate the time against with time with current
t=d/r
12 /(x- 5)= 20 /(x+ 5)
12 ×(x+ 5 )= 20 ×(x- 5 )
12 x + 60= 20 x -100
12 x -20 x = -100 + -60
-8x = -160
x = -160/-8
x= 20 mph
Therefore the speed of the boat in still water is 20mph.
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pls helppppppp dont worry abt the one i have marked
Select the correct answer.
Which graphs represent functions?
-4
14
●
4-
24
+2
4
Graph A
y
4
-2
4
2
2
2
4
X
-2
F
4
2
2
>4
Graph B
4
•
2
2 4
x
►x
Answer:
Graph B
Step-by-step explanation: No point has the same x-axis
40 points Is this correct please help
is 3(5a + 9) + 5a and 4(5a+6) equivalent
Answer:
no although they were close
Step-by-step explanation:
1. During photosynthesis, what is the path or flow of energy among biotic and abiotic parts of an ecosystem?
During photosynthesis, what is the path or flow of energy among biotic and abiotic parts of an ecosystem?
35 POINTS TO WHOEVER CAN ANSWER THIS!!!!!
Answer:
y = (-5/2)x + 8
slope = -2.5
b = 8
x-intercept = 3.2
y-intercept = 8
Step-by-step explanation:
Find the area. Simplify your answer.
x+8
x+5
Answer:
9x+5
Step-by-step explanation:
Add x and 8x
Which expressions have a product of 10? Options A. 5/14x14, B. 5/6 x14, C. 3/5 x 20, 2/3 x 15. Please help this assignment is due tomorrow
The expressions that have a product of 10 are:
Option B: 5/6 x 14 = (5/6) * 14 = (514)/6 = 70/6 = 10
Option C: 3/5 x 20 = (3/5) * 20 = (320)/5 = 60/5 = 12
Option A: 5/14 x 14 = (5/14) * 14 = (514)/14 = 70/14 = 5
Option D: 2/3 x 15 = (2/3) * 15 = (215)/3 = 30/3 = 10
So options B and D have a product of 10.
What is equality in Class 8?
In Class 8, equality is likely to be defined as a state of being where all individuals are treated fairly and have the same rights, opportunities, and access to resources, regardless of factors such as race, gender, socioeconomic status, and other characteristics. Equality is often contrasted with inequality, which is a state of being where individuals or groups have unequal access to resources and opportunities.
In class 8, students may learn about different types of equality, such as:
Political equality: This refers to the idea that all individuals have an equal say in the political process and equal representation in government.
Social equality: This refers to the idea that all individuals have the same rights and opportunities, and are treated with the same respect and dignity, regardless of their background or characteristics.
Economic equality: This refers to the idea that all individuals have access to the same economic opportunities and resources, regardless of their background or characteristics.
Educational equality: This refers to the idea that all individuals have access to quality education, regardless of their background or characteristics.
Equality of opportunity: This refers to the idea that all individuals have an equal chance to succeed and reach their full potential, regardless of their background or characteristics.
In class 8, students may also learn about the importance of equality and the ways in which inequality can negatively impact individuals and society. They may also learn about different strategies for promoting equality and reducing inequality in society.
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To get snacks for a class trip, Clare went to the “bulk” section of the grocery store, where she could buy any quantity of a product and the prices are usually good.
Clare purchased some salted almonds at $6 a pound and some dried figs at $9 per pound.
She spent $75 before tax.
If she bought 1 pounds of figs, how many pounds of almond did she buy?
How many images are formed If two plane mirror are position at an angle of 450?
There are seven images are formed If two plane mirrors are positioned at an angle of 45°.
The formula for the number of images formed by two plane mirrors placed at a certain angle is
⇒360/n
where n=angle between the mirrors
The most important thing to remember is that every time the number of images formed is only ODD numbers, if the result is even then you need to subtract 1 from that.
if it is odd u need not subtract, since the number of images ever formed is only odd.
so for an angle, of 45° the number of images formed are:
⇒360/45=8
in this case, since the images formed are even u need to subtract 1 from it i.e
8–1=7
So, the number of images formed is 7.
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What is the degree of the polynomial 3x² 5x¹ 7?
The degree of the polynomial 3x² 5x¹ 7 is 2.
This can be calculated by examining the highest exponent of the variable in the polynomial. Since the highest exponent is 2, the degree of the polynomial is 2.
Formula: Degree = Highest Exponent of the Variable
To calculate the degree of the polynomial 3x² 5x¹ 7, we first need to examine the highest exponent of the variable. The highest exponent is 2, therefore, the degree of the polynomial is 2.
3x² = 3x2 = 3x2 = Degree = 2
5x¹ = 5x1 = 5x1 = Degree = 1
7 = 7 = 7 = Degree = 0
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Given f(x) = x^2 - 6x + 1, what is the domain of f(x)?
A,B,C or D?
Any values of x will result in a real value of the quadratic equation f(x), which is written as f(x). Your minimum score is therefore (3,1).
what is domain ?A function's acceptable range of values is referred to as its domain. These numbers serve as representations of the x-values of a function like f. (x). The set of potential values that a function can be applied to is known as its domain. When the x value is inserted, the method returns the value that is contained in this set. A function's definition is Y = f, where x is the independent variable and y is the dependent variable (x). If it is possible to successfully use a value of x to generate a single value of y using the value of x, that value of x .
given
Let y = f(x) be a function where x and y are the independent and dependent variables.
then that selected x-value is said to belong to the domain of f.
Any values of x will result in a real value of the quadratic equation f(x), which is written as f(x). Your minimum score is therefore (3,1).
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(-6) to the power of -2
Answer: -1/36
Step-by-step explanation:
Answer: - 1/36
Step-by-step explanation: exponet rule a^-b=1/a^
What is the difference between expression and polynomial class 8?
Expression and polynomial are both mathematical terms that are used to represent mathematical equations.
An expression is any combination of symbols, numbers, and operations that represent a value or a quantity. Expressions can be represented as algebraic equations, numerical equations, or verbal phrases. A polynomial is a specific type of expression that is composed of one or more terms with each term containing a power of a single variable. A polynomial can be represented with a single variable or multiple variables.
A formula is an expression or equation that represents an idea or a relationship between two or more variables. A calculation is the actual process of evaluating an expression or equation to determine a numerical value or answer.
In summary, an expression is any combination of symbols, numbers, and operations that represent a value or quantity, while a polynomial is a specific type of expression composed of one or more terms with each term containing a power of a single variable. A formula is an expression or equation that represents an idea or relationship between two or more variables, while a calculation is the actual process of evaluating an expression or equation to determine a numerical value or answer.
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What is a domain of the functions Mcq?
A function's range is the collection of all possible values that it can accept, and its domain is the set of all values for which the function is defined.
A value for x is said to be in the domain of a function f if it successfully allows the production of a single value y using another value for x. The following requirements are frequently verified when a function's output y-value is required to be a real number:
1. The numerator cannot be zero.
2. Even roots (square roots, etc.) radicands (expressions denoted by a radical symbol) cannot have a negative value.
3. Only positive values can be used in logarithms.
4. Physical or other real-world constraints may be placed on word problems, such as the fact that time is not negative or that the number of items is an integer.
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Find the value of the remainder. Enter the value of the remainder (only the number, you do not need to place it over the divisor)
It is simple to locate the rest. Simply divide the number by another number with its multiples to obtain the remaining.
What is a remainder?Always less than the divisor is the remainder. It indicates that the division is not complete if the remainder exceeds the divisor. It might either be higher or lower than the quotient. For instance, the remaining is six when the number seven is divided by 41, leaving five as the quotient.
It's simple to locate the rest of them. To obtain the remainder, we simply divide the number by a different integer that is one of its multiples.
the surplus after division (happens when the first number does not divide exactly by the other). Example: Dividing 19 exactly by 5 is not possible. The closest you can go to 19 without going over is 3 x 5, which is 15 and 4 less. 4 remains as a result.
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Which doesn't belong and
why?
64
289 81 363 121
The number 363 does not belong to the set.
What is Square Root?Square root of a number is the value such that the value when multiplied to itself two times gives the original number.
It is denoted by the symbol √.
For example, square root of 16 is 4 since 4 × 4 = 16
The given numbers are 64, 289, 81, 363 and 121.
The numbers 64, 289, 81 and 121 is a perfect square of integers.
64 = 8²
289 = 17²
81 = 9²
121 = 11²
But 363 is not a perfect square of any integer.
Hence the number which is odd in the list is 363.
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Taylor writes an expression with 5 terms. All 5 terms are like terms. How many terms are in the equivalent expression with the least number of terms
Answer:
An equivalent expression with the least number of terms will have 1 term if all 5 original terms are like terms. This is because like terms can be combined using the distributive property and addition, resulting in a single term with a combined coefficient.
Step-by-step explanation:
How do you solve a 3d triangle?
If the base of the prism has dimensions x and y, and the diagonal along the base is represented by c, then x² + y² = c². The longest diagonal in the solid, s, is the hypotenuse of the triangle formed by the sides c and the height of the solid, z. So we know that, c² + z² = s².
What is a triangle in 3D called?
Triangular Prism (3D shape with identical triangle bases) Cone (3D triangle with a circular base)
In the case of a cylinder, one or several of the following data must be provided: the total area/lateral area and the circumference/area of the base or the area of the base and the volume. Ex: provided the circumference of the base = 10π inches, and the lateral area = 50π inches², the equation can be written as 50π ÷ 10π = 5 inches.
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The graph shows the location of the entrance to a state park each unit on the graph represents 1 km Rodrigo uses the graph to locate a waterfall the waterfall is located 3 km to the left and 4 km down from the park entrance Rodrigo says we ordered pair for the waterfall is three, nine use the drop-down menus to explain Rodrigo
The location of the waterfall on the graph will be (3,1). Instead of going down down, Rodrigo goes 4 units upward direction.
What are the coordinates of a point?
Coordinates are a pair of integers (Cartesian coordinates), or sporadically a letter and a number, that identify a certain place on a grid, also referred to as a coordinate plane. The x-axis (horizontal) and y-axis are the two axes that make up a coordinate plane (vertical).
Given that the location of the entrance is at (6,5).
To reach the waterfall, you need to go 3 km to the left and 4 km down from the park entrance.
The x-coordinate decreases to the left side of a point and increases to the right side of a point.
The y-coordinate decreases to a downward point and increases upward a point.
The x-coordinate of the waterfall is 6 - 3 =3
The y-coordinate of the waterfall is 5 - 4 = 1.
The coordinate that represents the location of a point is (3,1).
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What is the exponential of 0?
Answer:
The exponential form of 0 is 0.
In simple words, the exponents can said to be the powers of the base. For example, in 2³, the exponent will be 3. If a number is given without any power then its power is 0. The exponential form is simply the number with exponents.