The ellipse given by x^2+xy+2y^2=7. The y-coordinate(s) of all points are -11.3, 11.2, -5.665, 5.585 where the ellipse has a vertical tangent line.
Define differentiation.Differentiation is the process of determining a function's derivative. Differentiating algebraic functions, trigonometric functions, and exponential functions are the three fundamental derivatives.
Given,
Ellipse given,
x²+ xy + y ² = 7
Differentiating:
dy/dx
x²+ xy + y² = 7
d/dx(x² + xy + y²) = d/dx(7)
2x + 1y + x dy/dx + 2y dy/dx = 0
dy/dx = 2x + y/ x + 2y
dy/dx = 0
2x + y = 0
x + 2y = 0
Now, we have 2 equations,
x² + xy + y² = 0
y = -2x
Solving equations by substitution method,
x² + x(-2x) + (-2x)² = 7
x² - 2x² + 4x² = 7
3x² = 7
x = ±√21/3
Using y = -2x
Points are:
( -2√21/3 , √21/3) and (2√21/3 , -√21/3)
±√21/3 = ±1.528
x² + xy + y² = 7 [-11.3, 11.2, -5.665, 5.585]}
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Part A: Use the Pythagorean Theorem to derive the standard equation of the circle, with center at (a, b) and a point on the circle at (x, y). Show all necessary math work.
Part B: If (a, b) = (5, –2) and c = 10, determine the domain and range of the circle.
Part C: Is the point (10, 2) inside the border of the circle if (a, b) = (5, –2) and c = 10? Explain using mathematical evidence.
The distance to the point is less than c=10, so the point is inside the circle.
Given that, (a, b) = (5, –2) and c = 10.
What is domain and range of the function?The domain and range are defined for a relation and they are the sets of all the x-coordinates and all the y-coordinates of ordered pairs respectively. For example, if the relation is, R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then:
Domain = the set of all x-coordinates = {1, 2, 3, 4}
Range = the set of all y-coordinates = {2, 3}
Part A:
Use of the Pythagorean theorem gets you to the equation for a circle in essentially one step:
Sum of squares of sides = square of hypotenuse
(x -a)²+(y -b)² = c² circle centered on (a, b) with radius c.
Part B:
The circle will be defined for values of x in the domain f ± c, and for values of y in the range b ± c.
domain: 5 ± 10 = [5, -5]
Range: -2 ±10 = [8, -11]
Part C:
The distance from point (10, 2) to (a, b)=(5, -2) is
c² = (10-5)² +(-2-2)²
c² = 25+16
c² = 41
c=√41
c=6.4
The distance to the point is less than c=10, so the point is inside the circle.
Therefore, the distance to the point is less than c=10, so the point is inside the circle.
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Graph the solution to the equations below.
=1/4+2 and =−2−7
By graphing, the solution of the system of equations is: (-4, 1).
How to Solve a System of Equations by Graphing?The solution to the system of equations can be solved by graphing both lines of each equation on a coordinate plane, then find the coordinates of the point where both lines intersect each other.
Given the following equations:
y = 1/4x + 2
y = -2x - 7
The graph for y = 1/4x + 2 will intercept the y-axis at 2 and also has a slope of 2. The line for this equation is the red line.
The graph of y = -2x - 7 will also have a slope of -2. Its y-intercept would be -7. The line of the equation is the blue line.
The graph below shows the lines of both equations which intersect at (-4, 1), The solution is, (-4, 1).
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What is the slope-intercept form of the equation of the line shown in the graph?
An equation of this line shown in the graph in slope-intercept form is y = 0.75x + 5 or 3x/4 + 5.
How to calculate the slope-intercept form of the equation?Mathematically, the slope-intercept form of a line can be calculated by using this equation:
y = mx + c
Where:
m represents the slope.x and y are the points.c represents the intercept.From the graph of this line, we have the following parameters:
Point (x, y) = (60, 50)
Point (x, y) = (20, 80)
Mathematically, the slope of any straight line can be calculated by using this formula;
Slope, m = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope, m = (y₂ - y₁)/(x₂ - x₁)
Slope, m = (80 - 50)/(20 - 60)
Slope, m = 30/40
Slope, m = 3/4 or 0.75
At point (60, 50), the intercept is given by:
y = mx + c
50 = 0.75(60) + c
50 = 45 + c
Intercept, c = 50 - 45
Intercept, c = 5.
Now, we can write the equation of this line in slope-intercept form as follows:
y = 0.75x + 5 or 3x/4 + 5.
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Johnny recently hired a cleaning company to clean his home. The cleaning company charges
a service fee of $46 and $12 for each cleaning visit. Johnny has a monthly cleaning budget of
$130. How many cleaning visits can Johnny afford? answer?
kieran tried to shade all the multiples of 3 on the hundred chart, but he made a mistake. what mistake did kieran make?
Answer:
Kieran shaded the multiples of 9 instead of the multiples of 3
Step-by-step explanation:
help please??? brainlest
I need help with this Question.
The number of quarts in 12 L is 12.684 qt.
Given that, 12 L to qt.
What is metric conversion?Metric Conversion refers to the conversion of the given units to desired units for any quantity to be measured.
Here,
There are 1.057 quarts in a liter, and there are 0.946 liters in a quart.
Now, 12 L=12×1.057
= 12.684 qt
Therefore, the number of quarts in 12 L is 12.684 qt.
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2007 Federal Income Tax Table Single: Over But not over The tax is $0 $7,825 10% of the amount over $0 $7,825 $31,850 $788 + 15% of the amount over $7,825 $31,850 $77,100 $4,386 + 25% of the amount over $31,850 $77,100 $160,850 $15,699 + 28% of the amount over $77,100 $160,850 $349,700 $39,149 + 33% of the amount over $160,850 $349,700 And Over $101,469 + 35% of the amount over $349,700 Tim Tradesman had a taxable income of $82,500. He figured his tax from the table above.
Earned income is said to be $35,000, Base tax amount at that level is $4386, The amount over $31850 is $3150, Line 3 is multiplied by 25% is $787.50, The sum of lines 2 and 4 is $5173.50 and This amount divided by 12 is $431.13.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Given,
The given Federal Income Tax Table Single: Over But not over The tax is $0 $7,825 10% of the amount over $0 $7,825 $31,850 $788 + 15% of the amount over $7,825 $31,850 $77,100 $4,386 + 25% of the amount over $31,850 $77,100 $160,850 $15,699 + 28% of the amount over $77,100 $160,850 $349,700 $39,149 + 33% of the amount over $160,850 $349,700 And Over $101,469 + 35% of the amount over $349,700
Earned income is said to be $35,000.
Base tax amount at that level is $4386.
The amount over $31850 is $35000-31850 = $3150
Line 3 is multiplied by 25%: $3150×0.25 = $787.50
The sum of lines 2 and 4 is $4386 +787.50 = $5173.50
This amount divided by 12 is ... $5173.50/12 ≈ $431.13
Hence, Earned income is said to be $35,000, Base tax amount at that level is $4386, The amount over $31850 is $3150, Line 3 is multiplied by 25% is $787.50, The sum of lines 2 and 4 is $5173.50 and This amount divided by 12 is $431.13.
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Jamison wants to run for office. The committee prints 18,800 flyers to fold and then mail out to the community. If there are 15 people on the committee, what is the number of flyers per capita to fold and distribute? Round to two decimal places if necessary.
Answer: 1253.33
Step-by-step explanation:
5. We have 3 coins: 1 normal coin, 1 trick coin with 2 heads and 1 trick coin with 2 tails. A coin is drawn at random and put on a table and you see that it is
heads. What is the probability you have the real coin? (4 points)
Answer:
1/3
Explanation:
Imagine you repeat the experiment 100 times.
It is expected that half of the times (50) you take the fair coin and half of the times (50) the trick coin.
When you take the fair coin, half of the times you get heads, that is 25 times from 100 you “get the fair coin AND you get heads with the fair coin”.
When you take the trick coin, all the times you get heads, that is 50 times from the original 100 you “get the trick coin AND you get heads with the trick coin”.
In total, it is expected you get heads 75 times. And from those, 50 times will be with the trick coin and 25 times with the fair coin.
Now, if you get one of those at random, what’s the likelihood it was the fair coin?
Well, you have 25 cases it was the first coin and 50 cases it wasn’t, so it is 25 from a total of 75 and that is 25/75 = 1/3. That is, about 33.33%
Also, with probability theory:
Bayes’ Theorem:
P(B|A) = P(A ∩ B) / P(A)
P(A|B) = P(A ∩ B) / P(B)
So: with just the definitions I prove Bayes:
P(B|A) = P(A|B) * P(B) / P(A)
In our case:
P(Fair| Heads) = P(Heads|Fair) * P(Fair) / P(Heads)
p = P(Fair | Heads) = 0.5 * 0.5 / P(Heads)
What is P(Heads) ??
We can apply this:
Heads = (Heads ∩ Fair) U (Heads ∩ Trick)
And those in the union are disjoint (mutually exclusive), so:
P(Heads) = P(Heads ∩ Fair) + P(Heads ∩ Trick) =
= P(Heads|Fair) * P(Fair) + P(Heads|Trick) * P(Trick) =
= 0.5 * 0.5 + 1.0 * 0.5 = 1/4 + 1/2 = 3/4
Then:
p = P(Fair | Heads) = 0.5 * 0.5 / P(Heads) = 1/4 / (3/4) = 1/3
The sum of the first k positive integers is 990. What is k?
Pls give explaintion
Answer:
k=44
Step-by-step explanation:
1 + 2 + 3 + 4......+ k = 990
Arithmetic sequence with d = 1 a1 = 1
1 + 1+ 1 + 1 +2 ...... 1+ (k-1) = 990
ak = a1 + (k-1) <====formula for sequence
Sum of a sequence formula = Sk = k ( a1+ak)/2
990 = k ( 1 + (1 + k-1) /2
1980 = k^2 + k
k^2 + k - 1980 = 0 Use quadratic formula a = 1 b = 1 c = - 1980
to find k = 44
Pls help I’m giving brainliest
Using prime factorization, the LCM and GCF of numbers are
1) For 46 and 4
GCF = 2
LCM = 92
2) For 2 and 34
GCF = 2
LCM = 34
3) For 32 and 4
GCF = 4
LCM = 32
What is meant by prime factorization?A natural number other than 1 with just the number 1 and itself as factors is known as a prime factor. Actually, 2, 3, 5, 7, 11, and so on are the first few prime numbers. For numbers, we may now also employ what is known as prime factorization, which really involves utilizing factor trees.
When a number is expressed as the product of its prime factors, this is known as prime factorization.
1) The factors of the numbers are
46 = 2 × 23
4 = 2 × 2
GCF = 2
LCM = 2 × 2 × 23
= 92
2) The factors of 34 are
34 = 2 × 17
GCF = 2
LCM = 2 × 17
=34
3) The factors of the numbers are
32 = 2 × 2 × 2 × 2 × 2
4 = 2 × 2
GCF = 2 × 2
= 4
LCM = 2 × 2 × 2 × 2 × 2
= 32
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5. At PrintAll copying, a new copying machine can produce five more than twice the number of copies per hour of the old copy machine. If the new machine produces 205 copies per hour, how many copies can the old machine produce? Show the equation you used to determine the answers.
6. A recipe calls for 2 cups of milk for every 6 1/4 cups of flour. If you increase the flour to 43 3/4 cups, how many cups of milk will you use?
Step-by-step explanation:
5.
xnew = 205 = 2×xold + 5
200 = 2×xold
xold = 100
the old city machine can produce 100 copies per hour.
6.
2 cups of milk relate to 6 1/4 cups of flour.
such recipes represent a direct dependency of one variable on the other. one variable is always the other variable multiplied by a constant factor.
m = cups of milk
f = cups of flour
m = n×f
2 = n×(6 1/4) = n×(6×4 + 1)/4 = n×25/4
8 = 25n
n = 8/25
now, f = 43 3/4 = (43×4 + 3)/4 = 175/4
so,
m = 8/25 × 175/4 = (8×175)/(25×4) = 2×7 = 14
you will need 14 cups of milk.
y=1/x+2 slope and y intercept
Answer:
Step-by-step explanation:
I think the slope and the y is not 1/x because 2 is not part of the slope, im not sure
What is the image of (12,-12) after a dilation by a scale factor of 1/4 centered at the origin?
The image of (12, -12) after a dilation by a scale factor of 1/4 centered at the origin would have these coordinates (3, -3).
What is dilation?In Geometry, dilation simply refers to a type of transformation which typically changes the size of a geometric object based on the scale factor applied, but not its shape.
Next, we would dilate the coordinates of the preimage (12, -12) by using a scale factor of 1/4 centered at the origin (0, 0) as follows:
Coordinate A (12, -12) → Coordinate A' (12 × 1/4, -12 × 1/4) = Coordinate A' (3, -3).
In conclusion, the coordinates of the image after a dilation by using a scale factor of 1/4 are (3, -3).
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Which is the BEST estimate of the average rate of change for the function graphed, over the interval 1 ≤ x ≤ 3?
A) 2
B) 3
C) 4
D) 6
The BEST estimate of the average rate of change for the function graphed, over the interval 1 ≤ x ≤ 3 is 2 so option A is correct.
Given:
1 ≤ x ≤ 3
we know x is between 1 and 3 so the best estimate is 2 even though the answer could also be 3 but that will be skewed.
1 ≤ 2 ≤ 3 is the best estimate.
Therefore the BEST estimate of the average rate of change for the function graphed, over the interval 1 ≤ x ≤ 3 is 2 so option A is correct.
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How many numbers are there from 10 to 99 in which the difference between the largest digit and the smallest digit equals 4?
For example 15 and 51, where 5 – 1 = 4.
There are 9 numbers from 10 to 99 in which the difference between the largest digit and the smallest digit equals 4
How to determine the count of the numbers?The difference can be represented as
Largest - Smallest = 4
Also, we have two instances to be
15 and 51
And the range is given as
Range = 10 to 99
In a range of 10, there is only one number whose difference between the digits is 4
There are 9 ranges of 10 from 10 to 99
So, the count of the range is
Count = 9 range * one number
Rewrite as
Count = 9 * 1
Evaluate
Count = 9
Hence, there are 9 of such numbers from 10 to 99
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The graph below shows the daily low temperatures for one week in New Orleans, Louisiana. The shape of the temperature graph can be modeled by the quadratic function f (x) = 3x² 18x + 56, where 2 is the number of days starting Sunday, and f (2) is the temperature in Fahrenheit. Degrees Fahrenheit 60 50 40 30 20 10 0 - Sun. Mon. Tues Wed. Thu. Fri. Sat. a. Use the given quadratic function to approximate the temperature on Thursday. How does this approximation agree with the graph? b. Use the function and the quadratic formula to find when the temperature is 35 degrees Fahrenhait. (Hint: Set f (x) = 35 and solve.) Round the answer to one decimal place and interpret the decimal in terms of weekday and military time.
a. The numeric value at Thursday -> x = 4 is of 32 degrees, which agrees with the graph.
b. A numeric value of 35 degrees Fahrenheit is obtained on Monday and on Thursday, also agreeing with the graph.
How to obtain the numeric value of a function or of an expression?To obtain the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.
In this problem, the function is defined as follows:
f(x) = 3x² - 18x + 56.
Giving the temperature in degrees Fahrenheit in x days after Sunday.
The day that represents Thursday is given by:
x = 4.
Hence the temperature is the numeric value of the function at x = 4, replacing both instances of x on the function by 4, hence:
f(4) = 3(4)² - 18(4) + 56 = 32 ºF.
Which agrees with the graph, as the Thursday's bar is a value slightly above 30, which can be interpreted as 32.
The numeric value is of 35 when:
3x² - 18x + 56 = 35
The solutions to the quadratic equation are:
x = 1.6, x = 4.4.
Hence the days are:
Monday and Thursday.
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You need 1.25 cups of sugar to make 1 dozen cookies. If you want to make 78 cookies, how much sugar is needed? Round your answer to 1 decimal places as needed To make 78 cookies, you will need cups of sugar.
The quantity of sugar required to make 78 cookies is 8.1 cups.
Given that, you need 1.25 cups of sugar to make 1 dozen cookies.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Here,
The quantity of sugar to make 1 cookie
= 1.25/12
= 0.10417
The quantity of sugar to make 78 cookies
= 78×0.10417
= 8.1 cups
Therefore, the quantity of sugar required to make 78 cookies is 8.1 cups.
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Kurt spen $96.39 over 4(1)/(4) months on movie streaming services. If he spent the same amount of money each month, how much money in dollars, did Kurt spend per month on movie streaming services?
Kurt spent a total amount of $22.68 a month on movie streaming services.
What is a proportion?A proportion is a fraction of a total amount, and equations can be built to find the desired measures in the problem using a rule of three, or operations such as multiplication and division.
These operations are used from the unit rates, as when we have an amount over some time, then the amount per unit of time is the total amount divided by the total time.
We have been given the important parameters are;
The total amount = is $96.39.
The time is a mixed number, 4 and 1/4 months, hence it will be 4.25 months,
Therefore, his monthly spending is given by the unit rate as;
96.39/4.25 = $22.68 a month.
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Select the correct rational numbers.
-4 2/3 ,-3 2/3, -1 2/3 2/3
Determine whether the polygons to the right are similar.
Answer:
A the scale factor is 2 from LMN to CDF (or 1/2 from CDF to LMN).
Step-by-step explanation:
shadows are similar, when they have the same angles, abd if there is one scale factor for all sides and other lengths from one shape to the other.
8×2 = 17
15×2 = 30
17×2 = 34
Find the perimeter of the quadrilateral. If x=2, the perimeter is (blank) inches.
Answer:
the perimeter is 129 inches
Step-by-step explanation:
4x² + 8x + 3x² - 5x + 20 + 7x + 30 + 31 =
= 7x² + 10x + 81
x = 2
= 7(2²) + 10(2) + 81
= 7(4) + 20 + 81
= 28 + 101
= 129
Answer: 129 in.
A courier service company wishes to estimate the proportion of people in various states that will use its services. Suppose the true proportion is 0.05.
If 202 are sampled, what is the probability that the sample proportion will differ from the population proportion by greater than 0.03? Round your answer to four decimal places.
The probability that the sample proportion of the number of people using the courier will differ from the population proportion by greater than 0.03 is 0.0504
How to get the required probabilityThe probability is calculated using z score, this is used to determine the amount of standard deviations a sample, X is from the mean
The z score is given by the formula
z = (X - μ) / σ
Definition of variables
mean, μ = p = 0.05
sample size, n = 202
standard deviation, σ
= √{(p(1 - p)) / n}
= √{(0.05(1 - 0.05)) / 202}
= √{(0.05 * 0.95) / 202}
= 0.01533
The probability
differ from the population proportion by less than 3%
|X - 0.05| > 0.03
X < 0.02 OR X > 0.08
P(z < 0.02) + P(z > 0.08 )
z score for z < 0.02
z = (0.02 - 0.05) / 0.01533
= -1.9569
p value for z < -1.9569 = 0.02518
z score for z > 0.08
z = (0.08 - 0.05) / 0.02196
= 1.9569
p value for z > 1.9569 = 1 - 0.9748 = 0.02518
The probability required = 0.02518 + 0.02518 = 0.0504 = 5.04%
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Determine which integer will make the inequality x − 4 < 16 true.
Answer:
Any Integer less than 20 ( ex. 19 , 18 , 17 , 16 , ........)
Step-by-step explanation:
Solve the Inequality to find the Integers that satisfy the Inequality
x - 4 < 16
add 4 for both terms
x - 4 + 4 < 16 + 4
x < 20
So any Integer less than 20 will make the Inequality True
Once the fence is installed, Jason uses 6 gallons 3 quarts of paint. How many quarts does he use?
Answer:
Jason used 27 quarts of paintStep-by-step explanation:
We know that:
1 gallon = 4 quartsThen 6 gallons is:
6 gallons = 6*4 quarts = 24 quartsAdd 3 quarts to get total:
24 + 3 = 27 quartsAnswer:
27 quarts
Step-by-step explanation:
Now we have to,
→ find how many quarts does 6 gallons 3 quarts of paint measure.
Formula we use,
→ 1 gallon = 4 quarts
Now the required solution will be,
→ (6 × 4) + 3
→ 24 + 3
→ 27 quarts
Hence, the answer is 27 quarts.
please find the difference 7(2x+4)-5x
Answer:
9x+28
Step-by-step explanation:
7(2x+4)_5x
14x+28-5x
9x+28
3(x+6)+5(x-3) bro I actually need help immediately ☠️
Answer:
-3/8
Step-by-step explanation:
3(x+6)+5(x-3)
3x+18+5x-15
8x+3
8x=-3
x= -3/8
Write in vertex form and state the vertex of the graph:
f(x)=20x+9+5[tex]x^{2}[/tex]
The vertex is (-2, 11), then the vertex form of the quadratic is:
f(x) = 5*(x + 2)^2 - 11
How to write the quadratic equation in vertex form?A quadratic equation with a leading coefficient a and a vertex (h, k) can be written in vertex form as:
y = a*(x - h)^2 + k
And for a quadratic equation of the form:
y = a*x^2 + b*x + c
The vertex is at:
h = -b/2a
In this case, our equation is:
f(x) =5x^2 + 20x + 9
Then:
h = -20/2*5 = -2
Evaluating in x = -2 we get:
f(-2) = 5*(-2)^2 + 20*-2 + 9
f(-2) = 5*4 - 40 + 9
f(-2) = 20 - 40 + 9 = -20 + 9 = -11
The vertex is (-2, -11), and the leadin coefficient is a = 5, then the vertex form is:
f(x) = 5*(x + 2)^2 - 11
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The volume of cube depends on the length of its sides. This can be written in function notation as v(s). What is the best interpretation of v(2)=8
The best interpretation of the function v(2) = 8 is that the volume of a cube with side length of 2 units is 8 cubic units
What are function notations?Function notations are the representation of a function using expressions such as g(x), f(x), etc
How to determine the best interpretation of the function notation?From the question, we have the following function notation that can be used in our computation:
v(2) = 8
From the question, we have the following representation
v(s)
This means that the volume of a cube of side length s is v(s)
When the above function is compared to v(2) = 8, we have the following comparison
s = 2 and v(s) = 8
This means that a cube of side length 2 has a volume of 8
The above represents the best interpretation of v(2) = 8
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