Answers
The given equation represents a hyperbola. Its main features are:
center - (1,-3)vertices - V1= (1,-3+[tex]\sqrt{2}[/tex]) / V2= (1,-3-[tex]\sqrt{2}[/tex])foci - F1= (1 , -3+ [tex]2\sqrt{5}[/tex] )/ F2= (1 , -3 - [tex]2\sqrt{5}[/tex] )asymptotes = [tex]\pm \frac{1}{3}\left(x-1\right)-3[/tex]HyperbolaA hyperbola can be defined by its center, vertices, foci and asymptotes. And it is represented algebraically by the standard equation: [tex]\frac{\left(y-k\right)^2}{a^2}-\frac{\left(x-h\right)^2}{b^2}=1[/tex], where:
h= x-coordinate of center
k= y-coordinate of center
a and b= semi-axis
First, you need to rewrite the given equation 9y²-x²+2x+54y+62=0 in the standard equation hyperbola:
[tex](-x^2+2x+?)+9(y^2+6y+?)=-62\\(-x^2+2x+?)+9(y^2+6y+9)=-62\\(-x^2+2x-1)+9(y^2+6y+9)=-62\\(-x^2+2x-1)+9(y^2+6y+9)=-62-1+81\\((-x^2+2x-1)+9(y^2+6y+9)=18 ) \div 18\\ \frac{(-x^2+2x-1)}{18} +\frac{9(y^2+6y+9}{18}= \frac{18}{18} \\\frac{-(x-1)^2}{18} +\frac{(y+3^2)}{2}=1\\\frac{-(x-1)^2}{(3\sqrt{2})^2} +\frac{(y+3^2)}{(\sqrt{2})^2 }=1\\\frac{\left(y+3)^2}{\left(\sqrt{2}\right)^2}-\frac{\left(x-1\right)^2}{\left(3\sqrt{2}\right)^2}=1[/tex]
Comparing the previous equation with the standard form ([tex]\frac{\left(y-k\right)^2}{a^2}-\frac{\left(x-h\right)^2}{b^2}=1[/tex]), you have:
h=1, k=-3, a=[tex]\sqrt{2}[/tex] and b=[tex]3\sqrt{2}[/tex] . From now, it is possible to find that the question asks:
Find the centerThe coordinates for center is (h,k). Thus, the center is (1,-3).
Find the verticesThe vertices (V1 and V2) of hyperbola can be found from the coordinates of center (h,k) and the semi-axis (a).
V1= (h,k+a)= (1,-3+[tex]\sqrt{2}[/tex])
V2= (h,k-a)= (1,-3-[tex]\sqrt{2}[/tex])
Find the fociThe foci or the focus points can be found from the coordinates of center (h,k) and the c ([tex]\sqrt{a^2+b^2}[/tex]) which represents the distance from the center to the focus.
[tex]c=\sqrt{a^2+b^2}\\ c=\sqrt{(3\sqrt{2} )^2+(\sqrt{2} )^2}\\c=\sqrt{18+2} \\c=\sqrt{20}=2\sqrt{5}[/tex]
Thus,
F1= (h,k+c)= (1 , -3+ [tex]2\sqrt{5}[/tex] )
F2= (h,k-c)= (1 , -3 - [tex]2\sqrt{5}[/tex] )
Find the asymptotesThe asymptotes are the lines the hyperbola tends to at ±∞. For hyperbola, the asymptotes are defined as: [tex]y=\pm \frac{a}{b}\left(x-h\right)+k[/tex]. Then, for this question:
[tex]y=\pm \frac{\sqrt{2}}{3\sqrt{2}}\left(x-1\right)-3\\y=\pm \frac{1}{3}\left(x-1\right)-3[/tex]
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Related Questions
what digit 5 sits in this number 725,069
Answers
Answer:
The digit 5 is equal to is 5,000 :)
Have an amazing day!!
Please rate and mark brainliest!!
Answer:
The thousandth digit!
Step-by-step explanation:
solve these quetions......
Answers
Step-by-step explanation:
a.) 9
When you divide with a fraction you do the reciprocal of the fraction (basically flipping numerator and denominator) then multiplying (since the opposite of division is multiplication.
So if you flip
[tex]\frac{1}{9} [/tex]
It would be:
[tex] \frac{9}{1} [/tex]
9/1 is the same as 9 so:
1 times 9 = 9
b.) 15
Same thing like we did above, just do the reciprocal the multiply.
[tex] \frac{1}{5} [/tex]
Becomes
[tex] \frac{5}{1} [/tex]
5/1 is the same as 5.
3 times 5 = 15
Forces with magnitudes of 2000 newtons and 900 newtons act on a machine part at angles of +45degrees and -60degrees respectively, with the right side of the x-axis. Find the direction and magnitude of the resultant of these forces.
Answers
Answer:
First,you need to find the components of each force.
Sanjay is planting a rectangular garden. He wants the length of his garden to be 12 feet. He wants the area to be at least 60 square feet. Which graph best shows all the possible widths, in feet, for his garden?
Answers
Answer:
The answer is b
Step-by-step explanation:
Answer: B
I had to help my brother with this
solve the following inequality
4x+15≥x+6
Answers
Answer:
[tex]\boxed{\sf{x\geq -3}}[/tex]Step-by-step explanation:
The following inequality can be solved by separating the term x from one side.
4x+15≥x+6
First, subtract by 15 from both sides.
4x+15-15≥x+6-15
Solve.
4x≥x-9
Then, you subtract by x from both sides.
4x-x≥x-9-x
Solve.
3x≥-9
Divide by 3 from both sides.
3x/3≥-9/3
Solve.
Divide the numbers from left to right.
-9/3=-3
[tex]\Longrightarrow: \boxed{\sf{x\geq -3}}[/tex]
Therefore, the correct answer is x≥-3.I hope this helps, let me know if you have any questions.
Answer:
x ≥ -3
Step-by-step explanation:
To determine the solution to the inequality, we need to isolate the variable and its coefficient on one side of the equation. Furthermore, isolate the variable by dividing the coefficient by both sides of the equation. The inequality obtained after doing these steps is the solution to the inequality.
Given inequality:
4x + 15 ≥ x + 6As said above, let's isolate "x" and it's coefficient on one side of the equation. This can be done by subtracting "x + 15" to both sides of the equation.
⇒ 4x + 15 - (x + 15) ≥ x + 6 - (x + 15)⇒ 4x + 15 - x - 15 ≥ x + 6 - x - 15⇒ 3x ≥ -9As said above, let's further isolate "x" by dividing the coefficient of "x" to both sides of the equation.
⇒ 3x/3 ≥ -9/3⇒ x ≥ -3Therefore, the solution is x ≥ -3.
Find the maximum rate of change of the function f (x, y) = x^2y^2 + xy −10 at the
point (1, 1), and the direction in which it occurs.
Answers
Answer:
The maximum rate of change of the given function is equal to 3√2.
The direction at which the vector points is given by the unit vector [tex]\displaystyle \frac{\sqrt{2}}{2} \hat{\i} + \frac{\sqrt{2}}{2} \hat{\j}[/tex].
General Formulas and Concepts:
Pre-Calculus
Vectors
Unit Vector:[tex]\displaystyle \frac{\overline{v}}{|\overline{v}|}[/tex]
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Multivariable Calculus
Partial Derivatives
Gradient:
[tex]\displaystyle \nabla f(x, y, z) = \frac{\partial f}{\partial x} \hat{\i} + \frac{\partial f}{\partial y} \hat{\j} + \frac{\partial f}{\partial z} \hat{\text{k}}[/tex]
Gradient Property [Addition/Subtraction]:
[tex]\displaystyle \nabla \big[ f(x) + g(x) \big] = \nabla f(x) + \nabla g(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify given.
[tex]\displaystyle f(x, y) = x^2y^2 + xy - 10[/tex]
[tex]\displaystyle \text{Point} \ (1, 1)[/tex]
Step 2: Find Maximum Rate of Change
[Function] Take gradient:[tex]\displaystyle \nabla f(x, y) = \frac{\partial}{\partial x} \bigg[ x^2y^2 + xy - 10 \bigg] \hat{\i} + \frac{\partial}{\partial y} \bigg[ x^2y^2 + xy - 10 \bigg] \hat{\j}[/tex][Gradient] Rewrite [Gradient Property - Addition/Subtraction]:
[tex]\displaystyle \nabla f(x, y) = \bigg[ \frac{\partial}{\partial x} x^2y^2 + \frac{\partial}{\partial x} xy - \frac{\partial}{\partial x} 10 \bigg] \hat{\i} + \bigg[ \frac{\partial}{\partial y} x^2y^2 + \frac{\partial}{\partial y} xy - \frac{\partial}{\partial y} 10 \bigg] \hat{\j}[/tex][Gradient] Differentiate [Derivative Rule - Basic Power Rule]:
[tex]\displaystyle \nabla f(x, y) = \bigg( 2xy^2 + y \bigg) \hat{\i} + \bigg( 2yx^2 + x \bigg) \hat{\j}[/tex][Gradient] Substitute in point:
[tex]\displaystyle \nabla f(1, 1) = 3 \hat{\i} + 3 \hat{\j}[/tex][Gradient] Take magnitude:
[tex]\displaystyle \bigg| \nabla f(1, 1) \bigg| = \boxed{3 \sqrt{2}}[/tex]
∴ the maximum rate of change of the given function f(x, y) is equal to 3√2.
Step 3: Find Direction
[Gradient] Convert to unit vector:[tex]\displaystyle \text{Direction} = \boxed{ \frac{\sqrt{2}}{2} \hat{\i} + \frac{\sqrt{2}}{2} \hat{\j} }[/tex]
∴ we have found the direction of the gradient.
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---
Topic: Multivariable Calculus
Unit: Directional Derivatives
What type of function is shown below?
y=6/19x
Answers
Answer:
That is a linear function
Step-by-step explanation:
A linear function is a function which forms a straight line in a graph. It is generally a polynomial function whose degree is utmost 1 or 0.
If you try to graph that function on a graphing calculator, you will get a straight line.
Therefore, it is a linear function.
mark brainliest please!
Answer:
I think it's a linear one(function)
y=6x÷19
4) What is the measure of angle MLO?
(Equilateral Triangle)
Answers
Answer:
m<MLO = 120°
Step-by-step explanation:
Equilateral Triangles have all 60° angles.
m<MLN = m<OLN = 60°
60°+60°=120°
If an arithmetic sequence has a4 = 107 and a8 = 191, what is a₁?
PLEASE HELP ME
Answers
[tex]\text{nth term of an arithmetic sequence,}\\\\a_n = a_1 + (n-1)d\\\\a_4 = a_1 +3d=107~~~......(i)\\ \\a_8=a_1+7d = 191~~~......(ii)\\\\\\\text{(ii) -(i):}\\\\~~~~~~a_1+7d-a_1-3d = 191 -107\\\\\implies 4d = 84\\\\\implies d = \dfrac{84}4\\\\\implies d= 21\\\\\text{Substitute d= 21 in eq~(i):}\\ \\~~~~~ a_1 +3(21) = 107\\\\\implies a_1 + 63 = 107\\\\\implies a_1 = 107-63\\\\\implies a_1 =44[/tex]
I WILL GIVE 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT NOOOO SCAMS AND EXPLAIN WHY THAT IS THE ANSWER
Answers
Answer:
24:37
Simplify the ratio by dividing by the GCF, in this case, two.
[tex]\frac{48 / 2}{74 / 2} = \frac{24}{37}~also~known~as~24:37[/tex]
I WILL GIVE 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT NOOOO SCAMS AND EXPLAIN WHY THAT IS THE ANSWER
Answers
Answer:
here
Step-by-step explanation:
if you need more explanation comment on this
Y=-3×-2 graph can someone please help me with this please
Answers
Answer:
6
Step-by-step explanation:
Negative ×negative always gives gives a positive
A 9 ft. ladder leans against the side of a house. The bottom of the ladder is 6 ft. from the side of the house. How high is the top of the ladder from the ground? If necessary, round your answer to the nearest tenth.
Answers
Answer:
15 ft
Step-by-step explanation:
I'm not sure but I think u add them together to get ur answer
hope it helps
n -1 1/2 for n= 1 1/7 cant figure this out
Answers
Answer:-5/14
Step-by-step explanation:
make then equal and insert N, 16/14-21/14=-5/14
Refer to attachment for question
Answers
Answer:
Below!
Step-by-step explanation:
Let the irrational number be known as "x".
[tex]\implies -\sqrt{41} \times x = 1[/tex]
Divide both sides by -√41.
[tex]\implies \dfrac{(-\sqrt{41} \times x)}{-\sqrt{41} } = \dfrac{1}{\sqrt{-41} }[/tex]
[tex]\implies x = \dfrac{1}{-\sqrt{41} }[/tex]
Take the "-" to the numerator:
[tex]\implies x = \dfrac{-1}{\sqrt{41} }[/tex]
Use parenthesis to isolate the "-"
[tex]\implies x = -\huge\text{(}\dfrac{1}{\sqrt{41} } \huge\text{)}[/tex]
Note: 1 can also be written as √1. (√1 = √1 × √1 = 1)
[tex]\implies {x = -\huge\text{(}\dfrac{\sqrt{1}}{\sqrt{41} }\huge\text{)}}[/tex]
Combine the roots in the equation:
[tex]\implies {x = -\huge\text{(}\sqrt{\dfrac{1}{41} }} \huge\text{)}}[/tex]
Remove the parenthesis:
[tex]\implies {x = -\sqrt{\dfrac{1}{41} }}[/tex]
Thus, Option C is correct.
OLZ HELP ILL MARK AS BRAINLEST
Answers
Answer:
-2
Step-by-step explanation:
The slope is rise over run. If we look at the points we can see the rise is -2 and the run is 1. -2/1 = -2, so my answer is correct.
Answer:
Hi I’m the friend
Step-by-step explanation:
by the way I also need brainliest, so you get to choose who to mark as brainliest
If csc ( x ) = 4, for 90 ∘ < x < 180, then sin ( x /2 ) = ? cos ( x /2 ) = ? tan ( x/ 2 ) = ?
Answers
Answer:
see below
Step-by-step explanation:
cscx=4
sinx=1/4
cosx=-√(1-sin^2x)=-√(1-(1/16))=-√(15/16)=-√15/4
note: cos<0,sin>0 in quadrant II
sin(x/2)= [tex]\sqrt{\frac{1+cos(x)}{2} } =\sqrt\frac{1+\sqrt{\frac{15}{4} } }{2} =\sqrt{\frac{4+\sqrt{15} }{8} }[/tex]
cos(x/2)= [tex]\sqrt{\frac{1-cos(x)}{2} } =\sqrt{\frac{1-\sqrt{\frac{15}{4} } }{2} } =\sqrt{\frac{4-\sqrt{15} }{8} }[/tex]
tan(x/2)= [tex]\frac{sin(x)}{1-cos(x)} =\frac{\frac{1}{4} }{(1-\frac{\sqrt{15} }{4} )} =\frac{\frac{1}{4} }{\frac{4-\sqrt{15} }{4} } =\frac{1}{4-\sqrt{15} }[/tex]
Calculator check:
[tex]sinx=\frac{1}{4}[/tex]
[tex]x\neq 165.52[/tex]˚ in quadrant II
[tex]\frac{x}{2} \neq 82.76[/tex] ˚
sin(x/2)≈sin(82.76)≈0.9920..
exact value=√(4+√15)/8≈0.9920..
..
cos(x/2)≈cos(82.76)≈0.1260..
exact value=√(4-√15)/8≈0.1260..
..
tan(x/2)=tan(82.76)≈7.872..
exact value=1/(4-√15)≈7.872..
Please help now, I will give Brainliest!!! :D
Peter is 19 years old. He lives at home with his parents and goes to college part time. He recently started working as a server, working 40 hours per week.
Where Peter lives, the minimum wage for tipped and non tipped employees is $7.25 per hour.
In the average week, he serves 90 tables whose typical bill is $21 with an average tip of 15%.
How much money does he make in a typical week??
Answers
Answer:
$573.50 per week
Step-by-step explanation:
wages = 40 hr/wk * 7.25 $/hr $290 per week
tips 90 tables * ( 21*.15) = 283.50
total 290 + 283.5 = 573.50
Consider the equation v+4+v=8. What is the resulting equation after the first step in the solution?
0v+4=8-v
Ov+4=8
04+v=8-v
O2v + 4 = 8
Answers
You must combine like terms v + v = 2v
Therefore, the answer is 2v+4=8
The resulting equation after the first step in the solution is:
2v + 4 = 8.
What is Equation?An equation is a mathematical statement that asserts the equality between two expressions. It typically contains one or more variables and may involve mathematical operations such as addition, subtraction, multiplication, division, or exponentiation.
we have the Equation v + 4 + v =8
Now, simplifying we get
2v + 4= 8
2v = 4
v =2
To obtain this equation, we combined the like terms "v" on the left side of the equation, which gives us 2v. The rest of the equation remains unchanged.
Therefore, the correct answer is : 2v + 4 = 8.
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friends needs this anwsered help
Answers
Answer:
3, 8, 14, 30, 58
Step-by-step explanation:
As with all formulas, put the numbers in place of the corresponding variables, and do the arithmetic.
[tex]a_1=3\qquad\text{given}\\\\a_2=8\qquad\text{given}\\\\a_3=2a_1+a_2=2(3)+8=14\\\\a_4=2a_2+a_3=2(8)+14=30\\\\a_5=2a_3+a_4=2(14)+30=58[/tex]
The first 5 terms of the sequence are 3, 8, 14, 30, 58.
Anybody know the slope of the line is … Help
Answers
Answer: The slope is -1
Step-by-step explanation:
When you are given a line on a graph all you need to do is pick two points on the line and calculate the change over both the x and y-axis.
Pick both points to be at a position on the line that is easy to read. IE the intersection between two grid lines. For this problem lets go with the first point at (0,1) and the second point at (1,0).
Now we want to calculate the change in rise (the y-component) and divide by the change in the run (the x-component). Point 1 has a y-component of 1 and point 2 has a y-component of 0.
1 - 0 = 1 = rise
Point 1 has an x-component of 0 and point 2 has an x-component of 1.
0 - 1 = -1 = run
rise/run = 1/-1
Slope = -1
Need Answer Please And thank you
Answers
Answer:
157.78
Step-by-step explanation:
Area of a trapezoid = ( sum of base lengths / 2 ) × height
the base lengths of the trapezoid shown are 8.7m and 16.3 + 7.2 = 23.5m
the height is 9.8m
so area = (( 8.7 + 23.5 ) / 2) × 9.8
==> add 8.7 and 23.5
area = (32.2 / 2) × 9.8
==> divide 32.2 by 2
area = 16.1 × 9.8
==> multiply 16.1 and 9.8
area = 157.78
A can in the shape of a cylinder is filled 75% of the way with liquid. The can has a height of 10 cm and a radius of 2 cm. Please determine following:
Answers
Answer:
um..is this finished question ????
Write an equation in slope-intercept form of the line shown.
An equation is
Answers
Using the slope intercept formula, the equation for the line in slope intercept form is y = -x - 4
Slope intercept equationThe slope intercept equation can be describe as follows:
y = mx + b
where
m = slopeb = y-interceptTherefore,
(-4, 0)(1, -5)
m = -5 - 0 / 1 + 4 = -5 / 5 = -1
Therefore, let's find b using (1, -5)
-5 = -1(1) + b
-5 = -1 + b
b = -5 + 1
b = -4
Therefore, the equation is y = -x - 4
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A sequence can be generated by using
an=an-1-5 where
a1 = 100 and n is a whole number greater
than 1.
(a) What are the first five terms in the sequence?
(b) Write an iterative rule for the sequence. Show your work
Answers
a) The first five terms in the sequence is 100, 95, 90, 85 and 80
b) The nth term of the sequence will be 100 + (n - 1)(5) = 5n + 95
Iterative rule for a sequenceSequences are set of values arranged in a particular order. Given the recursive formula for a sequence expressed as;
an=an-1-5
If the first term a1 = 100, then;
a2 = a1 - 5
a2 = 100 - 5
a2 = 95
For the third term
a3= a2 - 5
a3= 95 - 5
a3 = 90
The other two terms will be 85 and 80
a) The first five terms in the sequence is 100, 95, 90, 85 and 80
b) The nth term of the sequence will be 100 + (n - 1)(5) = 5n + 95
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what is the median of 7,3,4,7,5,6,5,5,4
Answers
Your median is 5 because that is the number that is mentioned the most.
A median is a number that i said the most.
And in the pic your median is 7.
Hope this helps you :0
p^2 + 3p - 18 is divided by p - 3.
Answers
Answer:
p + 6
Step-by-step explanation attached as image.
Which is zero of the polynomial expression 7x+21
Answers
7x+21=0
Now solve for x, start by isolating the variable (x) by subtracting 21 from both sides
7x = -21
The only thing stopping c from being by itself is 7, so we divide both sides by 7 to isolate x
x = -3
3.2] I can rearrange formulas to solve for a specified variable.
Solve for r.
h=r-m
Answers
Answer:
r = h + m
Step-by-step explanation:
h = r - m
Switch sides.
r - m = h
Add m to both sides.
r - m + m = h + m
Simplify the left side. -m + m = 0
r = h + m
Solve the system using elimination.
4X +5Y = 22
X +3Y =0
Answers
Answer:
x = 3
y = -2
--------------------------------------------------------------------------------------------------------
4x + 5y = 2........i
2x + 3y = 0........ii
---------------------------------------------------------------------------------------------------------
Let's start by eliminating x, so we multiply i by 2 and ii by 4
2|4x + 5y = 2
4|2x + 3y = 0
So that we have
8x + 10y = 4..........iii
8x + 12y = 0..........iv
---------------------------------------------------------------------------------------------------
Subtracting iv from III, we have
-2y = 4
y = 4/-2
y = -2
---------------------------------------------------------------------------------------------------
Going ahead and substituting this value of y in eqn i, we have
4x + 5(-2) = 2
4x - 10 = 2
4x = 2 + 10
4x = 12
x = 12/4
x = 3
Step-by-step explanation: hope this helps, brainliest please <3