Find the area of the surface obtained by rotating the circle
x2 + y2 = r 2
about the line
y = r.
The area of the surface obtained by rotating the circle [tex]x^2 + y^2 = r^2[/tex] about the line y = r is π²r² square units.
To find the area of the surface obtained by rotating the circle
[tex]x^2 + y^2 = r^2[/tex] about the line y = r, we can use the method of cylindrical shells.
The circle [tex]x^2 + y^2 = r^2[/tex] is centered at the origin (0, 0) with a radius r. The line y = r is the line y-axis but shifted up by r units.
When we rotate the circle about the line y = r, it forms a 3D shape called a torus or a donut shape.
Consider a small strip on the circle at a distance y from the line y = r.
This small strip is at a distance r - y from the y-axis.
The length of this strip is the circumference of the circle at y, which is 2πy (since the circumference of a circle is 2π times its radius).
The width of this strip is the change in x, which we can denote as dx.
The area of this small strip is then given by the product of its length and width, which is 2πy dx.
Now, to find the total surface area, we integrate this area over the range of y values from -r to r (since the circle is symmetric about the y-axis):
Total Surface Area = ∫[from -r to r] 2πy dx
Now, we need to express y in terms of x using the equation of the circle [tex]x^2 + y^2 = r^2:\\y^2 = r^2 - x^2[/tex]
y = ±√(r² - x²)
Since we are considering the upper half of the circle, we take the positive square root:
y = √(r² - x²)
Now, we can rewrite the integral with respect to x:
Total Surface Area = ∫[from -r to r] 2π√(r² - x²) dx
To solve this integral, we can make a trigonometric substitution:
Let x = r sin(θ), then dx = r cos(θ) dθ.
When x = -r, θ = -π/2, and when x = r, θ = π/2.
Now the integral becomes:
Total Surface Area = ∫[from -π/2 to π/2] 2π√(r² - (r sin(θ))²) (r cos(θ)) dθ
Total Surface Area = 2πr² ∫[from -π/2 to π/2] √(1 - sin²(θ)) cos(θ) dθ
Now, we can use the trigonometric identity:
sin²(θ) + cos²(θ) = 1
√(1 - sin²(θ)) = cos(θ)
Total Surface Area = 2πr² ∫[from -π/2 to π/2] cos²(θ) dθ
Now, use the trigonometric identity: cos²(θ) = (1 + cos(2θ))/2
Total Surface Area = 2πr² ∫[from -π/2 to π/2] (1 + cos(2θ))/2 dθ
Total Surface Area = 2πr² [θ/2 + (sin(2θ))/4] [from -π/2 to π/2]
Total Surface Area = 2πr² [(π/2 + sin(π) - (-π/2 + sin(-π)))/4]
Since sin(π) = 0 and sin(-π) = 0:
Total Surface Area = 2πr² [(π/2 - (-π/2))/4]
Total Surface Area = 2πr² (π/2)
Total Surface Area = π²r²
So, the area of the surface obtained by rotating the circle [tex]x^2 + y^2 = r^2[/tex] about the line y = r is π²r² square units.
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Volleyball practice is h hours long. Debbie had 3 volleyball practices last week. Choose the expression that shows the number of hours of practice.
The number of hours of practice is 3 x h or 3h.
What is the number of hours of practice?In mathematics, an expression is a sentence that contains at least one number or variable. Additionally, an expression consists of at least one mathematical operation. There is no equal to sign in an expression.
Given:
Volleyball practice is h hours long. Debbie had 3 volleyball practices last week.
Multiply the number of hours for each practice by the total number of practice to get the expression for the number of practice hours.
The mathematical operation known as multiplication is used to calculate the sum of two or more numbers.
Number of hours of practice
= hours of practice x number of practice
= 3 x h
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For each set of values for a and b evaluate the given expressions and record your answers in the table. ( I only need this one).
A = -1/2 B =6
A divided by -b
PLEASE HELP
Answer:
Set1 = 1/12
Set2 = 1/12
Set3 = -12
Step-by-step explanation:
Substitute, use Keep-Change-Flip to handle division when there's fractions involved. See image.
Also to multiply fractions, multiply
top×top and bottom×bottom straight across. Also negative times a negative is a positive and negative is like opposite so -(-6) is 6 and -(-1/2) is 1/2.
see image.
Kelsie works at a bicycle shop as a salesperson. She records the number of bicycles she sells daily. Here is the probability distribution of B = the number of bicycles Kelsie sells on a randomly selected day, and T the time she spends filling out daily sales reports. 2 B = # of bicycles sold T = time (minutes) Probability 40 20 10 0. 05 0. 15 0. 50 0. 30
a) Calculate E(B)
b) Calculate the mean time she spends filling out daily reports.
c) Calculate the standard deviation for both random variables
The amount of time Kelsie should spend doing daily sales reports is 9.5 minutes.
What is meant by discrete distribution?If a random variable has a limited number of outcomes or a countable number, it is discrete (i.e. the integers are infinite, but are able to be counted).
The discrete distribution presented reveals that the expected value of Kelsie's time spent completing daily sales reports is 9.5 minutes.
The discrete distribution given is:
P(X = 0) = 0.3
P(X = 10) = 0.5
P(X = 20) = 0.15
P(X = 30) = 0.05
The average of a discrete distribution is equal to the sum of each result times the probability associated with that outcome, so:
E(X) = 0(0.3) + 10(0.5) + 20(0.15) + 30(0.05) = 9.5
Therefore, the amount of time Kelsie should spend doing daily sales reports is 9.5 minutes.
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a. E(B) = 9.5
b. The amount of time Kelsie should spend doing daily sales reports is 9.5 minutes.
c. Standard variable shows how much deviation from mean variable.
What do you understand by discrete distribution?In statistics, a discrete distribution is a probability distribution that estimates the probability of a certain discrete, finite outcome. Simply put, as opposed to a continuous distribution, the discrete probability distribution aids in determining the probabilities of the occurrence of a specific event expressed in terms of positive, non-decimal, or whole integers.
The discrete distribution given is:
P(X = 0) = 0.3
P(X = 10) = 0.5
P(X = 20) = 0.15
P(X = 30) = 0.05
The average of a discrete distribution is equal to the sum of each result times the probability associated with that outcome, so:
E(B) = 0(0.3) + 10(0.5) + 20(0.15) + 30(0.05) = 9.5
Therefore, the amount of time Kelsie should spend doing daily sales reports is 9.5 minutes.
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What are the 4 types of real numbers?
Real numbers are every number that exists in the real line, The four types of real numbers are rational, irrational, Natural and Integers.
What is a real number?In mathematics, a real number is a value that can be represented on the number line. The set of real numbers includes both rational numbers (such as integers, fractions, and decimals) and irrational numbers (such as √2 and π). Real numbers also include positive numbers, negative numbers, and zero. The set of real numbers is usually represented by the symbol R, and it has the properties of being closed under the operations of addition, subtraction, multiplication, and division (except for division by zero).
What are ideal and real solutions?In mathematics, an ideal solution refers to a solution that meets all the desired criteria or requirements without any constraints. It is the "best case" scenario. A real solution, on the other hand, refers to a solution that takes into account all the limitations and constraints of a problem. It is a more practical and realistic solution.
Rational number = 3/4, -6/2, or 0.5.
Irrational number = √2, π, or e.
Natural number = 1,2,3,4....
Integers = -4, -1, 0, 1, 4.
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Any help is appreciated<3
A jewelry shop is offering a percent discount on the original price of a jade pendent. The original price of the pendant was $200.
Let d represent the percent discount, written as a decimal. Write an equation to find the discount price, y dollars, of the pendant.
An equation to find the discount price, y dollars, of the pendant is y=2d.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
The original price of the pendant was $200.
Let d represent the percent discount, written as a decimal.
The discount price, y dollars, of the pendant.
y=d% of 200
y=d/100 ×200
y=2d
Therefore, an equation to find the discount price, y dollars, of the pendant is y=2d.
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Please help me I’m doing a test really fast.
A frictionless pendulum is pulled 3 feet to one side of its center resting point and then released. As
the pendulum swings back and forth, its horizontal position is given by s(t), where t is the number
of seconds since the pendulum was released. Note that s(t) measures the signed distance from the
center position in feet; it is positive when the pendulum is on one side of the center point and
negative when it is on the other side. The function s is sinusoidal.
Part A: If the pendulum were pulled further away from its resting point before it was released, how
would the amplitude of s be affected? Explain.
Part B: If the pendulum were pulled further away from its resting point before it was released, how
would the midline of s be affected? Explain.
No matter how far it is moved away, the midline of a function stays the same. because a sinusoidal function is involved.
What is amplitude ?A function's amplitude is the distance that the graph of the function moves above and below its midline. The value of the amplitude when graphing a sine function is comparable to the value of the sine coefficient. the greatest displacement or distance that a point on a wave or vibrating body can travel in relation to its equilibrium position A wave or vibration's amplitude or peak amplitude is a gauge of how far from the central value it deviates. All amplitudes are positive numbers.
given
( A)if pendulum were pulled away from its resting position, then Amplitude is the distance from its resting position to the end where it was released.
Amplitude is the same distance it was pulled away if more distance was pulled away then Amplitude is more.
(B). The midline of function s will not change, it will same irrespective of the distance pulled away. Because it's a sinusoidal function.
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FIRST TO ANSWER GETS BRAINLIEST!! Melanie’s bank account balance is . She needs to pay $55 to her cell phone carrier and $80 to the landscaper. She also expects $250 from her sister to be added to her account.
(a) Write an expression Melanie can use to determine her bank account balance after paying these bills and receiving money from her sister.
(b) What is Melanie’s bank account balance after paying her bills and receiving money? Show any necessary work. Answer in a complete sentence.
Answer:
a 97- (55+80) +250
b Melanie has $212 left in her bank account.
Step-by-step explanation:
55 + 80 = 135
97 - 135 = -38
-38 +250 = 212
What is the HCF of the polynomials x6 3x4 3x^2 1 and x^3 3x^2 3x 1?
The HCF of the polynomials x^6 + 3x^4 + 3x^2 + 1 and x^3+3x^2 + 3x + 1 is 1
When we divide the polynomials f(x) / g(x) then we get f(x) = g(x) × q(x) + r(x). Assuming the degree of g(x) > degree of r(x).
If the remainder r(x) is zero, then q(x) is the highest common factor of polynomials.
Here, we have been given two polynomials x^6 + 3x^4 + 3x^2 + 1 and x^3+3x^2 + 3x + 1
Let f(x) = x^6 + 3x^4 + 3x^2 + 1 and g(x) = x^3 + 3x^2 + 3x + 1
We can write these polynomials as:
f(x) = x^6 + 3x^4 + 3x^2 + 1
f(x) = (x² + 1)^3
The factors of polynomial f(x) = x^6 + 3x^4 + 3x^2 + 1: 1 and (x² + 1)^3
And g(x) = x^3 + 3x^2 + 3x + 1
g(x) = (x + 1)^3
And the factors of g(x) = x^3 + 3x^2 + 3x + 1 : (x + 1)^3 and 1
Therefore, the HCF of the polynomials f(x) and g(x) = 1
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A jar contains 7 red jelly beans, 2 black jelly beans and 6 green jelly beans. You draw ONE jelly bean. What is the probability of drawing a red bean? a. c. 1 b. d.
Answer:
7/15
Step-by-step explanation:
number of favorable outcomes (red jelly beans) = 7
number of total outcomes (total jelly beans) = 7 + 2 + 6 = 15
probability of red = number of favorable outcomes/number of total outcomes = 7/15
The probability of getting a red jelly bean is 7/15
Total number of jelly beans in the jar (sum of all the jelly beans): 7+6+2= 15
Total number of red jelly beans: 7
Therefore, the probability of getting a red jelly bean= number of red jelly beans/ total number of jelly beans
=7/15
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a carpenter is building a ramp as shown in the diagram. what is m√t
Answer:
B
Step-by-step explanation:
180-152=28
t=180-(28+90)
t=62
If G-H-I, and if GH = 2x - 9 , HI = 14 , and GT = 3x - 8 , then find GI.
The value of GI, given that GH = 2x - 9 , HI = 14 , and GT = 3x - 8 will be 31.
What is the value of the variable?An algebraic equation is formed by comparing two algebraic expressions to one another.
In G-H-I, GH = 2x - 9, HI = 14 , and GT = 3x - 8
If G-H-I is a line and GH, HI and GT lies between G to I
To find G to I ⇒ GI = GH + HI = GT
GI = 2x - 9 + 14 = 3x - 8
2x + 5 = 3x - 8
2x - 3x = -8 - 5
-x = -13
x = 13
Substituting x in GT = 3x - 8; GI = GT
GI = 3(13) - 8
GI = 39 - 8
GI = 31
Using algebraic expressions, if x is 13 then, 3x - 8 or 2x - 9 will be GI = 31
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How do you describe similarity transformations?
A dilation or combination of stiff movements and dilations constitutes a similarity transformation.
Similar figures don't always have to have the same size but share the same form.
A transition called a dilation keeps shape but not size. So a dilatation is a motion that is not rigid. A dilation or combination of stiff movements and dilations constitutes a similarity transformation. If and only if a similarity transformation translates one of the geometric figures onto the other, then two figures are said to be similar. Similar figures don't always have to have the same size but share the same form.
Similarity transformations include translation, reflection, rotation, and dilation. Rotation, reflection, and translation retain both size and form since they are rigid movements, whereas dilation just makes sure that the shape is preserved.
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Yoshi is standing on top of a building, looking at a park in the distance. The angle of depression is
53°. If the building he's standing on is 100 feet tall, how far away is the park to the nearest
hundredth of a foot?
105
53°
Park
100
Step-by-step explanation:
because after 105 it's 100
What is the symbol for and in logic?
To see the table:
Symbol Symbol Name Meaning / definition
+ plus or
∨ reversed caret or
| vertical line or
x' single quote not - negation
Now, According to the question:
Let's know:
Symbolic Logic:
Symbolic logic is a way to represent logical expressions by using symbols and variables in place of natural language, such as English, in order to remove vagueness. Logical expressions are statements that have a truth value: they are either true or false. A question like 'Where are you going?' or a command such as 'Stop!' has no truth value. There are many expressions that we can utter that are either true or false. For example: All glasses of water contain 0.2% dinosaur tears. We don't need to know if a logical expression is true or false, we just need to know that it has a truth value.
Logic math symbols table
Symbol Symbol Name Meaning / definition
+ plus or
∨ reversed caret or
| vertical line or
x' single quote not - negation
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write a linear equation that passes through the points (2,-7) and (4,-9) and has a slope "-5"
Step-by-step explanation:
a line passing through 2 points get a slope just by these 2 points.
all we can do is to verify that the given slope is the same as the slope defined by the 2 points.
the slope is the ratio
y coordinate change / x coordinate change
when going from one point on the line to another.
for our 2 points that gives us
x changes by + 2 (from 2 to 4).
y changes by -2 (from -7 to -9).
the slope is therefore
-2/+2 = -1
the question asks for a slope of -5.
that is a contradiction. there is no line possible to pass through both points and have a slope of -5.
20 workers can paint a building in 21 days. If work has to be finished in 15 days, find the number of extra workers required.
Step-by-step explanation:
20 workers finish work in 21 days.
the necessary work effort is 20×21 = 420 workdays.
the same effort is needed even if the overall time is shorter.
x workers in 15 days.
x × 15 = 420
x = 420/15 = 28 workers.
so, 28-20 = 8 additional workers are needed to finish the work in 15 days.
Graph the following linear function. f(x)= 5/2x+5
The graph of the linear function f(x) = 2/(x + 5) is attached.
What is the general equation of a straight line?The general equation of a straight line is -
y = mx + c
{m} - slope.
{c} - intercept along the y - axis.
Given is the linear function as -
f(x) = 5/(2x + 5)
Refer to the graph of the linear function attached. This graph represents the function f(x).
Therefore, the graph of the linear function f(x) = 2/(x + 5) is attached.
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A bird flew from the ground onto a tree branch 5 feet above the ground.What could the bird do next that would make this situation a model of combining a number and its additive inverse
We are given that A bird flew from the ground onto a tree branch 5 feet above the ground.
We are supposed to find What could the bird do next that would make this situation a model of combining a number and its additive inverse
Additive inverse - What you add to a number to get zero is additive inverse.
The negative of a number.
Now we are told What could the bird do next that would make this situation a model of combining a number and its additive inverse
Since it already flew 5 feet above the ground .
Additive inverse of 5 feet is -5 feet
So, to make the situation of combining a number and its additive inverse
Bird has to flew 5 feet return to the ground i.e.-5 feet
So, a model of combining a number and its additive inverse = 5+(-5)
= 0.
Fly down 5 feet and return to the ground.
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Answer:
Fly down 5 feet and return to the ground.
Step-by-step explanation:
have a nice day.
I really need an answer for this FAST so if anybody can answer id really appreciate it :)
both Square and Rectangle
because all squares are rectangles, but not all rectangles are squares. Hope this helps. :)
What percent of 48 is 16?
Answer:
Step-by-step explanation:
hello follow me on zepeto if this helps
16 is 33.3333% of 48. 33 is the whole number hope you get a
The four-digit numeral $3AA1$ is divisible by $9$. What digit does $A$ represent?
When a four-digit number is divisible by nine and is written as 3AA1, the digit A stands for 7.
Properties of 9-divisible numbersA number is itself divisible by nine if the total of its digits is also divisible by nine.
Multiples in mathematics are the outcomes of multiplying an integer by a certain number. a multiple of 9 is a number that divides by nine without remainder.
By utilizing the aforementioned attribute, we can
1 + 3 + 4 + 5 = 9, 18, or 27.
4 +2A = 9, OR 18, OR 27
calculating based on sum equal to 18
2A = 18 - 4
2A = 14
A = 7
therefore 3771 is the number
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Please help quickly!
Use the figure below to answer questions 18-21
Answer:
Step-by-step explanation:
18. m∠1 = 104°
19. m∠2 = 79°
20. m∠3 = 65°
21. m∠5 = 73°
Which variation is represented by this situation:
Martha earns $375 when she works 5 hours. How much will she earn
if she works 12 hours? Is it direct variation or inverse variation?
Direct variation is represented by this situation and she earn $900
if she works 12 hours.
What are direct variation and indirect variation?Direct variation describes a relationship whereby one measure directly affects the other when the first one changes. This is completely opposed by inverse variation. Learn the definitions of direct and inverse variations in this article, along with some instances that have been solved.
How do inverse and direct variations differ from one another?When x is not equal to 0, y = kx is the equation that describes the linear function known as direct variation. An equation of the form xy = k, where x is not equal to zero and k is a nonzero real number constant, describes the nonlinear function known as inverse variation.
5 hours = earns $ 375
1 hours = 375/5 = earns $ 75
12 hours = 75 12 = earns $ 900
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A catering company is setting up tables for a big event that will host 764 people when they set up the tables they need 2 forks for each child and 5 forks for each adult the company ordered a total of 2992 forks set up a system of equations involving the number of adults,a and the number of children,c and solve to find out how many of each attended the event
Answer:
488 adults and 276 children
Step-by-step explanation:
So, we need two linear equations here to start, and we do have them.
Our first linear equation tells us how we get the number of forks:
2c + 5a = 2992
And the second tells us the number of people there:
c + a = 764
I will solve this using substitution. We start by minusing a from both sides in the second equation, giving us:
c = 764 - a
Then, we will substitute this into our first equation, giving us:
2(764 - a) + 5a = 2992
Then, we will open the parentheses, giving us:
1528 - 2a + 5a = 2992
Combining like terms and subtracting 1528 from both sides gives us:
3a = 1464
Dividing by 3:
a = 488
So there were 488 adults. Then, we take our original second equation and substitute 488 for a, giving us:
488 + c = 764
Minus 488 from both sides gives us:
c = 276
So, there were 276 children. In total, we had 488 adults and 276 children.
Hope this helped!
If total employee benefits are calculated as a percentage of their gross pay, which of the following employees receives
the largest percentage of their gross pay in employee benefits?
a. Employee A: gross pay $32,600, total job benefits $33,600
b. Employee B: gross pay $32,900, total job benefits $34,000
c. Employee C: gross pay $33,400, total job benefits $33,900
d. Employee D: gross pay $33,700, total job benefits $34,700
The employee who receives the largest percentage of their gross pay in employee benefits is Employee D.
What is percentage?
Percentage is a way of expressing a number as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviation "pct". For instance, 45% (read as "forty-five percent") is equal to 45/100, or 0.45. Percentage can be used to express part-to-whole relationships, such as "25% of the students in the class". It can also be used to compare two or more values, such as "the temperature rose by 3% over the course of the day". In some cases, percentage can be used to calculate a quantity from a given value, such as finding the percentage increase from one number to another.
This is because they have the highest total job benefits ($34,700) when compared to their gross pay ($33,700). This results in a percentage of 103.3%, which is the highest of the four employee options.
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Q4 ABC is an isosceles triangle with AB=AC and AD is one of its altitudes. i) State the three pairs of equal pairs in ADB and ADC.
ii) Is ADB = ADC? Why or why not?
iii) Is B=C? Why or why not?
iv) Is BD=CD? Why or why not?
Answer:
d
Step-by-step explanation:
Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The sum of two numbers is 71. One number is 27 more than the other. Find the numbers. D Let x be the smaller number and y be the larger number. Write an equation that represents the fact that the sum of the numbers is 71. =71 (Simplify your answer.)
Given that the sum of two numbers is 71 and one number is 27 more than the other, we can set up the following system of equations:
What is the straightforward definition of a system of equations solution?
A collection of values for a variable that simultaneously fulfill each equation is the solution to a system of equations. A system of equations must be solved by identifying all possible sets of variable values that make up the system's solutions.
x + y = 71 (equation 1 represents the fact that the sum of the numbers is 71)
y = x + 27 (equation 2 represents the fact that one number is 27 more than the other)
To solve the system of equations, we can use the substitution method. We'll substitute equation 2 into equation 1:
x + (x + 27) = 71
Combining like terms:
2x + 27 = 71
Subtracting 27 from both sides:
2x = 44
Dividing both sides by 2:
x = 22
Now we can substitute x = 22 back into equation 2 to find the value of y:
y = x + 27
y = 22 + 27
y = 49
So the two numbers are x = 22 and y = 49, and they satisfy the given conditions that the sum of the two numbers is 71 and one number is 27 more than the other.
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Each week Sara's science teacher gives a 15-point quiz.
Her current scores are shown in the table.
What is Sara's mean quiz score?
Answer: Sara's mean quiz score is 12
Step-by-step explanation:
Mean is the Sum of the Terms divided by the number of terms.
If we add up every score we get 60, we then will divide this by how many scores there were (5).
60/5 = 12