The standard equation for the circle is (x - 1 ) ^2 + (y + 3)^2 = 2 ^2
A circle is a closed curve that is drawn from a fixed point called the center in which all points on the curve are the same distance from the center of the center. The equation of a circle with center (h, k) and radius r is given by:
(x-h)^2 + (y-k)^2 = r^2
This is the standard form of the equation. So if we know the coordinates of the center of the circle and also its radius, we can easily find its equation.
Consider any point P(x, y) on the circle. Let "a" be the radius of the circle which is equal to OP.
We know that the distance between the point (x, y) and the origin (0,0) can be found using the distance formula which is equal to -
√[x^2+y^2]= a
Therefore the equation of a circle with center as origin is,
x^2+y^2= a^2
Where "a" is the radius of the circle.
An alternative method
Let's derive another way. Assume that (x,y) is a point on the circle and the center of the circle is at the origin (0,0). Now, if we draw a perpendicular from the point (x,y) to the x-axis, we get a right triangle, where the radius of the circle is the hypotenuse. The base of the triangle is the distance along the x-axis and the height is the distance along the y-axis. Applying the Pythagorean theorem here, we therefore obtain:
x^2+y^2 = radius^2
We need to find the standard equation for the circle with center (1,-3) that passes through the point (2,2)
Standard equation for circle is
(x-h)^2 + (y-k)^2 = r^2................(1)
Here h = 1 , k = -3 and r = 2
put this value of h, k and r in equation (1), we get
(x - 1 ) ^2 + (y + 3)^2 = 2^2
Hence the standard equation of the circle is
(x - 1 ) ^2 + (y + 3)^2 = 2 ^2
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chegg find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative f(x)
The derivative and domain of the function f(x) = x²-2x³ is :[tex](-\infty, \infty)[/tex].
What is derivative of the function (differentiation)?The slope of a function's graph or, more precisely, the slope of the tangent line at a point can be used to interpret a function's derivative.
Its computation actually stems from the slope formula for a straight line, with the exception that curves require the employment of a limiting procedure.
Calculation for the derivative:
Step 1: Use the definition of the derivative. Remember that f(x+h) means plug (x+h) into everywhere there is an "x" in f(x).
[tex]\begin{aligned}f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\&=\lim _{h \rightarrow 0} \frac{(x+h)^{2}-2(x+h)^{3}-\left(x^{2}-2 x^{3}\right)}{h} \\&=\lim _{h \rightarrow 0} \frac{\left(x^{2}+2 h x+h^{2}\right)-2(x+h)(x+h)^{2}-x^{2}+2 x^{3}}{h} \quad \text { cancel } x^{2}\end{aligned}[/tex]
[tex]\begin{aligned}&=\lim _{h \rightarrow 0} \frac{\left(2 h x+h^{2}\right)-2(x+h)\left(x^{2}+2 h x+h^{2}\right)+2 x^{3}}{h} \\&=\lim _{h \rightarrow 0} \frac{\left(2 h x+h^{2}\right)-2\left(x^{3}+2 h x^{2}+h^{2} x+h x^{2}+2 h^{2} x+h^{3}\right)+2 x^{3}}{h} \\&=\lim _{h \rightarrow 0} \frac{2 h x+h^{2}-2 x^{3}-4 h x^{2}-2 h^{2} x-2 h x^{2}-4 h^{2} x-2 h^{3}+2 x^{3}}{h}\end{aligned}[/tex]
[tex]=\lim _{h \rightarrow 0} \frac{2 h x+h^{2}-6 h x^{2}-6 h^{2} x-2 h^{3}}{h}[/tex]
Step 2: Cancel out a factor of h from each term in the numerator with the h in the denominator. Then direct substitute h=0.
[tex]\begin{aligned}&=\lim _{h \rightarrow 0}\left(2 x+h-6 x^{2}-6 h x-2 h^{2}\right) \\&=2 x+0-6 x^{2}-6(0) x-2(0)^{2} \\&=2 x-6 x^{2}\end{aligned}[/tex]
f and f' are polynomials, so their domains are all real numbers.
Therefore, for f(x) = x²-2x³ domain of f and f' :[tex](-\infty, \infty)[/tex].
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The complete question is -
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative f(x) = x²-2x³.
Suppose you vote in an election for a slate of candidates from a particular party.The percentage of votes the party wins in a district determines how many seats in a district the party will gain at the national legislature. These districts are best characterized as
These districts are best characterized as Proportional Representation districts.
What is Proportional Representation?
There are numerous types of proportional representation systems, but they all have two things in common. They start by using districts with multiple members.
Proportional Representation, abbreviated as PR, uses significantly bigger districts that elect multiple members at once, such as five or ten, rather than electing one member of the legislature in each local district. Second, the percentage of votes a party receives in these districts with several members determines which candidates gain the seats.
The Proportional Representation system of America overcomes the wide ranging drawbacks of single-district voting system like not representing the large number of voters, discrimination against third parties, etc.
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How do you find
arcsin (-1)
By the definition of inverse functions, we will get:
[tex]arcsin(-1) = -pi/2[/tex]
How to find the arcsin of -1?The function arcsin(x) is the inverse of the sine function, such that, if:
[tex]sin(x) = y\\\\arcsin(y) = x[/tex]
Now, we know that:
[tex]sin( -pi/2) = -1[/tex]
Where pi = 3.14
Then, using the above rule:
[tex]arcsin(-1) = -pi/2[/tex]
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PLEASE HELP GEOMETRY
answer both parts
1) vertical angles theorem
2) transitive property of congruence
P(x) · Q(x) = R(x); if P(x) = x + 1 and R(x) = 2x³ + 2x²- 3x-3, what is Q(x)?
A. 2x² - 3
B. 2x² - 2
C. 2x² + 2
D. 2x² + 3
Answer: A
Step-by-step explanation:
[tex]P(x) Q(x)=R(x)\\\\(x+1)Q(x)=2x^3 + 2x^2 - 3x-3\\\\Q(x)=\frac{2x^3 + 2x^2 - 3x-3}{x+1}\\\\Q(x)=\frac{2x^2 (x+1)-3(x+1)}{x+1}\\\\Q(x)=2x^2 - 3[/tex]
The function f is defined by f(x)=2-x. If 2f(p)=8, what is f(2p)?
If function 'f' is defined by f(x)=2-x and 2f(p)=8 then the value of f(2p) is 6.
Given, f(x) = 2-x
Solve this problem by making a series of substitutions.
First, find the value of p.
Then use the definition of the function f to find the value of f(2p).
We are given
f(x) = 2 - x
Multiply both sides by 2, and simplify.
2f(x) = 2(-x +2)
2f(x) = -2x + 4
Let x = p.
2f(p) = -2(p) + 4
We are given that 2f(p) = 8.
Substitute 20 for 2f(p).
8 = -2p + 4
2p = -4
p = -2
Now we know that p = -2, so we can find f(2p).
f(x) = -x + 2
f(2p) = -2p + 2
Substitute -2 for p.
f(2p) = -2(-2) + 2
f(2p) = 4 +2
f(2p) = 6
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can someone please help me
Answer:
Cos (3pi/4) =0.999154554
Sin (3pi/4)=0.041111762
Step-by-step explanation:
On your calculator there should be SIN and COS next to each other
#1 tap COS
#2 put your given numbers, in this case
COS(3pi/4) = 0.999154554
same goes for SIN
SIN (3pi/4)=0.041111762
help me with this pls asap
Answer:
your picture is blank if you can fix it i will be glad to help you
Step-by-step explanation:
Answer:
First option
Step-by-step explanation:
The line corresponds to the f(x).
Hope this helps
How many non-square numbers lie between the 215to the power of 2 and 216 to the power of 2
There are 430 non square numbers that lie between [tex]215^{2}[/tex] and [tex]216^{2}[/tex].
Given two numbers [tex]215^{2}[/tex] and [tex]216^{2}[/tex] and we are require to find non square numbers that lie between them.
Numbers can be consecutive numbers or non consecutive numbers. In our question we will find the consecutive numbers.
Square numbers are those numbers whos square is in a proper number means decimal should not be there.
First number=[tex]215^{2}[/tex]
Second number=[tex]216^{2}[/tex]
First we have to find the exact value of numbers by finding the square of these numbers.
First number=215*215=46225
Second number=216*216=46656
Numbers that lie between them=(46656-46225)-1
=431-1
=430
Hence 430 numbers lie between [tex]215^{2}[/tex] and [tex]216^{2}[/tex].
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17. If13=xy-5 and y = 6, what is the value of x? (A) 0 (B) 3 (C) 6 (D) 8
Answer:
Option B: 3
Step-by-step explanation:
#1: Substitute in 6 for [tex]y[/tex] and 3 for [tex]x[/tex]: [tex]13=(3)(6)-5[/tex]
#2: Multiply 3 and 6: [tex]13=18-5[/tex]
#3: Evaluate the equation: [tex]13=18-5[/tex] → [tex]13=13[/tex]
#4: Check if the solution is equal: [tex]13=13[/tex] ✓
[tex]x[/tex] is equal to 3 (Option B).
A set of golf clubs and bag cost $225. The clubs cost $60 more than the bag. How much do the clubs cost?
Answer:
$142.5
Step-by-step explanation:
225-60=165
165/2=82.5
82.5+60=142.5
Two cans each hold 750ml by how many ml is their total volume less than 1.8 liters
If two cans hold 750ml ,then their voLume is 0.3 liters less than 1.8 liters.
Given volume of two cans being 750 ml each.
We are required to calculate the quantity in liters of how much the volume of two cans less than 1.8 liters.
Volume is the quantity of substance a container can hold in its capacity.
So volume of cone means quantity of liquid cone can hold if fully filled.
Volume of first cone=750 ml
Volume of second cone=750 ml
Total volume=750+750
=1500 ml
1 liter= 1000 ml
so 1500 ml will be equal to 1500/1000
=1.5 liter
We are required to do comparison with 1.8 liters.
Difference =1.8-1.5
=0.3 liters.
Hence two cans have 0.3 liters less than 1.8 liters.
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Evaluate. (-3 2/3)^2
Answer:
answer is 121/9
Step-by-step explanation:
13.4444444444 (decimal form)
121/9 (improper fraction form)
13 4/9 (mixed fraction form)
Select the correct answer. Identify the expression equivalent to 4(x + x + 7) - 2x + 8 - 4 by substituting x= 1 and x = 2.
A. 6x + 11
B. 3(x + 7)
C. 2(3x + 16)
D. 3x + 16
Answer:
C is correct
Step-by-step explanation:
I'm not sure what the substituting part has to do anything, but here is how I got the answer
4(x+x+7)-2x+8-4
4(2x+7)-2x+4
8x+28-2x+4
6x+32
Factor and you get
2(3x+16), which is answer choice C
You can check that the answer is right by plugging in x=1 and x=2 into the original equation and the answer choice, which may be what that is asking.
Maria is creating a round stained glass window like the one below. She needs to cut the dark pink glass to the correct size to fit into the frame. At what angle (x) should she cut the glass so that it fits if she knows the arc measure of a is 35º and the arc measure of b is 45º. What does x equal?
Applying the angle of intersecting chords theorem, the value of x is: 40º.
What is the Angle of Intersecting Chords Theorem?According to the angle of intersecting chords theorem the following equation is true:
x = 1/2(a + b).
Given the following:
a = 35º
b = 45º
Plug in the values
x = 1/2(35 + 45)
x = 1/2(80)
x = 40º
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If Avogadro's number of pennies is divided equally among the 246 million men, women, and children in the United States, how many dollars would each receive
USD per person = $[tex]2.31[/tex] × [tex]10^{13}[/tex].
What is the US dollar?The US dollar is the official currency of the United States and a number of other countries. The Coinage Act of 1792 established the United States dollar on equal footing with the Spanish silver dollar, divided it into 100 cents, and authorized the minting of coins denominated in dollars and cents.To find how many dollars would each receive:
Avogadro's number of pennies is divided equally among 261 million people.
Avogadro's number is [tex]6.022[/tex] × [tex]10^{23}[/tex].
To find the number of pennies each person will receive, we divide [tex]6.022[/tex] × [tex]10^{23}[/tex] by 261 million.
Pennies per person = [tex]\frac{6.022 * 10^{23} }{261 million} = \frac{6.022 * 10^{23} }{261 * 10^{6} } = 2.31 * 10^{15}[/tex]
This is equal to the number of cents each person receives.
To convert pennies into dollars, 1 USD = 100 pennies, divide [tex]2.31 * 10^{15}[/tex] by [tex]100[/tex].
USD per person = [tex]\frac{2.31 * 10^{15} }{100} = $2.31 * 10^{13}[/tex]
Therefore, USD per person = $[tex]2.31[/tex] × [tex]10^{13}[/tex].
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The question you are looking for is here:
If Avogadro's number of pennies is divided equally among the 261 million men, women, and children in the United States, how many dollars would each receive?
Given: Circle X with Radius r and circle Y with radius s
Prove: Circle X is similar to Circle Y
It can be proved that Circles X and Y are similar.
How can we prove that the circles are similar?We were told that circle Y have a radius of s
And X have the Radius r
Going with the theorem that Similar shapes are proportional, we can find the scale factor from circle X to circle Y by making a division on the radii of both circles.
Then the scale factor is [tex]\frac{s}{r}[/tex] which implies that circle Y will gotten by dilation of circle Y.
Therefore, both circles are similar
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Solve the following quadratic by
completing the square.
f(x) = x² + 2x - 12
+√[]
x = [?] +
Answer:
{x = -1 + √13 , x = -1 - √13}
Step-by-step explanation:
Completing the square:
f(x) = x² + 2x - 12
= x² + 2x + 1 - 1 - 12
= (x² + 2x + 1) - 1 - 12
= (x + 1)² - 13
Solving:
f(x) = 0
⇔ (x + 1)² - 13 = 0
⇔ (x + 1)² = 13
⇔ x + 1 = √13 or x + 1 = -√13
⇔ x = -1 + √13 or x = -1 - √13
Solve this system of equations: y = x2 – 3x 12 y = –2x 14 1.isolate one variable in the system of equations, if needed. y = x2 – 3x 12 y = –2x 14 2.use substitution to create a one-variable equation. –2x 14 = x2 – 3x 12 3.solve to determine the unknown variable in the equation. what are the values of x? {–2, 1} {–2, 4} {–1, 2} {–1, 4}
Answer:
our answer is X=2 or X= -1
Answer: C. {-1,2}
Correct on Edge 2023
The compound inequality 8.00 ≤ x < 9.50 represents all values, x, for which college students are paid hourly as teacher assistants. What is another way of writing this compound inequality?
x ≤ 8.00 and x < 9.50
x ≤ 8.00 or x < 9.50
x ≥ 8.00 and x < 9.50
x ≥ 8.00 or x < 9.50
Answer:
The 3rd answer down
Step-by-step explanation:
If I broke apart the initial answer, I would see that x is equal to or greater than 8 and x is less than 9.5
Meredith found the following partial containers of detergent in the basement: 12 quart, 114 gallons, and 34 gallon. how many quarts of detergent did meredith find in all?
Quarts of detergent 1208
First, we convert each unit to its pint equivalent.
1 gallon =eight US Pint
114 gallons =114 X eight =912 Pint
34 gallon =34 X 8 =272 Pint
further, 1 quart =2 Pint
12 quart = 12 X 2 Pint =24 Pint
including the three values
912+272+24=1208 Pint
therefore, Meredith will want 1,208 Pint boxes.
The unitary technique is a way of solving trouble by way of first finding the fee of a single unit, and then finding the essential value with the aid of multiplying the unmarried unit price.
The system of the unitary technique is to locate the fee of an unmarried unit after which multiply the fee of a single unit to a wide variety of devices to get the vital fee.
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The programmer at a local public radio station is compiling data on listeners who stream podcasts. an executive for the national broadcaster has communicated that the overall population mean is 20,500 with a standard deviation of 2,180. the local programmer has a sample of 30 podcasts for her station. by the central limit theorem, which interval do about 99.7% of the sample means fall within?
The interval in which 99.7% of the sample means fall within 3 standard deviations of the mean is 19,306 to 21,694.
What is the central limit theorem?According to the central limit theorem,
Sample mean = Population mean = μ
Sample standard deviation = (Population standard deviation)/√n = σ/√n
where n is at least 30.
Calculation:It is given that,
The overall population mean = 20,500
Population standard deviation σ = 2180
Sample size n = 30
Since n = 30, the central limit theorem can be applied.
According to the Empirical rule(68-95-99), 99.7% of the sample means fall within the 3 standard deviations of the mean.
So, the interval is calculated by
Sample mean = Population mean = μ = 20500
Standard deviation(sample) = σ/√n
⇒ S = 2180/√30 =398.011 ≅ 398
Thus,
The lower bound = 20500 - 3 × 398 = 19,306
The upper bound = 20500 + 3 × 398 = 21,694
Therefore, the required interval is from 19,306 to 21,694.
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How much wood is needed to replace the entire roof of this gazebo that is in the
shape of a right pyramid on top of a regular octagon? (Note: The roof's base does
not require wood.)
what’s ten times ten
Decide whether the triangles are similar. if so, determine the appropriate expression to solve for x. triangles abc and edf; triangle abc has angle a measuring 53 degrees, angle c measuring 62 degrees, side ac labeled as y, side ab labeled as w, and side bc labeled as x; triangle edf has angle d measuring 61 degrees, angle f measuring 53 degrees, side de labeled z, side ef labeled u, and side df labeled r.
Answer:
Step-by-step explanation:
Similar Triangle:
Two triangles are comparable if they have the same ratio of corresponding sides and equal pairs of corresponding angles.If more figures have the same shape, but their sizes are different, then such objects are referred to as similar figures. Consider a hula hoop and wheel of a cycle, the shapes of each of those gadgets are much like every other as their shapes are equal.
Triangle ABC has angles of 53 and 62 degrees....the sum is 115 degrees. The third angle must be 180 - 115 = 65 degrees.
A triangle with one angle = 61 degrees cannot be similar
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Answer: The triangles are not similar; no expression for x can be found.
A science test, which is worth 100 points, consist of 24 questions. Each question is worth either 3 points or 5 points. If X is the number of 3 point questions and why is the number of 5 point questions, the system shown represents this situation. X+y=24, 3x+5y=100. What does the solution of this system indicate about the questions on the test?
Answer:
x= 10 y=14 : 10 3 point, 14 5 point
Step-by-step explanation:
Write your equation down vertically,
3x + 5y = 100
x + y = 24
Next find multiply bottom equation by one number from the tops opposite,
3x + 5y = 100
(-3)x +(-3)y = (-3)24
3x + 5y = 100
-3x - 3y= -72
Add the two functions to isolate y
2y= 28 y=14
Plug y back into original and solve
3x +5(14) =100 --> 3x + 70 =100 --> 3x = 30 x=10
The citizens of a state may only register to vote after proving to have submitted their census survey.
The citizens of a state may only register to vote after proving to have submitted their census survey is a "false" statement.
What is census survey?In order to create statistics that define populations and their characteristics, such as age, education, housing, and income, the U.S. Census Bureau compiles data from households.
The purpose of census survey are-
We obtain this information by posing questions to members of the household regarding the occupants of the home, apartment, mobile home, or group housing.The census helps our communities decide where to build everything from homes to hospitals, from schools to supermarkets, and it provides information about who we are as a country and where we are headed. It aids in the government's decision-making about the allocation of cash and assistance to cities and states.It aids in predicting the nation's economic needs, such as those for electricity, housing, food, etc. figuring out the country's population size, standard of life, and number of unemployed people.To know more about census, here
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The complete question is-
The citizens of a state may only register to vote after proving to have submitted their census survey.(True/False)
uppose you want to test the value of the mean in a population. You collect a random sample of 60 which yields a sample mean and sample standard deviation. You only know sampled information to perform the test. What type of procedure is appropriate for your data
To test the value of the mean in a population when only sample mean and sample deviation are known to perform the test then it is appropriate to use z-test procedure.
What is a Z-test?
A statistical method of testing a hypothesis is the z-test where either:
We are aware of the population variance, which is equal to 2.Despite the fact that we are unsure of the population variance, our sample size, n 30, is sizable. In this instance, we estimate the population variance using the sample variance.A t-test unlike of z-test must be used if the sample size is less than 30 and the population variance is unknown.Additional prerequisites for using the z-test include the following:
Data must follow a normal distribution.Independent data points are required.The variances for each sample must be equal.To know more about the z-test..........
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Question 20
Points 3
The height of a lighthouse is 98 meters and the distance between the top of the lighthouse and boat is 125 meters. Find the distance
between the bottom of the lighthouse and the boat.
O 125 m
O 77.6 m
The distance between the bottom of the lighthouse and the boat is 77.6m.
What is the distance between the bottom of the lighthouse and the boat?
The lighthouse and the boat would form a right triangle. The height would be the height of the lighthouse. The distance between the top of the lighthouse and boat would be the hypotenuse. The distance between the bottom of the lighthouse and the boat would be the base.
In order to determine the required distance, Pythagoras theorem would be used.
The Pythagoras theorem: a² + b² = c²
where:125² = 98² + b²
b² = 125² - 98²
b = √(25² - 98² )
b = 77.6m
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write an equation for the graph of a linear function that has slope -1/3 and a y intercept -3
Answer: y=-(1)/(3)x-3
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