Answer:
In vertex form it's [tex]y=(x+7)^2-9[/tex]
In standard form it's [tex]y=x^2+14x+40[/tex]
Step-by-step explanation:
We can use the vertex form to solve for a in
[tex]y=a(x-h)^2+k[/tex]
"a" is the number out front that dictates the steepness, or lack thereof, in a parabola. That means that we need h and k (which we have in the vertex) and we need x and y (which we have in the form of one of the zeros). Filling in using a vertex of (-7, -9) and a coordinate point (-7, 0):
[tex]0=a(-4-(-7))^2-9[/tex] simplifies a bit to
[tex]0=a(-4+7)^2-9[/tex] simplifies a bit more to
[tex]0=a(3)^2-9[/tex] and
[tex]0=a(9)-9[/tex] so
[tex]0=9a-9[/tex],
[tex]9=9a[/tex] and finally,
a = 1
Phew! So there is the a value. Now we can simply fill in the formula completely, using the vertex as our guide:
[tex]y=(x+7)^2-9[/tex]
In standard ofrm that is
[tex]y=x^2+14x+40[/tex]
We can check ourselves for accuracy by factoring that standard polynomial using whatever method of factoring you like for quadratics and get that the roots are in fact x = -10, -4
The only reason we needed the zeros is to use one of them as the x and y to solve for a.
What is the value of P for the following solid figure?
PLEASE HELP ME SOLVE. So lost rn
Answer:
Step-by-step explanation:
yes 30
The revenue (in thousands of dollars) from producing x units of an item is modeled by R(x) = 12x -0.005x^2.
Find the average rate of change in revenue as x changes from 1001 to 1008.
The average rate of change in revenue is ____ dollars per unit.
(Do not round until the final answer. Then round to the nearest integer as needed.)
I just need to know how to solve this problem, I know the answer. Thank you!
Answer:
See below.
Step-by-step explanation:
Average rate of change =( R(1008) - R(1001) ) / (1008 - 1001)
=[ 12(1008) - 0.005(1008)^2 - (12(1001) - 0.005(1001)^2) ] / 7
Answer:
2
Step-by-step explanation:
Average rate of change is just the slope.
You want to find the slope of the line that goes through x=1001 and x=1008 while on R(x)=12x-0.005x^2.
So y=R(x)... R(x) will give your corresponding y values for the x's mentioned.
Just plug them in.
R(1001)=12(1001)-0.005(1001)^2=7001.995
R(1008)=12(1008)-0.005(1008)^2=7015.68
So now the question just becomes find the slope of the line that goes through
(1001,7001.995) and (1008,7015.68).
To find the slope of a line given two points: I just lined the points up and subtract vertically. Afterwards I put the second number on top of the first number.
So let's do that!
(1008 , 7015.68)
-(1001, 7001.995)
------------------------------
7 13.685
So the slope also known as the average rate of change here is 13.685/7.
Putting 13.685 divided by 7 into my calculator returns the value 1.955.
This rounded to the nearest integer is 2.
The ceiling function maps any number to the least integer that is _[blank]_ the number.
Which phrase correctly fills in the blank of the previous statement?
A less than or equal to
B less than
C greater than or equal to
D greater than
Answer:
C. greater than or equal to
Step-by-step explanation:
For example,
ceiling(5) = 5
ceiling(5.1) = 6
ceiling(-5) = -5
ceiling(-5.1) = -5
Please give an example of another function whose inverse is only defined if we restrict the domain of the original function.
(In general, a function must be one-to-one in order to have an inverse function. Some functions, though, have inverses that are very useful but require us to restrict the original function to an interval where it IS one-to-one. This is the case with all of our trigonometric functions.)
Answer:
f(x) = x^2
Step-by-step explanation:
The square root function is defined to have a non-negative range only. That corresponds to restricting the domain of f(x) = x^2 to positive values of x.
_____
The attached graph shows the domain-restricted f(x)=x² in solid red and the corresponding f⁻¹(x) = √x in solid blue. The other halves of those curves are shown as dotted lines (and are inverse functions of each other, too). The dashed orange line is the line of reflection between a function and its inverse.
Answer:
OH NANANA
Step-by-step explanation:
Find all real numbers $x$ such that $3x - 7 \le 5x +9$. Give your answer as an interval.Find all real numbers $x$ such that $3x - 7 \le 5x +9$. Give your answer as an interval.Find all real numbers $x$ such that $3x - 7 \le 5x +9$. Give your answer as an interval.Find all real numbers $x$ such that $3x - 7 \le 5x +9$. Give your answer as an interval.Find all real numbers $x$ such that $3x - 7 \le 5x +9$. Give your answer as an interval.Find all real numbers $x$ such that $3x - 7 \le 5x +9$. Give your answer as an interval.Find all real numbers $x$ such that $3x - 7 \le 5x +9$. Give your answer as an interval.Find all real numbers $x$ such that $3x - 7 \le 5x +9$. Give your answer as an interval.
Identify the values of a, b, and c.
a=
b=
C=
Given y = (2x + 3)? choose the standard form of
the given quadratic equation
0 = 25x2
0 = 4x2 +9
0 = 4x2 + 10x + 6
✓ 0 = 4x2 + 12x + 9
COMPLETE
RETRY
Answer:
a=4b=12c=9You have correctly selected the standard form.Step-by-step explanation:
(2x +3)² = (2x)² + 2(2x)(3) +(3)²
= 4x² +12x +9
Comparing that to ax² +bx +c, we can identify ...
a = 4b = 12c = 9The values of a, b, and c are:
a = 4
b = 12
c = 9
The given quadratic equation is:
y = (2x + 3)²
A quadratic equation is of the form:
y = ax² + bx + c
Expand the equation y = (2x + 3)²
y = (2x + 3)(2x + 3)
y = 4x² + 6x + 6x + 9
y = 4x² + 12x + 9
Comparing y = 4x² + 12x + 9 with y = ax² + bx + c
a = 4
b = 12
c = 9
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If we put 5 math, 6 biology, 8 engineering, and 3 physics books on a bookshelf at random, what is the probability that all the math books are together?
Answer: [tex]\dfrac{3}{4389}[/tex]
Step-by-step explanation:
Given : Number of math books = 5
Total number of books = 5+6+8+3=22
Number of books except math = 17
Number of ways to arrange 22 books in bookshelf = [tex]22![/tex]
When all math books are together , then we count whole set as one
Now, the number of objects in bookshelf = 17+1=18
Number of ways to arrange books such that all math books are together = [tex]18!5![/tex]
Now, the probability that all the math books are together :-
[tex]\dfrac{5!18!}{22!}=\dfrac{3}{4389}[/tex]
Hence, the probability that all the math books are together [tex]=\dfrac{3}{4389}[/tex]
Out of 100 people at a tennis match, 66% were rooting for the local champion. If two people were chosen at random from the crowd, what is the probability that at least one of them was not rooting for the champion?
Question 16 options:
22.7%
56.7%
59.2%
43.3%
Answer:
Approximately 56.7%.
Step-by-step explanation:
Choose two people at random from the crowd and there will be two cases:
Zero or one out of the two person was rooting for the champion, or both were rooting for the champion.There's no third possible outcome. In other words, the two cases are mutually exclusive. Either the first or the second event is expected to happen. The sum of their probabilities shall equal to 1.
66 out of that 100 were rooting for the champion. The probability that both were rooting for the champion will be easier to find. The probability that the first person is rooting for the champion is equal to [tex]66/100[/tex].
After that first person was chosen from the crowd, the 65 out of the remaining 99 person in the crowd were chanting. The probability that the second person is rooting as well will equal to [tex]65/100[/tex].
Both event shall take place. The probability that both were rooting for the champion will equal to
[tex]\displaystyle \frac{66}{100} \times \frac{65}{99}[/tex].
The probability that one or zero out of the two persons were rooting will equal to
[tex]\displaystyle 1 - \frac{66}{100} \times \frac{65}{99} \approx \frac{17}{30} = 56.7\%[/tex].
Answer:
56.7% is correct.
Step-by-step explanation:
Determine whether the sequence converges or diverges. If it converges, give the limit.
60, -10, 5/3, -5/18
Diverges
Converges; 11100 (this is definitely the wrong answer.)
Converges; 72
Converges; 0
Answer:
Converges; 51 3/7
Step-by-step explanation:
The common ratio is -10/60 = (5/3)/-10 = (-5/18)/(5/3) = -1/6.
Then the sum of the sequence is given by ...
S = a1/(1 -r) = 60/(1 -(-1/6))
S = 60/(7/6) = 360/7
S = 51 3/7
_____
If you erroneously evaluate the formula for the sum using +1/6 as the common ratio, then you will get S=60/(1-1/6) = 60·6/5 = 72.
Describe how to derive the quadratic formula from a quadratic equation in standard form.
Answer:
The quadratic formula is derived from a quadratic equation in standard form when solving for x by completing the square.
Step-by-step explanation:
Answer:
The quadratic formula is derived from a quadratic equation in standard form when solving for x by completing the square. The steps involve creating a perfect square trinomial, isolating the trinomial, and taking the square root of both sides. The variable is then isolated to give the solutions to the equation.
Step-by-step explanation:
The number N = 100 + 100^2 + 100^3 + ... + 100^n . Find the least possible value of n such that the number N is divisible by 11. NEED QUICKLY! Thanks!!!
Answer:
Step-by-step explanation:
very interesting question. The temptation is to say that n should be 11 and that likely is divisible by 11 but it may not be the smallest.
100 + 100^2 = 100 + 10000 = 10100
The pattern of the series goes 101010101 ... 00...
100 / 11 = The remainder is 1/11
10100 / 11 = the remainder is 2/11
1010100 /11 the remainder is 3/11
The pattern suggests that the remainder will be 0 then n = 11
There might be other ways of doing this, but I don't know them.
In triangle ABC, mA=35, mB=40, and a=9. Which equation should you solve for b?
A. sin35/b=sin40/9
B. sin35/9=sin40/b
C. cos35/9=cos40/b
D.b sqaure=9 square-2(9)bcos40
Answer:
B. sin35/9=sin40/b
Step-by-step explanation:
The law of sines tells you ...
sin(A)/a = sin(B)/b
Filling in the given values, you get ...
sin(35°)/9 = sin(40°)/b
Answer:
B.[tex]\frac{sin 35}{9}=\frac{sin 40 }{b}[/tex]
Step-by-step explanation:
We are given that in a triangle ABC. [tex]m\angle =35^{\circ}[/tex]
[tex]m\angle B=40^{\circ}[/tex]
a=9
We have to find an equation which solve for b
We know that a sine law
[tex]\frac{a}{sine A}=\frac{b}{sinB}=\frac{c}{sinC}[/tex]
Using above formula of sine law
Substituting all given values in the above formula of sine law
Then we get
[tex]\frac{9}{sin 35}=\frac{b}{sin 40}[/tex]
By cross multiply then we get
[tex]sin 40\times 9=sin35 \times b[/tex]
[tex] \frac{sin 40 \times 9}{b}= sin 35[/tex]
Using division property of equality
[tex]\frac{ sin 40}{b}=\frac{sin 35}{9}[/tex]
Using division property of equality
Hence, option B is true option for solving b.
Answer:B.[tex]\frac{sin 35}{9}=\frac{sin 40 }{b}[/tex]
A manufacturer of yoga pants sells them for $28 each. They hired some consultants who determined that the cost of manufacturing x pants was C\left(x\right)=x^2-2x-9 C ( x ) = x 2 − 2 x − 9 . a)Write a function for the revenue (the amount of money the company brings in). b)Write a function for the profit (the revenue – cost). c)Find the number of t-shirts they should make to maximize the profit function. Round your answer to the nearest whole number.
Answer:
a) r(x) = 28x
b) p(x) = -x^2 +30x +9
c) 15
Step-by-step explanation:
a) Let x represent the number of items sold. Each sale results in $28 of revenue, so the revenue function r(x) is ...
r(x) = 28x
__
b) p(x) = r(x) - c(x) = 28x -(x^2 -2x -9)
p(x) = -x^2 +30x +9
__
c) The axis of symmetry of ax^2 +bx +c is -b/(2a). Here, the axis of symmetry of the profit function is ...
x = -30/(2(-1)) = 15
15 is the quantity of sales that maximizes profit.
The employees of a company were surveyed on questions regarding their educational background (college degree or no college degree) and marital status (single or married). Of the 600 employees, 400 had college degrees, 100 were single, and 60 were single college graduates. The probability that an employee of the company is married and has a college degree is?
Answer:0.68
Step-by-step explanation:
Given
Total of 600 employees out of which 400 had college degree ,100 are single
and 60 were single graduates
therefore out of 100, 60 were single and rest 40 are single undergraduate
and out of 400, 60 were single graduates thus 340 are married graduate.
Now out of 600, 100 were single i.e. 500 is married
thus Probability that an employee is married and has a college degree is
=[tex]\frac{Favourable outcome }{Total outcome}[/tex]
P=[tex]\frac{340}{500}[/tex]=0.68
You start your shift at 3:00 pm. On nursing assessment rounds, you find that Mr. Johnson has an IV of D5W that is infusing at 32 gtt/min. The IV tubing is calibrated for 15 gtt/mL. How many mL will Mr. Johnson receive during your 8-hr shift?
Answer:
Mr Johnson will receive 1024 mL IV in 8 hours.
Step-by-step explanation:
Mr Johnson has an IV that is infusing at 32 gtt per minute.
So in 1 hour patient will get the drug = 32×60 = 1920 gtt
Now in 8 hours drug received by the patient = 1920 × 8
= 15360 gtt
Since IV tube is calibrated for 15 gtt per mL which means in 1 mL amount of drug is 15gtt.
Therefore, total volume of infusion (in mL) will be
= [tex]\frac{\text{Total drug infused}}{\text{Total drug in 1 mL}}[/tex]
= [tex]\frac{15360}{15}[/tex]
= 1024 mL.
Therefore, 1024 mL IV will be infused in 8 hours.
8.39+(-2.00)+161
i got 167.39 but it isnt correct
so if yall could help me plsssss
Answer:
8
Step-by-step explanation:
167.39 is right but it can be simplified.
1.61 was replaced by (161/100).
3 more similar replacement(s)
839 2 161
(——— + (0 - —)) +
100 1 100
639 + 161 8
—————————
100 1
Sorry if it looks confusing
A phone company offers two monthly plans. Plan A costs $11 plus an additional $0.17
for each minute of calls. Plan B costs $16 plus an additional $0.13 for each minute of calls.
For what amount of calling in minutes do the two cost the same?
What is the cost when the two plans cost the same?
Answer:
125 minutes of calling
It will cost $32.25 when the plans cost the same.
Step-by-step explanation:
The first plan's expression would be:
11+.17x x being the number of minutes
The second plan's expression would be:
16+.13x
You must set the expressions equal to one another. So:
11+.17x=16+.13x
Then solve for x:
.17x=5+.13x
.04x=5
x=125
So the plans will cost the same after 125 minutes. To find the cost of the plans at that time, substitute 125 in for the x in one of the equations.
11+.17(125)=y y being the overall cost of the plan
11+21.25=y
32.25=y
To check your answer, you can substitute again in the other equation:
16+.13(125)=y
16+16.25=7
32.25=7
The Fibonacci numbers are a sequence of integers in which the first two elements are 1, and each following element is the sum of the two preceding elements. The mathematical definition of each kth Fibonacci number is the following:
Answer:
[tex]F_{k^{th}}=F_{(k-2)^{th}}+F_{(k-1)^{th}}[/tex]
Step-by-step explanation:
Since each number is the sum of it's 2 preceding numbers thus mathematically it can be written as
[tex]F_{k^{th}}=F_{(k-2)^{th}}+F_{(k-1)^{th}}[/tex]
Fibonacci Series can be written as
1,1,2,3,5,8,13...
Which of the following statements about Pascal’s Triangle are true? It is symmetrical. The first diagonal is all 1’s. The second diagonal is the counting numbers. Any number in the triangle is the sum of the two numbers directly above it. Each row adds to a power of 2.
Answer: The following statements are true:
It is symmetrical.
The first diagonal is all 1’s.
The second diagonal is the counting numbers.
Any number in the triangle is the sum of the two numbers directly above it.
Each row adds to a power of 2.
Answer:
They are all correct
Step-by-step explanation:
Two sides of an isosceles triangle have lengths 7 and 14. Find the length of the third side.
PLS HELP
Answer:
14
Step-by-step explanation:
An isosceles triangle has two sides that are the same. If one side is 7 and another is 14, then the two possibilities are 7, 7, and 14, or 14, 14, and 7.
It can't be 7, 7, and 14, because the sum of the shortest sides of a triangle must be greater than the longest side.
Therefore, it must be 14, 14, and 7. So the third leg is 14.
The length of third side of the isosceles triangle is 14.
What is an isosceles triangle ?An isosceles triangle is a triangle that has two sides of equal length. Also the property of isosceles triangle states that the base angle of the isosceles triangle subtends angle of equal measure.
How to find the length of third side of given isosceles triangle ?It is given that the two sides of the triangle have lengths 7 and 14.
Thus for the triangle to be isosceles, the third side of the triangle will have the length as 7 or 14.
We know that the sum of any two sides of a triangle must be greater than the third side.
Thus if the third side is 7, then the sum of the sides will be (7 + 7) = 14 which is not greater than the other side. So, third side will not be of length 7 units.
The third side of the isosceles triangle is of 14 units .
Therefore, the length of third side of the isosceles triangle is 14.
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Solve each equation by graphing. Round to the nearest tenth.
-2x^2+2=-3x
Answer:
x = -0.5 or x = 2
Step-by-step explanation:
Finding solutions graphically is often easier if the equation can be put in the form f(x) = 0. Here, we can do that by subtracting the right-side expression to give ...
(-2x^2 +2) -(-3x) = 0
This could be put in standard form, but there is no need. A graphing calculator can deal with this directly.
The solutions are x = -0.5 and x = 2.
Select the correct answer.
Given: BC || DE, and ∠GAC ≅ ∠AFD.
----------------------------------------------------------
What is the missing step in the proof?
Answer:
B
Step-by-step explanation:
In step 1, we found ∠GAC ≅ ∠AFD.
In step 2, we found ∠GAC ≅ ∠AFE.
Therefore, by transitive property of equality, ∠AFD ≅ ∠AFE.
Connie, a marketing director, lost her job when her company downsized. This is an example of what type of unemployment?
Answer:
Cyclical unemployment.
Step-by-step explanation: It is not part of the natural unemployment rate.
It's caused by the contraction phase of the business cycle. That's when demand for goods and services generated by the company fall dramatically, forcing businesses to lay off large numbers of workers to cut or reduce costs.
A mechanic sells a brand of automobile tire that has a life expectancy that is normally distributed, with a mean life of 34 , 000 miles and a standard deviation of 2500 miles. He wants to give a guarantee for free replacement of tires that don't wear well. How should he word his guarantee if he is willing to replace approximately 10% of the tires?
Answer:
He should word his statement as "Free replacement for tires that wear before 30875 miles."
Step-by-step explanation:
If he is willing to replace 10% of tires he should find the life of tires that gives an area of 10% in the normal distribution graph.
Now for 10% of area standard normal deviate Z can be obtained from normal distribution table
Using normal distribution table for 10% area we have Z = -1.28
Thus we have [tex]Z=\frac{X-\overline{X}}{\sigma }\\\\\therefore X=\sigma Z+\overline{X}[/tex]
Applying given values we get
[tex]X=-1.28\times 2500+34000\\\\X=30875miles[/tex]
The mechanic would replace the tires that spoil without covering up to 30800 miles.
What is z score?Z score is used to determine by how many standard deviations the raw score is above or below the mean. It is given by:
z = (raw score - mean) / standard deviation
For a mean of 34000 miles and standard deviation of 2500. The probability of 10% correspond with a z score of -1.28. Hence:
-1.28 = (x - 34000)/2500
x = 30800
The mechanic would replace the tires that spoil without covering up to 30800 miles.
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Stacy rolls a pair of six-sided fair dice.
The probability that the sum of the numbers rolled is either a multiple of 3 or an even number is
, and the two events are exclusive.
Answer:
Pr(the sum of the numbers rolled is either a multiple of 3 or an even number)=[tex]\frac{2}{3}[/tex]
Step-by-step explanation:
Let A be the event "sum of numbers is multiple of 3"
and B be the event "sum is an even number".
As our dice has six sides, so the sample space of two dices will be of 36 ordered pairs.
|sample space | = 36
Out of which 11 pairs have the sum multiple of 3 and 18 pairs having sum even.
So Pr(A)= [tex]\frac{11}{36}[/tex]
and Pr(B)= [tex]\frac{18}{36}[/tex]
and Pr(A∩B) = [tex]\frac{5}{36}[/tex], as 5 pairs are common between A and B.
So now Pr(A or B)= Pr(A∪B)
= Pr(A)+Pr(B) - Pr(A∩B)
= [tex]\frac{11}{36}[/tex] + [tex]\frac{18}{36}[/tex] - [tex]\frac{5}{36}[/tex]
= [tex]\frac{24}{36}[/tex]
= [tex]\frac{2}{3}[/tex]
Answer:
2/3 and NOT mutually exclusive
Step-by-step explanation:
plato
If sine theta equals three over four, what are the values of cos θ and tan θ?
cosine theta equals plus or minus square root of seven over four, tangent theta equals plus or minus two times square root of seven over seven
cosine theta equals plus or minus seven over four, tangent theta equals negative three over seven
cosine theta equals plus or minus square root of seven over 4, tangent theta equals plus or minus three over seven
cosine theta equals plus or minus seven over four, tangent theta equals negative one over seven
Answer:
In words, Cosine theta equals plus or minus square root of seven over 4,tangent theta equals plus or minus three over root seven
Step-by-step explanation:
Given that sin ∅ =3/4 It means the ratio of the opposite side to the hypotenuse side is 3:4.
Using the Pythagoras theorem we can calculate the hypotenuse adjacent as follows.
a²+b²=c²
a²=c²-b²
a²=4²-3²
a²=16-9
a²=7
a=√7
Then Cos ∅= opposite/ adjacent
=√7/4
Then Tan ∅ = opposite/adjacent
=3/√7
In words, Cosine theta equals plus or minus square root of seven over 4,tangent theta equals plus or minus three over root seven.
Angle D is a circumscribed angle of circle O.
What is the perimeter of kite OBDE?
A) 17 units
B) 23 units
C) 27 units
D) 40 units
Answer:
The answer is 27 on edge as well!
Step-by-step explanation:
The answer is 27
This is because 15x15 is 225 plus 8x8 is 64 which is 289 and the square root of that is 17 and that is the diameter of the circle and the hypotenuse of the triangle and since the kite has two congruent sides which are both radii or half of a diameter times two would be the same as the length of the diameter which is 17 plus the two bottom sides which are both 5 and 5 plus 5 is 10 and 10 plus 17 is 27 or the perimeter of the kite.
Your Welcome!
Answer:
27
Step-by-step explanation:
im taking the test right now on edg. 2020
Which is the angle of elevation from C to B?
Answer:
∠4
Step-by-step explanation:
The angle of elevation is the measure of the angle from the horizontal upwards.
The angle of elevation from C to B is ∠4
Angle of elevation from C to B will be ∠4. Option (1) will be the answer.
Angle of elevation of an object from a point:Angle of elevation of an object from a point on the ground is defined by,
"Angle between the horizontal line and line of site (line joining the observer and the object above the horizontal line)"
Following the definition,
Angle of elevation of an object at B from C will be → ∠4
Therefore, Option (1) will be the answer.
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A nontoxic furniture polish can be made by combining vinegar and olive oil. The amount of oil should be three times the amount of vinegar. How much of each ingredient is needed in order to make 34 oz of furniture polish?
Answer:
V=8.5
Step-by-step explanation:
o=oil. vinegar=v. furniture polish=f
O=3v
34= 3v + v
Using a system of guessing and checking if that number fits equation you can tell that 8 causes the equation to be unequal and also 9. You can learn V must be between 8 and 9 so 8.5 might fit the equation. 8.5=V
Final answer:
To make 34 oz of furniture polish, 8.5 oz of vinegar and 25.5 oz of olive oil are needed, with the olive oil being three times the amount of vinegar.
Explanation:
To create 34 oz of nontoxic furniture polish, where the amount of olive oil should be three times the amount of vinegar, we need to solve a simple algebraic equation. Let's denote the amount of vinegar as v ounces. According to the conditions, the amount of olive oil will then be 3v ounces.
The total amount of furniture polish equals the amount of vinegar plus the amount of olive oil:
v + 3v = 34 oz
This simplifies to:
4v = 34 oz
Dividing both sides by 4 gives us:
v = 8.5 oz
Therefore, the amount of olive oil needed is:
3v = 3 Times 8.5 oz = 25.5 oz
To conclude, we need 8.5 oz of vinegar and 25.5 oz of olive oil to make 34 oz of furniture polish.
HELPPPP!!!
Drag each label to the correct location on the equation. Each label can be used more than once, but not all labels will be used.
Find the standard form of the equation of the hyperbola x2 − 4y2 − 2x + 16y − 31 = 0. Place the signs and values in the correct places in the standard form of the equation.
Answer:
(x - 1)²/4² - (y - 2)²/2² = 1 ⇒ The bold labels are the choices
Step-by-step explanation:
* Lets explain how to solve this problem
- The equation of the hyperbola is x² - 4y² - 2x + 16y - 31 = 0
- The standard form of the equation of hyperbola is
(x - h)²/a² - (y - k)²/b² = 1 where a > b
- So lets collect x in a bracket and make it a completing square and
also collect y in a bracket and make it a completing square
∵ x² - 4y² - 2x + 16y - 31 = 0
∴ (x² - 2x) + (-4y² + 16y) - 31 = 0
- Take from the second bracket -4 as a common factor
∴ (x² - 2x) + -4(y² - 4y) - 31 = 0
∴ (x² - 2x) - 4(y² - 4y) - 31 = 0
- Lets make (x² - 2x) completing square
∵ √x² = x
∴ The 1st term in the bracket is x
∵ 2x ÷ 2 = x
∴ The product of the 1st term and the 2nd term is x
∵ The 1st term is x
∴ the second term = x ÷ x = 1
∴ The bracket is (x - 1)²
∵ (x - 1)² = (x² - 2x + 1)
∴ To complete the square add 1 to the bracket and subtract 1 out
the bracket to keep the equation as it
∴ (x² - 2x + 1) - 1
- We will do the same withe bracket of y
- Lets make 4(y² - 4y) completing square
∵ √y² = y
∴ The 1st term in the bracket is x
∵ 4y ÷ 2 = 2y
∴ The product of the 1st term and the 2nd term is 2y
∵ The 1st term is y
∴ the second term = 2y ÷ y = 2
∴ The bracket is 4(y - 2)²
∵ 4(y - 2)² = 4(y² - 4y + 4)
∴ To complete the square add 4 to the bracket and subtract 4 out
the bracket to keep the equation as it
∴ 4[y² - 4y + 4) - 4]
- Lets put the equation after making the completing square
∴ (x - 1)² - 1 - 4[(y - 2)² - 4] - 31 = 0 ⇒ simplify
∴ (x - 1)² - 1 - 4(y - 2)² + 16 - 31 = 0 ⇒ add the numerical terms
∴ (x - 1)² - 4(y - 2)² - 16 = 0 ⇒ add 14 to both sides
∴ (x - 1)² - 4(y - 2)² = 16 ⇒ divide both sides by 16
∴ (x - 1)²/16 - (y - 2)²/4 = 1
∵ 16 = (4)² and 4 = (2)²
∴ The standard form of the equation of the hyperbola is
(x - 1)²/4² - (y - 2)²/2² = 1
Answer:
Refer to attachment below.
if sin(x) = squareroot 2 over 2 what is cos(x) and tan(x)
Answer:
cos(x) = square root 2 over 2; tan(x) = 1
Step-by-step explanation:
[tex]\frac{\sqrt{2} }{2}[/tex]
was, before it was rationalized,
[tex]\frac{1}{\sqrt{2} }[/tex]
Therefore,
[tex]sin(x)=\frac{1}{\sqrt{2} }[/tex]
The side opposite the reference angle measures 1, the hypotenuse measures square root 2. That makes the reference angle a 45 degree angle. From there we can determine that the side adjacent to the reference angle also has a measure of 1. Therefore,
[tex]cos(x)=\frac{1}{\sqrt{2} }=\frac{\sqrt{2} }{2}[/tex] and
since tangent is side opposite (1) over side adjacent (1),
tan(x) = 1