Answer:
B
Step-by-step explanation:
The equation of a circle in standard form is
(x - h)² + (y - k)² = r²
where (h, k) are the coordinates of the centre and r is the radius
To obtain this form use the method of completing the square
Given
x² - 8x + y² - 2y - 8 = 0 ( add 8 to both sides )
x² - 8x + y² - 2y = 8
add ( half the coefficient of the x/y terms )² to both sides
x² + 2(- 4)x + 16 + y² + 2(- 1)y + 1 = 8 + 16 + 1
(x - 4)² + (y - 1)² = 25 → B
Final answer:
The standard form of the given circle equation is (x - 4)^2 + (y - 1)^2 = 25, which is option (B).
Explanation:
To rewrite the given equation of a circle in standard form, we need to complete the square for both x and y terms. The equation is given as:
x² − 8x + y² - 2y - 8 = 0
We organize the equation by grouping x and y terms:
(x² − 8x) + (y ² - 2y ) = 8
To complete the square, we add to each side (½ the coefficient of x) ² for the x-terms and (½ the coefficient of y) ² for the y-terms:
For the x terms: (½ × -8)² = (-4)² = 16. Add 16 to both sides.
For the y terms: (½ × -2)² = (-1)² = 1. Add 1 to both sides.
Our equation becomes:
(x² − 8x + 16) + (y² - 2y + 1) = 8 + 16 + 1
Which simplifies to:
(x - 4)² + (y - 1)² = 25
Therefore, the standard form of the given equation is (x - 4)² + (y - 1)² = 25, and the correct answer from the options is (B).
I have no idea how to do this. I can’t cooperate with the imaginary number, please help me
Answer:
Step-by-step explanation:
This is a third degree polynomial because we are given three roots to multiply together to get it. Even though we only see "2 + i" the conjugate rule tells us that 2 - i MUST also be a root. Thus, the 3 roots are x = -4, x = 2 + i, x = 2 - i.
Setting those up as factors looks like this (keep in mind that the standard form for the imaginary unit in factor form is ALWAYS "x -"):
If x = -4, then the factor is (x + 4)
If x = 2 + i, then the factor is (x - (2 + i)) which simplifies to (x - 2 - i)
If x = 2 - i, then the factor is (x - (2 - i)) which simplifies to (x - 2 + i)
Now we can FOIL all three of those together, starting with the 2 imaginary factors first (it's just easier that way!):
(x - 2 - i)(x - 2 + i) = [tex]x^2-2x+ix-2x+4-2i-ix+2i-i^2[/tex]
Combining like terms and canceling out the things that cancel out leaves us with:
[tex]x^2-4x+4-i^2[/tex]
Remembr that [tex]i^2=-1[/tex], so we can rewrite that as
[tex]x^2-4x+4-(-1)[/tex] and
[tex]x^2-4x+4+1=x^2-4x+5[/tex]
That's the product of the 2 imaginary factors. Now we need to FOIL in the real factor:
[tex](x+4)(x^2-4x+5)[/tex]
That product is
[tex]x^3-4x^2+5x+4x^2-16x+20[/tex]
which simplifies down to
[tex]x^3-11x+20[/tex]
And there you go!
Which point does NOT lie on the graph of y = 2x 3?
(1, 8)
(-1, -2)
(0, 0)
(2, 16)
[tex]\bf (\stackrel{x}{1},\stackrel{y}{8})\qquad y=2x^3\implies 8=2(1)^3\implies 8\ne 2~~\bigotimes \\\\\\ (\stackrel{x}{-1},\stackrel{y}{-2})\qquad y=2x^3\implies -2=2(-1)^3\implies -2=-2~~\checkmark \\\\\\ (\stackrel{x}{0},\stackrel{y}{0})\qquad y=2x^3\implies 0=2(0)^3\implies 0=0~~\checkmark \\\\\\ (\stackrel{x}{2},\stackrel{y}{16})\qquad y=2x^3\implies 16=2(2)^3\implies 16=16~~\checkmark[/tex]
A square and a rhombus have the following in common EXCEPT:
A. Opposite pairs of sides are parallel
B. Opposite pairs of sides are congruent
C. Pairs of interior and exterior angles are supplementary
D. All interior angles are 90 degree
D.All interior angles are 90 degrees
Let's analyze each choice to determine which one is the exception, that is, a property that is not shared between a square and a rhombus:
A. Opposite pairs of sides are parallel:
This is true for both a square and a rhombus. By definition, both a square and a rhombus have opposite sides that are parallel.
B. Opposite pairs of sides are congruent:
This property also holds true for both squares and rhombi. In a square, all four sides are equal in length. In a rhombus, opposite sides are equal in length.
C. Pairs of interior and exterior angles are supplementary:
Again, this is a common feature of both squares and rhombuses. In any parallelogram (which includes both squares and rhombuses), each pair of interior and exterior angles on the same side are supplementary, totaling 180 degrees.
D. All interior angles are 90 degrees:
This is where we can find the exception. In a square, all four interior angles are indeed 90 degrees. However, this is not the case for a rhombus. A rhombus does not necessarily have right angles; its angles can be of any measure as long as opposite angles are equal and the sum of the angles is 360 degrees, which is true for any quadrilateral.
Therefore, the correct answer to the question is:
D. All interior angles are 90 degrees
This is the property that is not common between a square and a rhombus, making it the exception.
The circumcenter of a triangel is the point equidistant from the vertices of the triangel. True or false
Answer:
True
Step-by-step explanation:
The circumcenter is the center of the circumscribing circle, the circle that intersects each of the vertices. Since the points on a circle are equidistant from its center, the vertices of a triangle are equidistant from the circumcenter.
Final answer:
The circumcenter of a triangle is the point that is equidistant from the vertices of the triangle.
Explanation:
True
The circumcenter of a triangle is the point that is equidistant from the vertices of the triangle. This means that the distance from the circumcenter to each vertex of the triangle is the same.
For example, in an equilateral triangle, the circumcenter is the point where all the perpendicular bisectors of the sides intersect.
What should be the next number in the following series? 1 2 8 48 384 _ ?_
The answer should be 3840
Express this number in standard form.
3
.
6
4
3
×
1
0
−
1
=
?
3.643×10
−1
=?3, point, 643, times, 10, start superscript, minus, 1, end superscript, equals, question mark
Answer:
0.3643
Step-by-step explanation:
3.643×10^-1 = 3.643×(1/10) = 3.643×0.1 = 0.3643
For parametric equations x= a cos t and y= b sin t, describe how the values of a and b determine which conic section will be traced.
Step-by-step explanation:
x = a cos t, y = b sin t
cos t = x / a, sin t = y / b
cos² t + sin² t = 1
(x / a)² + (y / b)² = 1
If a = b, the conic section is a circle.
If a and b are different, the conic section is an ellipse.
URGENT! Use Gauss's approach to find the following sums 1+3+5+7+...997
Formula please and solve
Answer:
The sum of the first 499 terms is 249001
Step-by-step explanation:
* Lets revise the arithmetic sequence
- There is a constant difference between each two consecutive
numbers
- Ex:
# 2 , 5 , 8 , 11 , ……………………….
# 5 , 10 , 15 , 20 , …………………………
# 12 , 10 , 8 , 6 , ……………………………
* General term (nth term) of an Arithmetic sequence:
- U1 = a , U2 = a + d , U3 = a + 2d , U4 = a + 3d , U5 = a + 4d
- Un = a + (n – 1)d, where a is the first term , d is the difference
between each two consecutive terms n is the position of the
number
- The sum of first n terms of an Arithmetic sequence is calculate from
Sn = n/2[a + l], where a is the first term and l is the last term
* Now lets solve the problem
∵ The terms of the sequence are 1 , 3 , 5 , 7 , ......... , 997
∵ The first term is 1 and the second term is 3
∴ The common difference d = 3 - 1 = 2
∵ The first term is 1
∵ The last term is 997
∵ The common difference is 2
- Lets find how many terms in the sequence
∵ an = a + (n - 1) d
∴ 997 = 1 + (n - 1) 2 ⇒ subtract 1 from both sides
∴ 996 = (n - 1) 2 ⇒ divide both sides by 2
∴ 498 = n - 1 ⇒ add 1 for both sides
∴ n = 499
∴ The sequence has 499 terms
- Lets find the sum of the first 499 terms
∵ Sn = n/2[a + l]
∵ n = 499 , a = 1 , l = 997
∴ S499 = 499/2[1 + 997] = 499/2 × 998 = 249001
* The sum of the first 499 terms is 249001
Final answer:
The sum of the series 1+3+5+7+...+997 is calculated using Gauss's approach for arithmetic series, resulting in a sum of 249500.
Explanation:
To find the sum of the series 1+3+5+7+...+997 using Gauss's approach, we first recognize that this is an arithmetic series where each term increases by a constant difference, in this case, 2. The first term, a, is 1 and the last term, l, is 997. We can find the number of terms, n, in the series using the formula: n = (l - a) / common difference + 1, which in this case is (997 - 1) / 2 + 1 = 499. Then, the sum of an arithmetic series can be found using the formula: Sum = n/2 * (a + l), so the sum of this series is 499/2 * (1 + 997) = 249500.
If 3(x+2)=5(x-8) what is the value of x+2?
A. 23
B.25
C.40
D.46
Answer:
B. 25
Step-by-step explanation:
In order to find out the value of x + 2, you need to know what x is. Let's solve for it then sub it back in to evaluate the expression.
3(x+2)=5(x-8) distributes out to give you
3x + 6 = 5x - 40. Get like terms together on opposite sides of the equals sign:
46 = 2x and divide to get x = 23. Now that we know that x = 23, that means that x + 2 is the same as 23 + 2 which is 25.
9. A gear of radius 6.1 cm turns at 11 revolutions per second. What is the linear velocity of the gear in meters per second?
Show your work
Answer:
4.22 meters per second.
Step-by-step explanation:
First multiply the revs per second by pi:
= 11pi = 34.558.
Now multiply this by the diameter which is 2*6.1 = 12.2 cm = 0.122 m.
Linear velocity = 0.122 * 34.558
= 4.22 m/s.
Linear velocity of the gear in meters per second is 4.22 m/s.
What is linear velocity?Linear velocity is the measure of “the rate of change of displacement with respect to time when the object moves along a straight path.” It is a vector quantity.
Given
Gear of radius = 6.1 cm
Gear of diameter = [tex]2 \times 6.1 = 12.2 cm[/tex] = 0.122 m
Revolutions per second = [tex]11\pi[/tex] = 34.558
Linear velocity of the gear = diameter × Revolutions per second
Linear velocity = 0.122 × 34.558
= 4.22 m/s
Linear velocity of the gear in meters per second is 4.22 m/s.
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Please help me out with this
Check the picture below.
Please help me out please
Answer:
h=20
Step-by-step explanation:
h=3 (V/lw)
h= 3 (960/12*12)
Neil has 3 partially full cans of white paint.They contain 1/3 gallon,1/5 gallon,and 1/2 gallon of paint.About how much paint does neil have in all
I don’t know what answer Is I wish I could help
Mikayla is a waitress who makes a guaranteed $50 per day in addition to tips of 20% of all her weekly customer receipts, t. She works 6 days per week. Which of the following functions best represents the amount of money that Mikayla makes in one week?f(t) = 50 + 20tf(t) = 300 + 20tf(t) = 50 + 0.2tf(t) = 300 + 0.2t
$50 per day x 6 days = 300
20% = 0.2, times t = 0.2t
The equation would be: f(t) = 300 + 0.2t
Answer:
Option D:[tex]f(t)=300+0.2 t[/tex]
Step-by-step explanation:
We are given that Mikayla is a waitress
She makes a guaranteed per day =$50
Her tips per day =20 % of weekly tips t
We have to find a function that represents the amount of money makes by Mikayla in one week
[tex]20% of t=\frac{20}{100}t=0.2t[/tex]
She works per week=6 days
She earns in 6 days=[tex]50\times 6[/tex]=$300
Total earning of Mikayla in one week =[tex]300+0.2t[/tex]
[tex]f(t)=300+0.2 t[/tex]
Hence, option D is true.
A circle with a diameter of 10 inches is spanned by a central angle of 180 degrees. What is the length of the subtended arc?
since the diameter of the circle is 10, then its radius must be half that or 5.
[tex]\bf \textit{arc's length}\\\\ s=\cfrac{\pi \theta r}{180}~~ \begin{cases} r=&radius\\ \theta =&angle~in\\ °rees\\ \cline{1-2} r=&5\\ \theta=&180 \end{cases}\implies s=\cfrac{\pi (180)(5)}{180}\implies s=5\pi \implies s\approx 15.71[/tex]
Answer:
The correct answer is 15.7 inches
Step-by-step explanation:
Points to remember
Circumference of a circle = 2πr
Where r is the radius of circle
To find the value of arc length
It is given that, diameter = 10 inches
Radius = diameter/2 = 10/2 = 5 inches
Circumference = 2πr
= 2 * 3.14 * 5
= 31.4
central angle of arc =180
Arc length = (180/360) * circumference
= (1/2) * 31.4
= 15.7 inches
Therefore the correct answer is 15.7 inches
Which polar coordinates represent the same point as the rectangular coordinate (2,-1?
[tex]\bf (\stackrel{a}{2}~,~\stackrel{b}{-1})\qquad \begin{cases} r=\sqrt{a^2+b^2}\\\\ \theta =tan^{-1}\left( \frac{b}{a} \right) \end{cases} \\\\[-0.35em] ~\dotfill\\\\ r=\sqrt{2^2+(-1)^2}\implies r=\sqrt{5} \\\\\\ \theta =tan^{-1}\left( \cfrac{-1}{2} \right)\implies \theta \approx -26.57^o\implies \theta \approx 333.43^o \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill (\sqrt{5}~~,~~333.43^o)~\hfill[/tex]
Answer:
[tex](r,\theta); (\sqrt{5} , tan^{-1}(\frac{x}{y}))\\(r,\theta); (-\sqrt{5} , -tan^{-1}(\frac{x}{y}))[/tex]
Step-by-step explanation:
Here we are given our rectangular coordinates as (2,-1) . We have to convert this into polar coordinates. The formula for conversion into polar form is
[tex]r=\sqrt{x^2+y^2}[/tex]
[tex]\theta=tan^{-1}(\frac{x}{y})[/tex]
Substituting the values of x and y in the above formulas we get
[tex]r=\sqrt{2^2+(-1)^2}\\r=\sqrt{4+1}\\r=\sqrt{5}\\r=-\sqrt{5}\\[/tex]
[tex]\theta=tan^{-1}(\frac{-1}{2})[/tex]
Hence our polar coordinates are
[tex]r=(\sqrt{5},tan^{-1}(\frac{-1}{2}) )\\r=(-\sqrt{5},tan^{-1}(\frac{-1}{2}) )\\[/tex]
Time sensitive question. Find the sum of the first 26 terms of an arithmetic series whose first term is 7 and 26th term is 93.
ANSWER
[tex]S_{26}=1300[/tex]
EXPLANATION
The sum of an arithmetic sequence whose first term and last terms are known is calculated using
[tex]S_{n}= \frac{n}{2} (a + l)[/tex]
From the given information, the first term of the series is
[tex]a = 7[/tex]
and the last term of the series is
[tex]l = 93[/tex]
The sum of the first 26 terms is
[tex]S_{26}= \frac{26}{2} (7 + 93)[/tex]
[tex]S_{26}= 13 (100)[/tex]
[tex]S_{26}=1300[/tex]
The sum of the first 26 terms of the given arithmetic series is 1300, obtained using the sum formula S = n/2 * (a1 + an) for arithmetic series.
Explanation:To find the sum of the first 26 terms of an arithmetic series whose first term (a1) is 7 and the 26th term (a26) is 93, you can use the formula for the sum of an arithmetic series: S = n/2*(a1 + an). Here, n is the number of terms, a1 is the first term, and an is the nth term. In this case, we have n = 26, a1 = 7, and a26 = 93.
The sum, S, of the series is calculated as follows:
S = 26/2 * (7 + 93) = 13 * (100) = 1300.
Therefore, the sum of the first 26 terms of the arithmetic series is 1300.
Brainliest + Points! Can someone please HELP ME
A spring has spring constant 0.3 m/newton. What force is necessary to stretch the spring by 1.5 meters?
Use Hooke's Law, F = kx. Show all your steps.
In the equation F=kx, F is the force, k is the spring constant, and x is the displacement. Plug in and solve:
F=0.3(1.5)
F=0.45N
Hope this helps!!
Find the component of a velocity vector of an airplane that is traveling 150 mph at 45° below horizontal.
Answer:
none of the above
Step-by-step explanation:
All of the answer choices agree that the magnitude of the components is 150/√2 ≈ 106.1. The units of these components should be "mph", the same as the units of the magnitude of the given vector.
We only know the angle with respect to horizontal. We don't know whether that angle is measured with respect to the positive x-axis or the negative x-axis. Both of those are horizontal, and there is nothing in the problem statement that restricts the airplane to be traveling in one direction or the other.
Possible answers are ...
<106.1 mph, -106.1 mph> . . . . . . "horizontal" is +x direction
<-106.1 mph, -106.1 mph> . . . . . "horizontal" is -x direction
(The units are not degrees (°).)
Could you Help me there was 395 Lenin ice cups at thesnack shop. People bought 177 lemon ice cups . How many lemon ice cups are still all the snack shop.
Answer:
There are 288 cups left
Step-by-step explanation:
Take the amount that were there (395) and subtract the amount that were sold (177). that is the amount that are left
395-177 = 218
There are 288 cups left
after selling 177 cups, there are 218 lemon ice cups still at the snack shop.
To calculate how many lemon ice cups are still at the snack shop after some were sold, we need to perform a simple subtraction operation. The initial quantity of lemon ice cups was 395. After selling 177 cups, we subtract 177 from 395 to find out how many are left.
Here's the calculation:
Start with the initial number of lemon ice cups: 395.Subtract the number of lemon ice cups sold: 395 - 177.Calculate the remaining number of lemon ice cups: 218.So, after selling 177 cups, there are 218 lemon ice cups still at the snack shop.
If the mean of a normal distribution is 18, what is the median of the distribution?
A) 22
B) 26
C) 14
D) 18
D. Normal distributions are symmetric, so the mean is the same as the median.
Answer:
18
Step-by-step explanation:
Evaluate 7 − (−1).
6
−6
8
−8
Answer:
8
Step-by-step explanation:
The minus sign outside parentheses changes the sign of what's inside parentheses when parentheses are eliminated:
7 -(-1) = 7 +1 = 8
_____
Alternate way to think about it
Subtraction is the same as addition of the opposite. The opposite of -1 is +1, so subtracting -1 is the same as adding +1.
please help me with this geometry question
image attached
For this problem, you need to know how to use sin, cos, and tan or 'SOH-CAH-TOA' sin- opposite/hypotenuse cos-adjacent/hypotenuse toa- opposite/adjacent
in this case, you can use tan or toa
the equation should be set up like this:
4/x=1/0.848
solve, and you will get 3.39...
(you need to use the sin,cos,tan degree chart to find 0.848)
Ashley is thinking of two numbers. The first number is four more than twice the second number. The sum of the two numbers is 16. Which system of equations can be used to determine the first number, x, and the second number, y?
x + y = 16
x = y^2 x 4
i hope this helps :)
good luck
Which value for x makes the following equation TRUE?
-2x - 4 = - 6
Answer:
1
Step-by-step explanation:
-2(1) = -2
then -2 - 4 = -6
-6 = -6
Answer:
1
Step-by-step explanation:
Move all terms that don't contain x to the right side & solve.
if f(x)=x-2 which of the following is the inverse of f(x) brainly
Answer:
The inverse of f(x) is [tex]f ^ {- 1}(x) = x + 2[/tex]
Step-by-step explanation:
To find the inverse of the function [tex]f (x) = x-2[/tex], perform the following steps:
1) do [tex]y = f (x)[/tex]
[tex]y = x-2[/tex]
2) Solve the equation for the variable x.
[tex]y + 2 = x -2 +2[/tex]
[tex]y + 2 = x[/tex]
3) exchange the variable x with the variable y
[tex]y + 2 = x[/tex] ----> [tex]x + 2 = y[/tex]
4) Change the variable y by [tex]f ^{- 1}(x)[/tex]
Finally the inverse function is:
[tex]f ^ {- 1} (x) = x + 2[/tex]
Answer:
f-1(x)=x+2
Step-by-step explanation:
Please please answer this correctly
Answer:
800,000,000
Step-by-step explanation:
Subtract the known numbers from the sum to find the missing number:
800,903,402 - 900,000 -2 -3,000 -400 = 800,000,000
The test to detect the presence of respiratory syncytial virus is 97% accurate for a person who has the virus and 99% accurate for a person who does not have the virus. In a given population, 0.55% of the people are infected. The probability that a randomly chosen person gets an incorrect result is .
[tex]\textrm{97 percent accurate for individual with the virus}[/tex]
[tex]\text{3 percent inaccurate}[/tex]
[tex]\textrm{99 percent accurate for an individual}[/tex]
[tex]\textrm{1 percent inaccurate probability}[/tex]
[tex]\textrm{Probability of infected:}[/tex]
[tex].0055[/tex]
[tex]\textrm{Probability of not being infected}[/tex]
[tex]1 - .0055 = .9945[/tex]
[tex]\textrm{Combine it all together}[/tex]
[tex]0.0055 * 0.03 + 0.9945 * 0.01 = 0.01011[/tex]
[tex]\textrm{The probability that a randomly chosen person gets an incorrect result is 0.01011}[/tex]
[tex]\textbf{Answer}[/tex]
[tex]\textrm{0.01011}[/tex]
The probability of an incorrect result for the test to detect respiratory syncytial virus is calculated by considering the test's sensitivity and specificity, along with the prevalence of the virus in the population.
Explanation:In order to calculate the probability that a randomly chosen person gets an incorrect result, we first need to understand how test accuracy works. Sensitivity refers to how often the test correctly identifies the presence of a disease when it is indeed there, while specificity refers to how often it correctly identifies the absence of a disease when it isn't there. In this case, the sensitivity is 97%, meaning it correctly identifies the virus in 97% of individuals who have the virus; the specificity is 99%, meaning it correctly identifies no virus in 99% of individuals who do not have the virus.
For the given population, 0.55% are infected. Hence, the probability of an incorrect result for a person with a virus is 3% (100% - 97%), and for a person without the virus is 1% (100% - 99%). Therefore, the total probability of getting an incorrect result is (0.55%/100 * 3%) + [(1 - 0.55%/100) * 1%].
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PLEASE HELP ASAP 36 PTS + BRAINLIEST TO RIGHT/BEST ANSWER
Answer:
(b) 3x -5
Step-by-step explanation:
You can slog through the entire long division process, or you can realize that the product of the quotient constant and the divisor constant must match the dividend constant. If we let q represent the quotient constant, you must have ...
4q = -20
q = -20/4 = -5
The only answer choice with a constant of -5 is choice B, 3x -5.
Lisa lives in Atlanta. She is going to a baseball game with her two sisters and her parents. They have $100 to spend. After they buy their tickets, how much money will they have left to buy refreshments?
Answer:
27.90
Step-by-step explanation:
Multiplication is the process of multiplying, therefore, adding a number to itself for the number of times stated. The amount of money that is left with Lisa after buying tickets is $27.90.
What is multiplication?Multiplication is the process of multiplying, therefore, adding a number to itself for the number of times stated. For example, 3 × 4 means 3 is added to itself 4 times, and vice versa for the other number.
The total number of members in the family is 5 (Parents+2sisters+Lisa). Also, the cost of a single ticket in Atlanta is $14.42, therefore, the cost of five tickets will be,
The cost of 5 tickets = $14.42 × 5 = $72.10
The amount Lisa has to spend is $100, therefore, the money left with Lisa after buying the tickets is,
Amount left = $100 - $72.10 = $27.90
Hence, the amount of money that is left with Lisa after buying tickets is $27.90.
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