Answer:
3rd grade
Step-by-step explanation:
Given that the values are different types (fractions, decimals, and percentages), it would be helpful to convert them to the same type of value.
Converting everything to percentages seems more convenient.
Since 61.24% is already a percentage, we simply have to convert 0.52, 25/36, and 0.5274444 (I wrote 0.5274 like this since the 4 is a repeating value).
To convert 0.52, we simply multiply by 100:
0.52 * 100 = 52%
For 25/36, we need to know its decimal form and multiply by 100 to find the percentage:
25 / 36 = 0.6944 * 100 = 69.44%
For 0.5274444, we also multiply by 100:
0.5274444 * 100 = 52.74444
Thus, we have 52% (2nd grade), 69.44% (3rd grade), 61.24% (4th grade), and 52.74444% (5th grade).
3rd grade has the highest portion of students
Evaluate The quantity of x cubed plus 3x squared minus 2x plus 7 divided by the quantity of x minus 2 end quantity period.
1. x squared minus 5x plus 2 plus 23 divided by the quantity of x minus 2
2.x cubed plus 5x plus 8 plus 11 divided by the quantity of x minus 2 end quantity
3.x squared minus 5x plus 6 plus 3 divided by the quantity of x minus 2 end quantity 4. x squared plus 5x plus 8 plus 23 divided by the quantity of x minus 2 end quantity
The value of x^3 + 3x^2 - 2x + 7 divided by x- 2 is (x^2 + 5x + 8) + 23/(x - 2)
What is a quotient?Quotients involve the result of dividing a dividend by a divisor.
In other words, quotient means division or the result of a division operation
How to solve the quotient?The quotient expression is given as:
(x^3 + 3x^2 - 2x + 7)/x- 2
Expand the numerator in the above expression
(x^3 + 5x^2 - 2x^2 + 8x - 10x - 16 + 23)/(x - 2)
Rearrange the terms of the numerator in the above expression
(x^3 + 5x^2 + 8x - 2x^2 - 10x - 16 + 23)/(x- 2)
Factorize the numerator in the above expression
[x(x^2 + 5x + 8) - 2(x^2 + 5x + 8) + 23]/(x - 2)
Factor out x^2 + 5x + 8
[(x -2)(x^2 + 5x + 8) + 23]/(x - 2)
Split the fractions
(x -2)(x^2 + 5x + 8)/(x - 2) + 23/(x - 2)
Divide the common factors
(x^2 + 5x + 8) + 23/(x - 2)
Hence, the value of x^3 + 3x^2 - 2x + 7 divided by x- 2 is (x^2 + 5x + 8) + 23/(x - 2)
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Complete question
Evaluate (x^3 + 3x^2 - 2x + 7)/x- 2
Answer:
4 / D | x^2 + 5x + 8 + 23/x - 2Step-by-step explanation:
The correct answer is D because when you divide & simplify the original question's equation, you get the results that are equivalent to answer D.
△ABC has vertices A(-2, 0), B(0,8), and C(4,2) Find the equations of the three altitudes of △ABC
The equations of the three altitudes of triangle ABC include the following:
3y - 2y - 4 = 0.y + 3x - 8 = 0.4y + x - 6 = 0.What is a triangle?A triangle can be defined as a two-dimensional geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.
What is a slope?A slope is also referred to as gradient and it's typically used to describe both the ratio, direction and steepness of the function of a straight line.
How to determine a slope?Mathematically, the slope of a straight line can be calculated by using this formula;
[tex]Slope, m = \frac{Change\;in\;y\;axis}{Change\;in\;x\;axis}\\\\Slope, m = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}[/tex]
Also, the point-slope form of a straight line is given by this equation:
y - y₁ = m(x - x₁)
Assuming the following parameters for triangle ABC:
Let AM be the altitudes on BC.Let BN be the altitudes on CA.Let CL be the altitudes on AB.For the equation of altitude AM, we have:
Slope of BC = (2 - 8)/(4 - 0)
Slope of BC = -6/4
Slope of BC = -3/2
Slope of AM = -1/slope of BC
Slope of AM = -1/(-3/2)
Slope of AM = 2/3.
The equation of altitude AM is given by:
y - y₁ = m(x - x₁)
y - 0 = 2/3(x - (-2))
3y = 2(x + 2)
3y = 2x + 4
3y - 2y - 4 = 0.
For the equation of altitude BN, we have:
Slope of CA = (2 - 0)/(4 - (-2))
Slope of CA = 2/6
Slope of CA = 1/3
Slope of BN = -1/slope of CA
Slope of BN = -1/(1/3)
Slope of BN = -3.
The equation of altitude BN is given by:
y - y₁ = m(x - x₁)
y - 8 = -3(x - 0)
y - 8 = -3x
y + 3x - 8 = 0.
For the equation of altitude CL, we have:
Slope of AB = (8 - 0)/(0 - (-2))
Slope of AB = 8/2
Slope of AB = 4
Slope of CL = -1/slope of AB
Slope of CL = -1/4
The equation of altitude CL is given by:
y - y₁ = m(x - x₁)
y - 2 = -1/4(x - 4)
4y - 2= -(x - 4)
4y - 2= -x + 4
4y + x - 2 - 4 = 0.
4y + x - 6 = 0.
In conclusion, we can infer and logically deduce that the equations of the three altitudes of triangle ABC include the following:
3y - 2y - 4 = 0.y + 3x - 8 = 0.4y + x - 6 = 0.Read more on point-slope form here: brainly.com/question/24907633
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Instructions: Identify the vertices of the feasible region for the given linear programming constraints.
Optimization Equation:
z=−3x+5y
Constraints:
x+y≥−2
3x−y≤2
x−y≥−4
Fill in the vertices of the feasible region:
(0, )
(−3, )
(3, )
The vertices of the feasible region are (0, -2), (-3, 1) and (3, 7)
How to identify the vertices of the feasible region for the given linear programming constraints?The optimization equation is given as
z=−3x+5y
The constraints are given as:
x+y≥−2
3x−y≤2
x−y≥−4
Next, we plot the constraints on a graph and determine the points of intersections
See attachment for the graph
From the attached graph, the points of intersections are
(-3, 1), (3, 7) and (0, -2)
So, we have:
(0, -2)
(-3, 1)
(3, 7)
Hence, the vertices of the feasible region are (0, -2), (-3, 1) and (3, 7)
So, the complete parameters are:
Optimization Equation:
z=−3x+5y
Constraints:
x+y≥−2
3x−y≤2
x−y≥−4
Vertices of the feasible region
(0, -2)
(-3, 1)
(3, 7)
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Find the value of x.
Answer:
x = 52
Step-by-step explanation:
Since the inscribed angle measures 64 degrees, the minor arc is 64*2 = 128 degrees. Therefore the major arc is 180-128 = 232 degrees. Angle X is the difference of major and minor arc divided by 2, so X = (232-128)/2 = 52 degrees.
Working together a small pipe and large pipe can fill a big pool in 6 hour. It takes the smaller pipe 5 hours longer than the large pipe to fill the big pool working alone. How long does it take the smaller pipe to fill the pool by itself ?
The time taken for the smaller pipe to fill the pool by itself is 15.71 hours
Rate of workTime taken for both pipes = 6 hoursTime taken for long pipe = xTime taken for small pipe = x + 6Rate of work of both pipes = 1/6Rate of work of long pipe = 1/xRate of work of small pipe = 1/x + 61/6 = 1/x + 1/(x+6)
1/6 = (x+6)+(x) / (x)(x+6)
1/6 = (x+6+x) / x²+6x
1/6 = (2x+6)(x² + 6x)
cross product1(x² + 6x) = 6(2x+6)
x² + 6x = 12x + 36
x² + 6x - 12x - 36 = 0
x² - 6x - 36 = 0
Using quadratic formulax = 9.71 or -3.71
The value of x cannot be negative
Therefore, the
Time taken for long pipe = x
= 9.71 hours
Time taken for small pipe = x + 6
= 9.71 + 6
= 15.71 hours
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HELP!
ANGLE 1 = 52°
(Show work)
The measure of the unknown angles are as shown below;
m<2 = 128 degrees = m<4
m<5 = 128 degrees
m<6 = 52 degrees
m<3 = 52 degrees = m<8
m<7 = 128 degrees
Lines and anglesAn angle is the point where the two lines meet. From the given diagram, the lines B and D are parallel to each other with a transversal.
Given the following parameters
m<1 = 52 degrees
From the given diagram, the following are true
m<1 = m<3 (corresponding angle)
m<3 = m<8 (vertical angles)
m<1 = m<6 (vertical angles)
m<2 = m<5 (vertical angles)
m<4 = m<7 (vertical angles)
Also;
m<1 + m<2 = 180
m<2 = 180 - m<1
m<2 = 180 - 52
m<2 = 128 degrees = m<4
m<5 = 128 degrees
m<6 = 52 degrees
m<3 = 52 degrees = m<8
m<7 = 128 degrees
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Ellen read a book a day for 10 weeks. How many books did
she read in all?
1 week = 7 days
10 weeks = 10×7 days
= 70 days
if Ellen reads 1 book in a day
Therefore,
1 day = 1 book
70 days = 70×1
= 70 books.
Thus Ellen reads 70 books in 10 weeks.
Hope helps
Answer:
70 books
Step-by-step explanation:
Ellen reads one book per day for 10 weeks.
All we need to do to know the number of books Ellen read overall is to multiply 1 (number of books he read each day) and 10 weeks (total number of days).
Each week is 7 days, so 10 weeks is equal to 70 days.
The equation will be: 1 x 70
We can conclude that Ellen read 70 books overall within 10 weeks.
If one angle in a parallelogram is a right angle prove the other are the same.
Answer: If one angle of a parallelogram is a right angle, then all other angles would also be right angles and the parallelogram would be a rectangle.
Anthony travels from Newcastle to Manchester at an average speed of 65 miles per hour.
The journey takes him 2 hours and 15 minutes.
Declan makes the same journey in 2 hours and 35 minutes.
(a) Work out Declan's average speed for the journey.
Answer:
See below
Step-by-step explanation:
Distance = rate * time
= 65 m/hr * 2 1/4 hr = 146.25 miles
rate = distance / time
for Declan : rate = 146.25 miles / (2 hrs + 35/60 min) = 56.61 mph
CAN SOMEONE HELP PLEASE!
Answer:
None of these
Step-by-step explanation:
[tex]\frac{360}{n}=24 \implies n=15[/tex]
This is called a pentadecagon.
write and solve an inequality that means a number plus four than or equal to twelve.
The correct answer for the solution of inequality is [tex]x\geq 8[/tex].
Let the unknown number be "[tex]x[/tex]."
The statement "a number plus four" can be written as "[tex]x+4[/tex]."
The phrase "is greater than or equal to twelve" can be expressed as "[tex]\geq 12.[/tex]
Put these together, the inequality becomes:
[tex]x + 4\geq 12[/tex]
solve for "[tex]x[/tex]," isolate the variable on one side of the inequality:
Subtract -[tex]4[/tex] from both side of the expression:
[tex]x + 4 - 4 \geq 12 - 4[/tex]
[tex]x\geq 8[/tex]
The correct expression for the inequality is [tex]x\geq 8[/tex]
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can someone please help me
The formula that shows the relationship between the distance, rate and time is t = d/r and t = 5.
What is an equation?
An equation is an expression that shows the relationship between two or more numbers and variables.
An independent variable is a variable that does not depend on any other variable for its value while a dependent variable is a variable that depends on other variable.
Given that:
d = rt
Hence:
t = d/r
For d = 40, r = 8:
t = 40 / 8
t = 5
The formula that shows the relationship between the distance, rate and time is t = d/r and t = 5.
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select the correct answer. What is the area of the triangle in the diagram
Answer:
B
Step-by-step explanation:
The area of the triangle is half the base time the height. The height is y sub 1. The length is the distance between the x values.
Use the normalcdf function on a calculator to find the probability that battery life is 20 ± 2 hours (between 18 and 22 hours) for each phone.
Prior values (If needed)
.159
.50
The probability that the battery life is 20 ± 2 hours (between 18 and 22 hours) for each of the phones is:
Phone C = 15.9%Phone T = 50%What is the probability of the battery life?Using the normalcdf function on a calculator, the probability that Phone C's battery would last between 18 and 22 hours can be found by inputing the function:
normalcdf (20, IE99, 18, 2)
The result is 0.158655 or 15.9%.
For Phone T, the function is:
normalcdf (20, IE99, 20, 3)
The result is 0.5 or 50%.
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What number is six and four hundredths larger than two and five tenths?
Answer:
eight and thirty-seven fiftyths
Step-by-step explanation:
6 and 4/100 + 2 and 5/10
6 and 1/25 + 2 and 1/2
156 / 25 + 5 / 2
312 / 50 + 125 / 50
437 / 50
400 / 50 = 8
8 and 37 / 50
Find the future value and interest earned if $8704.56 is invested for 8 years at 4% compounded (a) semiannually and (b) continuously.
intorntic compounded semiannually is approximately
a) The future value, principal plus interest, with compound interest on a principal of $8,704.56 at a rate of 4% per year compounded 2 times per year over 8 years is $11,949.50.
b) The future value, principal plus interest, with compound interest on a principal of $8,704.56 at a rate of 4% per year compounded continuously over 8 years is $11,987.29.
How is the future value determined?The future value can be determined using an online finance calculator.
Data and Calculations:
a) Compounded Semiannually:Principal (P): $8,704.56
Annual Rate (R): 4%
Compound (n): Compounding Semi-Annually
Time (t in years): 8 years
Result:
A = $11,949.50
A = P + I where
P (principal) = $8,704.56
I (interest) = $3,244.94
Calculation Steps:First, convert R as a percent to r as a decimal
r = R/100
r = 4/100
r = 0.04 rate per year,
Then solve the equation for A
A = P(1 + r/n)nt
A = 8,704.56(1 + 0.04/2)(2)(8)
A = 8,704.56(1 + 0.02)(16)
A = $11,949.50
b) Compounded Continuously:Using the formula A = Pert
Principal (P): $8,704.56
Annual Rate (R): 4%
Compound (n): Compounding Continuously
Time (t in years): 8 years
Result:
A = $11,987.29
A = P + I where
P (principal) = $8,704.56
I (interest) = $3,282.73
Calculation Steps:First, convert R as a percent to r as a decimal
r = R/100
r = 4/100
r = 0.04 rate per year,
Then solve the equation for A, using the mathematical constant, e = 2.71828
A = Pert
A = 8,704.56(2.71828)(0.04)(8)
A = $11,987.29
Thus, while the future value of $8,704.56 at a rate of 4% per year compounded semiannually over 8 years is $11,949.50, the future value of $8,704.56 at a rate of 4% per year compounded continuously over 8 years is $11,987.29.
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Find the experimental probability that only 2
of 4 children in a family are boys.
The problem has been simulated by tossing 4
coins (one to represent each child). Let "heads"
represent a boy and "tails" represent a girl. A
sample of 20 coin tosses is shown.
HTHH HTTH TTTT THTT HTHT
HHTT HHHT THHT
TTHH
HTTT HTHT TTHH THTH HTHH
TTHT HTTT HTHT HHHT HHHH
HTTH
Experimental Probability = [?]%
The experimental probability that only 2 of 4 children in a family are boys is 50%.
What is experimental probability?Experimental probability refers to the chance of an expected success being achieved in a series of experiments conducted.
Experimental probability is the number of times that the expected success occurs as a fraction of the total number of times the experiment was conducted.
Like all probabilities, the experimental probability is based on the likelihood that what the experimenter expects is achieved.
Expected number of boys = 2
The number of children in the family = 4
Experimental probability = 50% (2/4 x 100)
Thus, we can conclude, based on the experimental probability, that 50% (or 2) of the 4 children in the family are boys.
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Answer:
15%
Step-by-step explanation:
Step 1: Count samples with one H
Why; Only 1 of 4 children in families are boys. The problem is simulated with samples.
Step 2: Count samples with only one H.
Why; The H represents the one boy in the family.
Step 3: Add the samples you counted with one H.
Answer: Your answer should be 3/20 after you change it into a percentage your final answer is 15%
Find the gradient of the line passing through the points (– 2,– 4) and (3,5).
Answer:
Gradient of the line is choice D.9/5
Step-by-step explanation:
Hello!Slope between two points:slope=(y₂-y₁)/(x₂-x₁)
(x₁.y₁)=(-2,-4)(x₂.y₂)=(3,5)slope(m)=
[tex] \frac{5 - ( - 4)}{3 - ( - 2)} \\ refine \: (m )= \frac{9}{5} [/tex]
elsa sold 24 drawings for $12 each at the art fair. She is going to use 1/3 of the money to buy books. The rest of the money is going into her savings account. How much money will she put into her savings account?
if a rectangular piece of metal has 27.75 square inches what is the length and width?
We cannot get further information about the dimensions of the piece since the number of variables is greater than the number of equations.
What are the dimensions of a rectangular piece of metal?
By geometry we know that the area of the piece of metal is equal to the product of its length and width, then we must find two real numbers such that:
l · w = 27.75, where l, w > 0.
Unfortunately, we cannot get further information about the dimensions of the piece since the number of variables is greater than the number of equations. We need at least one equation to find an unique solution.
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Prove: In an equilateral triangle the three medians are equal
Step-by-step explanation:
Let ABC be the equilateral triangle. Let AE, BD and CF be the medians. A meridian divides a side into two equal parts. Hence, proved that medians of an equilateral triangle are equal .
Zoe is shopping for a new car and has to make some decisions. The model she’s chosen comes in two versions, hybrid and electric. For the exterior, she can choose red, blue, or green. For the interior, she can choose fabric, leather, or vinyl.
If Zoe randomly chooses from the options, the probability that Zoe picks a blue hybrid car with an interior that is not leather is
.
If Zoe randomly chooses from the options, the probability of Zoe picking an electric car in a color other than blue is
a) The probability is P = 1/9
b) The probability is P = 1/3.
How to get the probability?First, we need to count the total number of outcomes (different cars that can be made with the given options):
There are 2 versions (electric and hybrid).There are 3 exterior colors (red, blue, green).There are 3 interiors (fabric, leather, vinyl)Then there are 2*3*3 = 18 different cars.
a) Picking a blue hybrid car with an interior that is not leather.
There are 2 cars that meet that condition:
blue hybrid with fabric.blue hybrid with vinyl.2 out of the 18 cars meet the condition, then the probability is:
P = 2/18 = 1/9
b) picking an electric car in a color other than blue is
The cars that meet the condition are:
Green electric with any interior (3 options here)Red electric with any interior (3 options here)So 6 out of the 18 cars meet the condition, then the probability is:
P = 6/18 = 1/3
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what is 4,928 will rounded to the nearest hundred
Answer:
4900
Step-by-step explanation:
When rounding a number such as 4928 to the nearest hundred, we use the following rules:
A) We round the number up to the nearest hundred if the last two digits in the number are 50 or above.
B) We round the number down to the nearest hundred if the last two digits in the number are 49 or below.
C) If the last two digits are 00, then we do not have to do any rounding, because it is already to the hundred.
In this case, Rule B applies and 4928 rounded to the nearest hundred is:
4900
Answer:
Step-by-step explanation:
4,928, look at the last two numbers 28 if they are above 50 you round up, if below 50 round down,
The answer is 4,900
a) If x = a + 7 and y = b-a, show that x + y = b + 7.
Step-by-step explanation:
x = a + 7
y = b-a
To prove
x + y = b + 7.
x+y= (a+7) + (b-a)=a+7+b-a
=a-a+7+b
=0 + 7+b
= b+7
Proved ✅
1. You have learned several techniques for solving quadratic equations. Create an equation that would be best for each technique listed below. Then explain why that technique would be best for each equation that you created.
2.What is the discriminant and what happens if you are solving and the discriminant turns out to be negative?
The techniques that can be used to best explain for each equation that I have created are:
What are the techniques?A) For graphing - This is often used in checking if your answer is correct, if the math is known to be a little tedious, graphing is recommended
(b) For factoring: x^2 -3x +4 '
Where: (x-4)(x+1)=0
Then x=-1, 4
(c) For square root method: x^2 = 64
Therefore x = + or - [tex]-\sqrt{64}[/tex] = + or - 8
(d) For completing the square: x^2 + 4x = 11 and x^2 +4x + 4
Note that: x^2 +4x + 4
= x^2 + 4x = -4
= x^2 +4x + 4 = -4 + 4
= (x+2)^2 = 0
So, x = -2
(e) For quadratic formula: 6x² + 7x -19 =0
x= [-b + or - sqr(b² - 4ac)]/2a
Note that b=7
a=6
c=-19
Therefore
x= [-7+ or - sqr(49+4(6)(9)]/2
2. if the discriminant is said to be negative, then the equation is one that is made up of two imaginary or complex solutions.
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See full question below
You have learned several techniques for solving quadratic equations. Create an equation that would be best for each technique listed below. Then explain why that technique would be best for each equation that you created.
a. graphing
b. factoring
c. square root method
d. completing the square
e. quadratic formula
2. What is the discriminant and what happens if you are solving and the discriminant turns out to be negative?
If 1/2x = a= 2/3y and x + y= na, what is the value of n?
Answer:
Step-by-step explanation:
[tex]\frac{1}{2} x=a=\frac{2}{3} y\\\frac{1}{2}x=a\\x=2a \\\frac{2}{3} y=a\\y=\frac{3}{2} a\\x+y=2a+\frac{3}{2} a=\frac{4a+3a}{2} =\frac{7}{2} a=na\\n=\frac{7}{2}[/tex]
Identify what a, b, and c would equal in standard form.
Do not simplify the equation.
-14y= 28-10x
a=
b=
C=
Answer:
below
Step-by-step explanation:
-14y = 28 - 10x
10x - 14y - 28 = 0.
a = 10
b = -14
c = -28
Answer:
a = 10
b = -14
c = 28
Step-by-step explanation:
Goal form: Standard form
ax + by = c
Given form: linear form (not fully simplified)
-14y = 28 - 10x
Add 10x on both sides (to move the x-value to the left-hand side of the equation)
-14y + 10x = 28 - 10x + 10x
-14y + 10x = 28
10x - 14y = 28
Determine [ a ], [ b ], [ c ]
ax + by = c
10x - 14y = 28
[tex]\Large\boxed{a=10}[/tex]
[tex]\Large\boxed{b=-14}[/tex]
[tex]\Large\boxed{c=28}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Graph the following conditions. {(x, y): x + 2y > 6}
See attachment for the graph of the inequality expression x + 2y > 6
How to graph the conditions?The condition is given as:
{(x, y): x + 2y > 6}
This means that
x + 2y > 6
The above is an inequality expression, where the variables are x and y
So, we simply plot the graph of the inequality expression x + 2y > 6
See attachment for the graph of the inequality expression x + 2y > 6
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NO LINKS!! Please help me with this problem
[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
The given figure shows a vertical hyperbola with its centre at origin, and as we observe the figure, we can conclude that :
Length of transverse axis is :
[tex]\qquad \sf \dashrightarrow \: 2b = 12[/tex]
[tex]\qquad \sf \dashrightarrow \: b = 6[/tex]
length of conjugate axis is :
[tex]\qquad \sf \dashrightarrow \: 2a = 8[/tex]
[tex]\qquad \sf \dashrightarrow \: a = 4[/tex]
Equation of hyperbola ~
[tex]\qquad \sf \dashrightarrow \: \cfrac{ {y}^{2} }{ {b}^{2} } - \cfrac{ {x}^{2} }{ {a}^{2} } = 1[/tex]
plug in the values ~
[tex]\qquad \sf \dashrightarrow \: \cfrac{ {y}^{2} }{ {6}^{2} } - \cfrac{ {x}^{2} }{ {4}^{2} } = 1[/tex]
[tex]\qquad \sf \dashrightarrow \: \cfrac{ {y}^{2} }{ {36}^{} } - \cfrac{ {x}^{2} }{ {16}^{} } = 1[/tex]
Answer:
[tex]\dfrac{y^2}{36}-\dfrac{x^2}{16}=1[/tex]
Step-by-step explanation:
Standard form equation of a vertical hyperbola
[tex]\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1[/tex]
where:
center = (h, k)vertices = (h, k±a)co-vertices = (h±b, k)foci = (h, k±c) where c² = a² + b²[tex]\textsf{asymptotes}: \quad y =k \pm \left(\dfrac{a}{b}\right)(x-h)[/tex]Transverse axis: x = hConjugate axis: y = kFrom inspection of the graph:
center = (0, 0) ⇒ h = 0, k = 0vertices = (0, 6) and (0, -6) ⇒ a = 6co-vertices = (4, 0) and (-4, 0) ⇒ b = 4Substitute the found values into the formula:
[tex]\implies \dfrac{(y-0)^2}{6^2}-\dfrac{(x-0)^2}{4^2}=1[/tex]
[tex]\implies \dfrac{y^2}{36}-\dfrac{x^2}{16}=1[/tex]
"Solve the following first order differential equation for x(t):
x'=-9tx"
How do I do this?
I'm not sure if the last two apostrophes are part of the quote - "Solve ... " - or if you mean the second derivative [tex]x''[/tex]. I think you mean the first interpretation, but I'll include both cases since they are both solvable.
If the former is correct, separate variables to solve.
[tex]x' = -9tx \implies \dfrac{dx}{dt} = -9tx \implies \dfrac{dx}x = -9t\,dt[/tex]
Integrate both sides to get
[tex]\ln|x| = -\dfrac92 t^2 + C[/tex]
Solve for [tex]x[/tex].
[tex]e^{\ln|x|} = e^{-9/2\,t^2 + C} \implies \boxed{x = Ce^{-9/2\,t^2}}[/tex]
If you meant the latter, then the ODE can be rewritten as
[tex]9t x'' + x' = 0[/tex]
Reduce the order of the equation by substituting [tex]y(t) = x'(t)[/tex] and [tex]y'(t) = x''(t)[/tex].
[tex]9t y' + y = 0[/tex]
Solve for [tex]y'[/tex] and separate variables.
[tex]y' = -\dfrac y{9t} \implies \dfrac{dy}{dt} = -\dfrac y{9t} \implies \dfrac{dy}y = -\dfrac{dt}{9t}[/tex]
Integrate.
[tex]\ln|y| = -\dfrac19 \ln|t| + C[/tex]
Solve for [tex]y[/tex].
[tex]e^{\ln|y|} = e^{-1/9 \,\ln|t| + C} \implies y = Ct^{-1/9}[/tex]
Solve for [tex]x[/tex] by integrating.
[tex]x' = Ct^{-1/9} \implies x = C_1 t^{8/9} + C_2[/tex]