On solving the provided question we can say that the expressions are
a) 1000
b 10
c) 100000.001
d) 99000
What is expression?In mathematics, it is pοssible to multiply, divide, add, or remove. The construction of an expression is as follows: Expressiοn, number, and mathematical operatοr Numbers, variables, and functions are the building blocks of a mathematical expressiοn (such as addition, subtraction, multiplicatiοn or division etc.)
Expressiοns and phrases can be contrasted. Any mathematical statement with variables, numbers, and an arithmetic οperation between them is called an expression or an algebraic expressiοn. For instance, the expressiοn 4m + 5 has the terms 4m and 5 as well as the variable m of the supplied expressiοn, all of which are separated by the arithmetic sign +.
The expressiοns are
a)10¹⁵/10¹² = 10³ = 1000
b)10¹/10⁰ = 10/1 = 10
c)10⁵ + 10⁻³ = [tex]\dfrac{10^8 +1}{100}[/tex]= 100000.001
d)10⁵ - 10³ = 100000 - 1000 = 99000
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In a class of 50 students, 20 play hockey and 35 play football . 10 students play both hockey and football.
The number of students who either play hockey or football is given by the equation n ( A ∪ B ) = 45 students
What is Set Theory Formula?The set formula is given in general as n(A∪B) = n(A) + n(B) - n(A⋂B), where A and B are two sets and n(A∪B) shows the number of elements present in either A or B and n(A⋂B) shows the number of elements present in both A and B.
Given data ,
Let the total number of students in the class be n ( U ) = 50 students
Let the number of students who play hockey be n ( A ) = 20 students
Let the number of students who play football be n ( B ) = 35 students
Let the number of students who play hockey and football n ( A ∩ B ) = 10
So , from the set theory n(A∪B) = n(A) + n(B) - n(A⋂B)
Substituting the values in the equation , we get
The number of students who play hockey or football = 20 + 35 - 10
On simplifying the equation , we get
The number of students who play hockey or football = n( A ∪ B ) = 45 students
And , the number of students who do not play hockey or football is given by the equation = n ( U ) - n ( A ∪ B )
On simplifying the equation , we get
The number of students who do not play hockey or football = 50 - 45 = 5 students
Hence , the number of students who play hockey or football is 45 students
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A scientist has two solutions, which she has labeled Solution A and Solution B. Each contains salt. She knows that Solution A is 65% salt and Solution B is 90% salt. She wants to obtain 130 ounces of a mixture that is 70% salt. How many ounces of each solution should she use?
On solving the provided question we can say that equations are = .60x + .85y = .75(180)
What is equation?A mathematical equatiοn is a formula that joins two statements and uses the equal symbol (=) tο indicate equality. A mathematical statement that establishes the equality of twο mathematical expressiοns is known as an equation in algebra. Fοr instance, in the equatiοn 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart.
The relationship between the twο sentences on either side of a letter is described by a mathematical formula. Often, there is οnly one variable, which also serves as the symbοl. for instance, 2x – 4 = 2.
t x = the number of ounces of Sοlution A
y = the number of ounces of Sοlution B
x + y = 180
y = 180 - x
Solving equation -
0.60 + 0.85y = 0.75(180)
0.60 + 0.85y = 135
60x + 85y = 13500
60x + 85(180-x) = 13500
60x + 15300 - 85x = 13500
-25x = -1800
x = 72ounces
y = 180 - 72
y = 108 ounces
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For a certain company, the cost function for producing x
items is C(x)=30x+200
and the revenue function for selling x
items is R(x)=−0.5(x−80)2+3,200
. The maximum capacity of the company is 130
items.
The profit function P(x)
is the revenue function R(x)
(how much it takes in) minus the cost function C(x)
(how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit!
Answers to some of the questions are given below so that you can check your work.
Assuming that the company sells all that it produces, what is the profit function?
P(x)=
Preview Change entry mode .
Hint: Profit = Revenue - Cost as we examined in Discussion 3.
What is the domain of P(x)
?
Hint: Does calculating P(x)
make sense when x=−10
or x=1,000
?
The company can choose to produce either 50
or 60
items. What is their profit for each case, and which level of production should they choose?
Profit when producing 50
items =
Number
Profit when producing 60
items =
Number
Can you explain, from our model, why the company makes less profit when producing 10 more units?
Answer: The profit function P(x) can be found by subtracting the cost function C(x) from the revenue function R(x), so:
P(x) = R(x) - C(x) = (-0.5(x-80)^2 + 3,200) - (30x + 200)
The domain of P(x) is the set of all possible values of x for which the profit calculation makes sense. In this case, the company has a maximum capacity of 130 items, so the domain of P(x) is 0 <= x <= 130.
To calculate the profit when producing 50 items and 60 items, we simply plug in these values for x in the profit function:
Profit when producing 50 items = P(50) = (-0.5(50-80)^2 + 3,200) - (30 * 50 + 200) = $2350
Profit when producing 60 items = P(60) = (-0.5(60-80)^2 + 3,200) - (30 * 60 + 200) = $2280
Since the profit is higher for producing 50 items, the company should choose to produce 50 items.
From our model, we can see that as the production increases, the cost also increases linearly with a slope of 30, while the revenue increases parabolically but with a negative slope, meaning that after reaching a certain point, the increase in revenue becomes slower compared to the increase in cost. This is why the profit decreases as the production increases, and therefore the company makes less profit when producing 10 more units.
Step-by-step explanation:
Looking for answer key to Gina Wilson All Things Algebra Unit 3 Parallel and Perpendicular Lines Homework 5 Slopes of lines; Parallel and Perpendicular Lines
It's important to work through the problems yourself to improve your understanding and problem-solving skills.
For the unit 3 homework on parallel and perpendicular lines, you should have learned about the slope of lines, how to find the slope of a line given two points, and the relationship between parallel and perpendicular lines.
To find the slope of a line given two points, you can use the slope formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.
To determine if two lines are parallel, you can compare their slopes. Parallel lines have the same slope.
To determine if two lines are perpendicular, you can use the fact that the product of their slopes is -1. That is, if m1 and m2 are the slopes of two lines, and m1 * m2 = -1, then the lines are perpendicular.
It's important to practice working through problems using these concepts and formulas to develop your skills and understanding. Additionally, you can seek assistance from your teacher or tutor if you have specific questions or concerns.
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The terminal side of θ in standard position contains the point (–14, 0). Find the exact values of the six trigonometric functions of θ.
sin(θ) =
cos(θ) =
tan(θ) =
csc(θ) =
sec(θ) =
cot(θ) =
The exact values of the six trigonometric functions of θ are:
sin(θ) = 0
cos(θ) = -√2/2
tan(θ) = 0
csc(θ) = undefined
sec(θ) = -√2
cot(θ) = undefined
The given point (-14,0) lies on the x-axis in the negative x-axis direction. Drawing a line from the origin to this point creates a right triangle with the hypotenuse being the line from the origin to (-14,0), the opposite side being the y-coordinate of (-14,0), and the adjacent side being the x-coordinate of (-14,0). The length of the hypotenuse can be found using the Pythagorean theorem:
h^2 = (-14)^2 + 0^2
h^2 = 196
h = 14√2
Using the definitions of the six trigonometric functions in terms of the opposite, adjacent, and hypotenuse sides of a right triangle, we can find the exact values of these functions for θ:
sin(θ) = opposite/hypotenuse = 0/14√2 = 0
cos(θ) = adjacent/hypotenuse = -14/14√2 = -√2/2
tan(θ) = opposite/adjacent = 0/-14 = 0
csc(θ) = 1/sin(θ) = undefined (since sin(θ) = 0)
sec(θ) = 1/cos(θ) = -√2
cot(θ) = 1/tan(θ) = undefined (since tan(θ) = 0)
Therefore, the exact values of the six trigonometric functions of θ are:
sin(θ) = 0
cos(θ) = -√2/2
tan(θ) = 0
csc(θ) = undefined
sec(θ) = -√2
cot(θ) = undefined
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There are 12 consecutive parking slots available in a hotel parking lot in how many ways can four distinct cars be packed so that at least one parking slot remains vacant between into cars
Answer:
4 different spots
Step-by-step explanation:
12 - 4 spaces in between = 8 spaces
8 spaces / 4 cars = 4 spots
4 spots and 4 cars = 4 different spots for each car
16
A sprinter starts from rest, and accelerates at 1.87 m/s^2 over a distance of 14.0 m. What is a her final velocity?
Describe the long run behavior of f(p)=(p+1)^3(p+4)^3(p-1)
Answer:
Step-by-step explanation:
The long-run behavior of a function can be determined by examining its asymptotes and the behavior of the function near these asymptotes.
For the function f(p)=(p+1)^3(p+4)^3(p-1), there are three points to consider as possible asymptotes: p = -1, p = -4, and p = 1.
At p = -1, the factor (p-1) is equal to zero, so the function has a vertical asymptote there. This means that as p approaches -1 from either side, the function will approach infinity.
At p = -4, the factor (p+4)^3 is equal to zero, so the function also has a vertical asymptote there. Similarly, as p approaches -4 from either side, the function will approach infinity.
At p = 1, the factor (p+1)^3 is equal to zero, so the function has another vertical asymptote there. In this case, as p approaches 1 from either side, the function will approach negative infinity.
So, the long-run behavior of the function can be described as follows:
The function approaches infinity as p approaches -1 or -4 from either side.
The function approaches negative infinity as p approaches 1 from either side.
Therefore, the long-run behavior of f(p)=(p+1)^3(p+4)^3(p-1) is characterized by three vertical asymptotes at p = -1, p = -4, and p = 1.
Answer:
[tex]\text{As } p \to -\infty, f(p) \to -\infty\\\\\text{As } p \to \infty, f(p) \to \infty[/tex]
====================================================
Explanation:
Each factor is a cubic of degree 3.
The degree of the entire polynomial is 3+3+3 = 9.
Since the degree is odd and the leading coefficient is positive, this means the graph falls to the left and rises to the right.
"falls to the left" means [tex]\text{As } p \to -\infty, f(p) \to -\infty[/tex]
"rises to the right" means [tex]\text{As } p \to \infty, f(p) \to \infty[/tex]
This is verified using a graphing tool like Desmos as shown in the screenshot below. I'm using x in place of p.
4x
Store A $25 for a 45 minute lesson
Store B. $30 for a 1 hour lesson
Store C $45 for a 1.5 hour lesson
Store D. $80 for three 1 hour lessons
Julle wants to purchase swim lessons for her son. There are four different stores that offer swim lessons. Based on the lowest cost per minute, which store has the BEST dea
4x A
4x B
x
C
4x D
Store B
Store A
Store D
Store C
The store with the best deal is given as follows:
Store D.
How to obtain the store with the best deal?The store with the best deal is obtained applying the proportions in the context of the problem.
The best deal is the store with the lowest cost per hour, and the cost per hour is obtained dividing the total cost by the number of hours.
Hence the hourly cost for each store is obtained as follows:
Store A: 25/0.75 = $33.3 per hour. (45 minutes = 45/60 hour = 0.75 hour).Store B: $30/1 = $30 per hour.Store C: $45/1.5 = $30 per hour.Store D: $80/3 = $26.7 per hour -> best deal.More can be learned about proportions at https://brainly.com/question/24372153
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6 There are 2 blue chips for every 3 red chips. If there are 20 total chips, how many blue chips and red chips are in the bucket?
Answer:
12 blue chips 8 red chips.
Step-by-step explanation:
Let's call the number of blue chips "b" and the number of red chips "r". We know that 2 blue chips for every 3 red chips, so we can set up the following equation:
2b = 3r
And we know that there are 20 total chips, so:
b + r = 20
Now we can use substitution to find the value of b. We'll substitute the expression for r from the first equation:
2b = 3(20 - b)
Expanding the right side:
2b = 60 - 3b
Combining like terms:
5b = 60
And finally, solving for b:
b = 12
So there are 12 blue chips and 20 - 12 = 8 red chips.
Which of the following is equivalent to the expression below?
√-98
A. -2i√7
B. 2i√7
c. 7i√2
D. -7i√2
Answer: Choice C) [tex]7i\sqrt{2}[/tex]
Work Shown:
[tex]\sqrt{-98} = \sqrt{-1*49*2}\\\\\sqrt{-98} = \sqrt{-1}*\sqrt{49}*\sqrt{2}\\\\\sqrt{-98} = i*7*\sqrt{2}\\\\\sqrt{-98} = 7i\sqrt{2}\\\\[/tex]
Which decimal is represented by the red portion of the picture?
Answer:
D
Step-by-step explanation: I did it
The ratio 1500:1200 in its lowest terms is
Answer:
Step-by-step explanation:
To put a ratio in its lowest terms, we need to divide both terms of the ratio by their greatest common divisor (GCD). The GCD of 1500 and 1200 is 300, which is the largest number that divides both numbers evenly.
Dividing both terms of the ratio by 300, we get:
1500 / 300 : 1200 / 300 = 5 : 4
So, the ratio 1500:1200 in its lowest terms is 5:4.
What is the measure of each angle in a regular polygon with 14 sides? If necessary, round your answer to the nearest tenth.
Answer:
154.3° (to the nearest tenth)
Step-by-step explanation:
1 exterior angle=360/number of sides
360/14=25.714285
To get the one angle within the 14 sided polygon, take 1 exterior angle away from 180, as they are on a straight line
180-25.714285=154.285714
To the nearest tenth:
154.3°
Only using positive exponents, what is the simplified form of the expression p−6q7r/p2q−2
Step-by-step explanation:
Here is it. I hope it will help
show that the following function are continuos and discontinuos
A.(f×)={×+1 ,for×less than or equai to 2
{ 2×–1 ,for 1<×<2
{×–1 ×<1
f(x) is discontinuous.
Since LHS ≠ RHS ≠ f(x).
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
f(x) = x + 1, for x ≤ 2
f(x) = 2x - 1, for 1 < x < 2
f(x) = x - 1, for x < 1
Now,
A x = 1
LHS of f(x) = [tex]\lim_{x\to1^-}[/tex] f((h - 1)) = [tex]\lim_{h \to 0}[/tex] (h - 1 - 1) = 0 -2 = -2
RHS of f(x) = [tex]\lim_{x\to1^+}[/tex] f(h + 1) = [tex]\lim_{h \to 0}[/tex] (2(h + 1) - 1) = 2h + 2 - 1 = 0 + 1
f(1) = x + 1 = 1 + 1 = 2
We see that,
LHS ≠ RHS ≠ f(x)
So,
f(x) is not continuous, it is discontinuous.
Thus,
f(x) is discontinuous.
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The complete question.
Show that the following functions are continuous or discontinuous at x = 1.
f(x) = x + 1, for x ≤ 2
f(x) = 2x - 1, for 1 < x < 2
f(x) = x - 1, for x < 1
A certain antihistamine is often prescribed for allergies. A typical dose for a 100-pound person is 19 mg every six hours. Complete parts (a) and (b) below. a. Following this dosage, how many 13.2 mg chewable tablets would be taken in a week? b. This antihistamine also comes in a liquid form with a concentration of 13.2 mg/ 10mL. Following the prescribed dosage, how much liquid antihistamine should a 100-pound person take in a week?
A 100-pound person could take mL in a week? this is the question i need answered the most...thank you
Answer: First, let's find out how many times a 100-pound person would take the antihistamine in a week:
Every 6 hours * 24 hours/day = 4 times a day
4 times a day * 7 days/week = 28 times a week
Next, let's find the number of 13.2 mg chewable tablets taken in a week:
A typical dose of 19 mg * 28 doses/week = 532 mg
532 mg / 13.2 mg/tablet = 40 tablets
So, a 100-pound person would take 40 chewable tablets in a week following the prescribed dosage.
Now, let's find the amount of liquid antihistamine to be taken in a week:
A typical dose of 19 mg * 28 doses/week = 532 mg
532 mg / 13.2 mg/10 mL = 40.15 mL
So, a 100-pound person would take 40.15 mL of liquid antihistamine in a week following the prescribed dosage.
Step-by-step explanation:
DATE
10. If you laid the radius end to end around the circumference of a circle A, how many radii would
it take to fit exactly?
Answer:
Step-by-step explanation:
Step 1: Understand the formula for the circumference of a circle
The formula for the circumference of a circle is given by:
C = 2 * π * r
where C is the circumference of the circle, π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the circle. This formula expresses the relationship between the radius and the circumference of a circle.
Step 2: Substitute the radius of the circle for "r" in the formula
Let's say the radius of the circle is "r". Substitute "r" for "r" in the formula for the circumference:
C = 2 * π * r
Step 3: Divide the circumference by the radius
Now that you have the formula for the circumference, divide the circumference by the radius:
C/r = 2 * π
Step 4: Simplify the expression
The expression on the right side of the equation can be simplified to 2 times pi:
C/r = 2 * π
C/r = 2 * 3.14 = 6.28
Step 5: Conclusion
Therefore, it would take 6.28 radii laid end to end to fit exactly around the circumference of the circle.
Note: This answer is only an approximation, as the value of π is an irrational number that cannot be expressed exactly as a fraction or a decimal.
A function is given.
g(x) =
2
x
; x = 1, x = a
(a) Determine the net change between the given values of the variable.
(b) Determine the average rate of change between the given values of the variable.
According to the given function net change and average rate of change between the given values of the variable.
A. 2
B. 2/a
What is the net change?A rate of change's integral is taken into account by the net change theorem. According to this, a quantity's new value equals its initial value plus the integral of its rate of change whenever it changes. There are two methods to formulate the equation. The second is more widely known and is just the definite integral.
What does a function's net change look like?The (definite) integral of a function's derivative, in other words, represents the overall change in a function. In specifically, the integral of velocity is the net distance travelled (final position minus initial position). The integral of acceleration is the difference between the final velocity and the starting velocity.
According to the given information:g(x) =2/x
x = 1
x = a
A.
g(x) =2/x
g(1) =2/x
g(1) =2
B.
g(x) =2/x
g(a) =2/x
g(a) =2/a
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Write an equation of the circle graphed below
Answer:
Below
Step-by-step explanation:
Center (h,k) is - 2, -2 radius is 1
Equation for a circle with center (h,k) and radius r is :
(x-h)^2 + (y- k )^2 = r^2 <====you have to learn/remember this
Plug in the values above to get
(x+2)^2 + (y+2)^2 = 1
Answer:
(x+2)^2+(y+2)^2=0.5625
Step-by-step explanation:
trust
Question 4 of 10
Which of the following is a root of the polynomial shown below?
f(x) = x³ + 2x²-x-2
OA. 0
OB. 2
O C. 1
OD. 3
SUBMIT
Answer:
1
Step-by-step explanation:
f(x)=x3+2x2-x-2
...................
The Roots of the given polynomials shown below f(x) = x^3+2x^2-x-2 are -1, 1, -2.
What is a polynomial?They are mathematical expressions involving variables raised with non negative integers and coefficients(constants who are in multiplication with those variables) and constants with only operations of addition, subtraction, multiplication and nonnegative exponentiation of variables involved.
Example: x² + 3x + 3 is a polynomial.
Since, The given polynomials is,
f(x) = x³ + 2x²-x-2
Now, Solve as;
f(x) = x³ + 2x²-x-2
f (x) = x² (x + 2) - 1 (x + 2)
f (x) = (x² - 1) (x + 2)
Hence, The Roots of the polynomials : -1, 1, -2
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Solve the quadratic equation by the square root method and write the solution in radical form. simplify the solution. 4y^2=80
Answer:
[tex]y=\pm2\sqrt{5}[/tex]
Step-by-step explanation:
[tex]4y^2=80\\y^2=20\\y=\pm\sqrt{20}\\y=\pm2\sqrt{5}[/tex]
The square root method for solving a quadratic equation involves isolating the squared term, and then taking the square root of both sides of the equation.
Starting with the equation 4y^2 = 80, we can isolate the squared term by dividing both sides of the equation by 4:
y^2 = 20
Next, we take the square root of both sides of the equation:
y = ±√20
The solution in radical form is y = ±√20. To simplify the radical, we can factor 20 into 2 * 10. Then, we can simplify the square root as follows:
y = ±√2 * √10 = ±2√10
So, the simplified solution is y = ±2√10.
Multiply. (Simplify your answer
(-5y4z)(-7y6z³)
Answer:
35y^(10)z^(4)
Step-by-step explanation:
Solution Given:
(-5y^(4)z)(-7y^(6)z³)
Here while opening bracket
If their is like terms they can be multiplied ,divided,subtracted as well as added.
So
Here
opening bracket and keeping in serially
-5*-7*y^(4)*y^(6)*z*z^(3)
here if their is like terms their power is added.
And solving remaining one.
35y^(4+6)z^(1+3)
Here -5*-7=35
Again
Solving it.
35y^(10)z^(4)
Answer:
[tex]35y^{10}z^4[/tex]
Step-by-step explanation:
Given expression:
[tex](-5y^4z)(-7y^6z^3)[/tex]
Apply the rule: (-a) · (-b) = ab
[tex]\implies 5y^4z \cdot 7y^6z^3[/tex]
Multiply the numbers: 5 · 7 = 35
[tex]\implies 35y^4zy^6z^3[/tex]
Collect like terms:
[tex]\implies 35y^4y^6z^3z[/tex]
[tex]\textsf{Apply exponent rule:} \quad a^b \cdot a^c=a^{b+c}[/tex]
[tex]\implies 35y^{(4+6)}z^{(3+1)}[/tex]
Simplify the exponents:
[tex]\implies 35y^{10}z^4[/tex]
This question I can not figure out Someone please help.
Answer:
x y
0 -2
4 -1
graph: should look like the picture attached
explanation:
Mary plans to buy a kettle for R180-00, excluding vat.vat is charged at 15%.what is the total cost of the kettle vat is added
Answer:
R207
Step-by-step explanation:
R180 times 15% =27
27+180=207, therefore, R270 ir correct
Alba Isabel bought a Bluray disc player on sale for $219.99. What is the sales tax if Alba lives in Kingston, New York, where the state tax is 4% and the county tax is 4%?
The sales tax on the Bluray disc player is $17.60.
What is percentage ?
Percentage can be defined as product of ratio of given value, total value and hundred.
To calculate the sales tax on the Bluray disc player, we need to add the state tax and county tax to the sale price of the player.
First, we can calculate the amount of the state tax. The state tax rate is 4%, which can be expressed as a decimal by dividing by 100:
State tax rate = 4% = 0.04
To find the amount of state tax, we can multiply the sale price by the state tax rate:
State tax = 0.04 x $219.99 = $8.80
Next, we can calculate the amount of the county tax. The county tax rate is also 4%, so we can use the same method as above to find the county tax:
County tax rate = 4% = 0.04
County tax = 0.04 x $219.99 = $8.80
Now, we can find the total sales tax by adding the state tax and county tax:
Total sales tax = State tax + County tax
= $8.80 + $8.80
= $17.60
Therefore, the sales tax on the Bluray disc player is $17.60.
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Line CD passes through points (0, 2) and (4, 6). Which equation represents line CD?
Oy=2x-2
Oy = 2x + 2
Oy=x+2
Oy=x-2
The required equation of the line representing line CD is y = x + 2. Option C is correct.
The equation of the line passing through two points is given by the equation,
y - y₁ = (y₂ - y₁) / (x₂ - x₁) (x - x₁) - - - -- -(1)
Here,
Line CD passes through points (0, 2) and (4, 6)
Let (0, 2) and (4, 6) equal to (x₁, y₁) and (x₂, y₂) respectively.
Substitute this point in equation 1 to get the equation of the line representing CD,
So,
y - 2 = (6 - 2)/(4-0) (x - 0)
y = x + 2
Thus, the required equation of the line representing line CD is y = x + 2. Option C is correct.
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If X and Y are independent and identically distributed uniform random variables on (0, 1), compute the joint density of
(a) U = X + Y, V = X/Y
(b) U = X, V = X/Y
(c) U = X + Y, V = X/(X+Y)
If X and Y are independent and identically distributed uniform random variables on (0, 1), then the joint density of
(a) U = X + Y, V = X/Y is u/v² for v > 1 and 0 ≤ u ≤ 1.
(b) U = X, V = X/Y is 1/v for v > 1 and 0 ≤ u ≤ 1.
(c) U = X + Y, V = X/(X+Y) is v/(1+v)² for v > 0 and v/(1+v) ≤ u ≤ 1.
In probability theory, joint density is a mathematical function that describes the probability distribution of two or more random variables. It represents the probability of occurrence of different values of those variables in a particular region of space. In this problem, we have to calculate the joint density of U and V, where U and V are functions of X and Y.
(a) For U = X + Y and V = X/Y, we can find the joint density as follows:
First, we need to find the distribution of U and V separately. Since X and Y are independent and identically distributed uniform random variables on (0, 1), their individual probability density functions are f(x) = 1 for 0 ≤ x ≤ 1.
To find the density of U, we can use the convolution formula, which states that the density of the sum of two independent random variables is the convolution of their individual densities. Thus,
fU(u) = ∫ fX(u - y)fY(y) dy
= ∫ 1 dy
= u for 0 ≤ u ≤ 1.
Next, to find the density of V, we need to transform X and Y using the change of variables formula. Let Z = X/Y, then
fV(v) = fZ(z)|dz/dv|
= fX(vz)/(z²) |z|
= 1/(v²) for v > 1.
Therefore, the joint density of U and V is given by the product of their individual densities:
fUV(u,v) = fU(u) fV(v)
= u/v² for v > 1 and 0 ≤ u ≤ 1.
(b) For U = X and V = X/Y, we can similarly find the joint density as:
fUV(u,v) = fX(u) fV(v)
= 1/v for v > 1 and 0 ≤ u ≤ 1.
(c) For U = X + Y and V = X/(X+Y), we can use the same method as in (a) to find the individual densities of U and V:
fU(u) = u for 0 ≤ u ≤ 1,
fV(v) = v/(1+v)² for v > 0.
Then, the joint density of U and V is given by:
fUV(u,v) = fU(u) fV(v) |du/dx|
= v/(1+v)² for v > 0 and v/(1+v) ≤ u ≤ 1.
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At the 2022 Winter Olympics, one country won a total of 150 medals. A circle graph of the medals is shown.
a circle graph titled 2022 Winter Olympic Medals, with three sections labeled gold 20 percent, silver 30 percent, and bronze 50 percent
How many gold and bronze medals were won?
50
70
105
120
Answer:
120
Step-by-step explanation:
The total number of medals won is 150, and the circle graph shows that the proportion of medals won as gold is 20 percent, as silver is 30 percent, and as bronze is 50 percent.
To find the number of gold and bronze medals won, we need to calculate the total number of medals won in each category by multiplying the total number of medals by the proportion of medals won in each category.
Gold medals: 150 * 20% = 30
Bronze medals: 150 * 50% = 75
So, a total of 30 + 75 = 105 medals were won as gold and bronze.
