The MLE of σ^2 for a random sample X1, X2, ..., Xn from a normal distribution with known mean μ is s^2/(n-1), and the MLE of σ is the square root of s^2.
Given a random sample X1, X2, ..., Xn from a normal distribution with mean μ and unknown variance σ^2, the likelihood function is:
L(σ^2) = (1/(2πσ^2)^(n/2)) * exp[-(1/(2σ^2)) * ∑(Xi - μ)^2]
To find the maximum likelihood estimator (MLE) of σ^2, we need to find the value of σ^2 that maximizes the likelihood function.
To do this, we take the natural logarithm of the likelihood function, since the logarithm is a monotonically increasing function and thus does not change the location of the maximum:
ln L(σ^2) = (-n/2) ln(2πσ^2) - (1/(2σ^2)) * ∑(Xi - μ)^2
To maximize this expression, we take the derivative with respect to σ^2 and set it equal to zero:
d/d(σ^2) ln L(σ^2) = (-n/2σ^2) + (1/(2(σ^2)^2)) * ∑(Xi - μ)^2 = 0
Solving for σ^2, we get:
σ^2 = (∑(Xi - μ)^2) / n
This means that the MLE of σ^2 is the sample variance, s^2 = (∑(Xi - μ)^2) / (n-1), since we usually use the sample variance to estimate the population variance when the population mean is known.
Therefore, the MLE of σ is the square root of the sample variance:
σ(hat) = sqrt(s^2)
To now more about MLE:
https://brainly.com/question/29842773
#SPJ4
____the given question is incomplete, the complete question is given below:
Suppose that X1, . . . , Xn form a random sample from a Normal distribution for which mean μ is known, but the variance σ
2
is unknown. Find the MLE (maximum likelihood estimation) of σ.
I’ll give brainliest don’t solve just check!
Answer:
Yep
Step-by-step explanation:
Yep. Checking this, all of these are correct. Range on parabolas and absolutes will always go to infinity. Nice work
10. What is the total surface area of the drawing?
A. 549 km²
B. 256 km²
C. 564 km²
D. 265 km²
Help please! I have no idea!!!!
Answer:
14,4,10
Step-by-step explanation:
The histogram below shows the average number of days per year in 117 Oklahoma cities where the high temperature was greater than 90 degrees Fahrenheit. Which is an accurate comparison? The mean is likely greater than the median because the data is skewed to the right.
The accurate comparison shown by Histogram is that the mean is likely greater than the median because the data is skewed to the right. So, the correct option is A).
The histogram shows a right-skewed distribution, with a tail extending to the right of the peak. This indicates that there are a few cities with very high values that are pulling the mean to the right. In a right-skewed distribution, the mean is always greater than the median. This is because the mean is sensitive to extreme values and the median is not.
Therefore, option A is the accurate comparison. Option B is incorrect because the data is not skewed to the left. Option C is incorrect because the median is always less than the mean in a right-skewed distribution. Option D is also incorrect because the median is always less than the mean in a left-skewed distribution. The correct answer is A).
To know more about histogram:
https://brainly.com/question/30354484
#SPJ4
_____The given question is incomplete, the complete question is given below:
3 4 5 8 9 10
The histogram below shows the average number of days per year in 117 Oklahoma cities where the high temperature was greater than 90 degrees Fahrenheit.
Which is an accurate comparison?
A The mean is likely greater than the median because the data is skewed to the right.
B The mean is likely greater than the median because the data is skewed to the left.
C The median is likely greater than the mean because the data is skewed to the right.
D The median is likely greater than the mean because the data is skewed to the left.
The Venn diagram here shows the cardinality of each set. Use this to find the cardinality of the given set.
cardinality of the given set n(A'∩B') is 19
Define Venn DiagramIn a Venn diagram, circles are used to represent connections between objects or small groups of objects. Circles that overlap have some properties, but circles that do not overlap do not have properties. Venn diagrams are very useful for showing how two concepts are related and different graphically.
This Venn diagram illustrates the relationship between the subsequent set of integers.
Whereas the other set comprises the numbers in the 5x table from 1 to 25, the first set only contains even numbers from 1 to 25.
The intersection component demonstrates that 10 and 20 are both multiples of 5 from 1 to 25 and even integers.
Complement of A =14+5+12+7=38
Complement of B=7+4+3+9+12+7=42
n(A'∩B')=19
To know more about integers, visit:
https://brainly.com/question/15276410
#SPJ1
Fai spend $9 on his lunch. This is 30% of the money he had in his wallet. How much money did Fai have in his wallet?
Answer:
Fai's wallet contained $30.
Step-by-step explanation:
$9 = 30%
x = 100%
If you cross multiply:
x * 30% = $9 * 100%
0.3x = $9
x = $9/0.3
x = $30
Create a random triangle,
ABC . Record the lengths of its sides.
The triangle ABC with the given dimensions:AB = 10, BC = 6, CA = 90 are constructed.
Explain about the side-side-side congruence?You may determine the third angle by deducting the first two angles from 180 if you know the angles of one SSS triangle and another SSS triangle.Triangles that have corresponding sides with the same measurements are subject to the SSS hypothesis. A triangle with sides of 3, 4, & 5 and a triangle with sides of 4, 3, and 5 are two examples. SSS triangles—triangles whose values of all three sides coincide also with parameters of the second triangle—are also known as SSS triangles.The dimension of the triangle ABC are:
AB = 10
BC = 6
CA = 90
Thus, the triangle with the given dimensions are constructed.
Know more about the side-side-side congruence
https://brainly.com/question/3999145
#SPJ1
Which graph matches the function given:
a chord 7cm long is drawn in a circle of radius 3.7cm. calculate the distance of the chord from the centre of the circle
Answer: To find the distance of a chord from the center of a circle, we need to use the following formula:
Distance from center = sqrt(r^2 - (c/2)^2)
Where r is the radius of the circle and c is the length of the chord.
In this case, the radius of the circle is 3.7cm and the length of the chord is 7cm.
So, substituting these values in the formula, we get:
Distance from center = sqrt(3.7^2 - (7/2)^2)
= sqrt(13.69 - 12.25)
= sqrt(1.44)
= 1.2 cm
Therefore, the distance of the chord from the center of the circle is 1.2 cm.
Step-by-step explanation:
The area of the intersection of a circle and a triangle is 45% of the area of their union. The area of the triangle outside the circle is 40% of the area of their union. What percentage of the circle lies outside the triangle?
Answer:
Percentage of the circle that lies outside the triangle = 15%
Step-by-step explanation:
Given,
The area of the intersection of circle and triangle is 45% of the area of the union.
The area of the triangle outside of the circle is 40% of the area of their union.
Required to find,
The percentage of the circle lies outside the triangle
Let us take 'C' as the area of the circle and 'T' as the area of the triangle.
The union of the area of the circle and triangle = C∪T
Let CUT be A
and the percentage of the circle that lies outside the triangle be 'x'
The intersection of the area of the circle and triangle = C∩T
Area of the triangle outside the circle = T - C
Area of the circle outside the triangle = C - T
Given,
C∩T = 45% of CUT = 45% of A
T - C = 40% of CUT = 40% of A
We know that,
The union of the area of the circle and triangle = Area of The intersection of the area of the circle and triangle + Area of the triangle outside the circle + Area of the circle outside the triangle
(CUT) = (C∩T) + (T - C) + (C - T)
[tex]A = 45\%A + 40\%A + x\%A[/tex]
[tex]A = (85+x)\% A[/tex]
[tex]85 +x = 100[/tex]
∴ [tex]x = 15[/tex]
Percentage of the circle that lies outside the triangle = 15%
(b) There are 40 students in a class. If the number of boys is 10 more than that of girls, find the number of boys and girls.
Answer:
Boys are 25
Girls are 15
Step-by-step explanation:
Total class population = 40
Let girls population = g
Then, boys population = (10+g)
Boys + Girls = 40
[tex]{ \sf{(10 + g) + g = 40}} \\ { \sf{2g = 30}} \\ { \sf{g = 15}}[/tex]
Girls = 15
Boys = (10 + g) = (10 + 15) = 25
Create a trigonometric function that models the ocean tide..
Explain why you chose your function type. Show work for any values not already outlined above.
Answer:
One possible function that models the ocean tide is:
h(t) = A sin(ωt + φ) + B
where:
h(t) represents the height of the tide (in meters) at time t (in hours)
A is the amplitude of the tide (in meters)
ω is the angular frequency of the tide (in radians per hour)
φ is the phase shift of the tide (in radians)
B is the mean sea level (in meters)
This function is a sinusoidal function, which is a common type of function used to model periodic phenomena. The sine function has a natural connection to circles and periodic motion, making it a good choice for modeling the regular rise and fall of ocean tides.
The amplitude A represents the maximum height of the tide above the mean sea level, while B represents the mean sea level. The angular frequency ω determines the rate at which the tide oscillates, with one full cycle (i.e., a high tide and a low tide) occurring every 12 hours. The phase shift φ determines the starting point of the tide cycle, with a value of zero indicating that the tide is at its highest point at time t=0.
To determine specific values for A, ω, φ, and B, we would need to gather data on the tide height at various times and locations. However, typical values for these parameters might be:
1. A = 2 meters (representing a relatively large tidal range)
2. ω = π/6 radians per hour (corresponding to a 12-hour period)
3. φ = 0 radians (assuming that high tide occurs at t=0)
4. B = 0 meters (assuming a mean sea level of zero)
Using these values, we can write the equation for the tide as:
h(t) = 2 sin(π/6 t)
We can evaluate this equation for various values of t to get the height of the tide at different times. For example, at t=0 (the start of the cycle), we have:
h(0) = 2 sin(0) = 0
indicating that the tide is at its lowest point. At t=6 (halfway through the cycle), we have:
h(6) = 2 sin(π/2) = 2
indicating that the tide is at its highest point. We can also graph the function to visualize the rise and fall of the tide over time:
Tide Graph
Overall, this function provides a simple and effective way to model the ocean tide using trigonometric functions.
(please mark my answer as brainliest)
An aquarium of height 1.5 feet is to have a volume of 12ft^3. Let x denote the length of the base and y the width.
a) Express y as a function of x.
b) Express the total number S of square feet of glass needed as a function of x.
The aquarium has six rectangular faces, four of which are identical (two sides and two ends), and two of which are identical to each other but different from the others (top and bottom).
What is the needed as a function?a) We can use the formula for the volume of a rectangular prism, which is given by V = lwh, where l is the length, w is the width, and h is the height. In this case, we have [tex]V = 12 ft^3 and h = 1.5 ft[/tex] . We want to express y as a function of x, so we need to eliminate w from the equation.
We can rearrange the equation for the volume to get [tex]w = V/(lh)[/tex] , and substitute in h [tex]= 1.5 ft[/tex] :
[tex]w = V/(1.5lx)[/tex]
Now we can substitute y for w to get:
[tex]y = V/(1.5lx)[/tex]
b) To find the total surface area, we need to find the area of each face and add them up.
The area of one of the identical sides or ends is lw, so the total area of these four faces is:
[tex]4lw = 4xy[/tex]
The area of the top and bottom faces is lx, so the total area of these two faces is:
[tex]2lx[/tex]
Therefore, the total surface area S is given by:
[tex]S = 4xy + 2lx[/tex]
We can express y in terms of x using the equation from part a):
[tex]y = V/(1.5lx)[/tex]
Substituting this into the expression for S, we get:
[tex]S = 4x(V/(1.5lx)) + 2lx[/tex]
Simplifying, we get:
[tex]S = (8/3)V/x + 2lx[/tex]
So the total surface area S is a function of x, and we can use this equation to find the value of S for any given value of x.
Learn more about function here:
https://brainly.com/question/14002287
#SPJ1
A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply 1 unit of protein, 2 units of carbohydrates, and 1 unit of fat. Each ounce of nuts will supply 1 unit of protein, 1 unit of carbohydrates, and 1 unit of fat. Every package must provide at least 8 units of protein, at least 11 units of carbohydrates, and no more than 10 units of fat. Let x equal the ounces of fruit and y equal the ounces of nuts to be used in each package.
Let x be the number of ounces of fruit and y the number of ounces of nuts. Referring to the chart, give the three inequalities that x and y must satisfy because of the package's requirements for protein, fat, and carbohydrate.
__ _ 8
__ _ 11
__ _ 10
Give the inequalities that x and y must satisfy because they cannot be negative.
y ≥ __
x ≥ __
Answer:
Step-by-step explanation:
The three inequalities that x and y must satisfy because of the package's requirements for protein, fat, and carbohydrate are:
Protein: 1x + 1y ≥ 8 (at least 8 units of protein)
Carbohydrates: 2x + 1y ≥ 11 (at least 11 units of carbohydrates)
Fat: 1x + 1y ≤ 10 (no more than 10 units of fat)
The inequalities that x and y must satisfy because they cannot be negative are:
x ≥ 0
y ≥ 0
Step-by-step explanation:
To satisfy the requirements for protein, fat, and carbohydrates, the following three inequalities must be satisfied:
1. 1x + 1y ≥ 8 (At least 8 units of protein)
2. 2x + 1y ≥ 11 (At least 11 units of carbohydrates)
3. 1x + 1y ≤ 10 (No more than 10 units of fat)
To ensure that x and y are non-negative, the following inequalities must be satisfied:
x ≥ 0y ≥ 0
Therefore, the complete set of inequalities for x and y are:
x + y ≥ 82x + y ≥ 11x + y ≤ 10x ≥ 0y ≥ 0
The windows to a Tudor-style home create many types of quadrilaterals. Use the picture of the window below to answer the following questions.
Please help me I will give literally anything
a. Determine which type of quadrilaterals you see. Name these quadrilaterals using the labeled vertices.
b. What properties of quadrilaterals would you have to know to identify the parallelograms in the picture? Be specific as to each type of parallelogram by using the properties between sides, angles, or diagonals for each.
Answer:
I'd be happy to help!
a. From the picture of the window, we can identify the following quadrilaterals:
Rectangle: ABCD (all angles are right angles and opposite sides are parallel and congruent)
Parallelogram: EFGH (opposite sides are parallel and congruent)
Trapezoid: BCGH (at least one pair of opposite sides are parallel)
b. To identify the parallelograms in the picture, we would need to know the following properties of parallelograms:
Opposite sides are parallel and congruent
Opposite angles are congruent
Diagonals bisect each other
Using these properties, we can identify the following parallelograms in the picture:
Parallelogram EFGH: Opposite sides EF and GH are parallel and congruent, and opposite sides EG and FH are also parallel and congruent. Additionally, angles E and G are congruent, and angles F and H are congruent.
Rectangle ABCD: Opposite sides AB and CD are parallel and congruent, and opposite sides AD and BC are also parallel and congruent. Additionally, angles A and C are congruent, and angles B and D are congruent. The diagonals AC and BD bisect each other, meaning that they intersect at their midpoints.
Step-by-step explanation:
What is the equation of the circle in the standard (x, y) coordinate plane that has a radius of 4 units and the same center as the circle determined by x^2 + y^2 - 6y + 4=0?
A. x² + y^2 = -4
B. (x+3)^2 + y^2 = 16
C. (x-3)^2 + y^2 = 16
D. x^2 + (y+3)^2 = 16
E. x^2 + (y-3)^2 = 16
Answer:
E. x² + (y - 3)² = 16
Step-by-step explanation:
The equation of a circle in the standard (x, y) coordinate plane with center (h, k) and radius r is given by:
[tex]\boxed{(x - h)^2 + (y - k)^2 = r^2}[/tex]
To find the equation of the circle with a radius of 4 units and the same center as the circle determined by x² + y² - 6y + 4 = 0, we need to first write the equation of the second circle in the standard form.
We can complete the square for y to rewrite this equation in standard form. To do this move the constant to the right side of the equation:
[tex]\implies x^2 + y^2 - 6y + 4 = 0[/tex]
[tex]\implies x^2 + y^2 - 6y = -4[/tex]
Add the square of half the coefficient of the term in y to both sides of the equation:
[tex]\implies x^2 + y^2 - 6y +\left(\dfrac{-6}{2}\right)^2= -4+\left(\dfrac{-6}{2}\right)^2[/tex]
[tex]\implies x^2 + y^2 - 6y +9= -4+9[/tex]
[tex]\implies x^2 + y^2 - 6y +9=5[/tex]
Factor the perfect square trinomial in y:
[tex]\implies x^2+(y-3)^2=5[/tex]
[tex]\implies (x-0)^2 + (y-3)^2=5[/tex]
So the center of this circle is (0, 3) and its radius is √5 units.
Since the new circle has the same center, its center is also (0, 3).
We know its radius is 4 units, so we can write the equation of the new circle as:
[tex]\implies (x - 0)^2 + (y - 3)^2 = 4^2[/tex]
[tex]\implies x^2 + (y - 3)^2 = 16[/tex]
Therefore, the equation of the circle in the standard (x, y) coordinate plane with a radius of 4 units and the same center as the circle determined by x² + y² - 6y + 4 = 0 is x² + (y - 3)² = 16.
To find:-
The equation of circle which has a radius of 4units and same centre as determined by x² + y² - 6y + 4 = 0.Answer:-
The given equation of the circle is ,
[tex]\implies x^2+y^2-6y + 4 = 0 \\[/tex]
Firstly complete the square for y in LHS of the equation as ,
[tex]\implies x^2 + y^2 -2(3)y + 4 = 0 \\[/tex]
Add and subtract 3² ,
[tex]\implies x^2 +\{ y^2 - 2(3)(y) + 3^2 \} -3^2 + 4 = 0 \\[/tex]
The term inside the curly brackets is in the form of a²-2ab+b² , which is the whole square of "a-b" . So we may rewrite it as ,
[tex]\implies x^2 + (y-3)^2 -9 + 4 = 0 \\[/tex]
[tex]\implies x^2 + (y-3)^2 - 5 = 0 \\[/tex]
[tex]\implies x^2 + (y-3)^2 = 5\\[/tex]
can be further rewritten as,
[tex]\implies (x-0)^2 + (y-3)^2 = \sqrt5^2\\[/tex]
now recall the standard equation of circle which is ,
[tex]\implies (x-h)^2 + (y-k)^2 = r^2 \\[/tex]
where,
(h,k) is the centre.r is the radius.So on comparing to the standard form, we have;
[tex]\implies \rm{Centre} = (0,3)\\[/tex]
Now we are given that the radius of second circle is 4units . On substituting the respective values, again in the standard equation of circle, we get;
[tex]\implies (x-h)^2 + (y-k)^2 = r^2 \\[/tex]
[tex]\implies (x-0)^2 + (y-3)^2 = 4^2 \\[/tex]
[tex]\implies \underline{\underline{\red{ x^2 + (y-3)^2 = 16}}}\\[/tex]
and we are done!
When Bazillium released its signature trading-card game, Wandering Wizards, the value of a first-edition deck decreased at first. As the game got more popular and the deck became more rare, its value started to increase. The value of a first-edition deck in dollars can be modeled by the expression 0.25t2–4t+28, where t is the time in years after it was first released.
Yes, that is correct. The expression 0.25t2–4t+28 shows that the value of a first-edition deck of Wandering Wizards will decrease initially when it is first released, then increase as the deck becomes more rare. The maximum value of the deck is 28, which is when t=0 (when the game is first released).
3. The following questions refer to the section “Building the Client Presentation” in the case file. They loosely follow the bullet points in that section (though with more detail). Use Excel for the calculations.
a. First, convert the index values in local currencies to US dollars (see Appendix A). Note that exchange rates are quoted as foreign currency per dollar, i.e., 100 Japanese Yen would buy 1 US dollar.
b. Calculate the average monthly returns and the standard deviations for all country indexes in both local currency and US dollars for the entire sample. Annualize the statistics. Repeat for the two sub-periods (before and after 2002). Present your results in a table (or tables) that allows for easy comparisons.
c. Estimate the correlation matrix of the country index returns in both local currency and US dollars for the three time periods under consideration (Tip: Check under the Data Analysis module in Excel).
d. Calculate how much an investor would have earned if he or she had invested $1 in the US (S&P 500), the Developed ex-US (EAFE), and the Emerging Market (EM) indexes in both local currency and US dollars from 1991 to 2012 (see Appendix B).
e. Calculate the average monthly returns and the standard deviations of the US (S&P 500), the Developed ex-US (EAFE), and the Emerging Market (EM) indexes (see Appendix C). Annualize the statistics. Calculate their respective Sharpe Ratios. The average risk-free rate was 3% (annualized) over the same time period.
Answer:
Identify the graph represent a linear relationship
Ben knows that a line passes through the point (-3, 8) and has a slope of -3/4, but he needs to find the equation of the line. Therefore, he should substitute 8 for Response area and he should substitute -3 for Response area, and he should substitute -3/4 for Response area into the Point-Slope Formula. He will need to Response area and simplify before solving for y = mx + b form.
Answer:
y = -3/4 + 10 1/4
Step-by-step explanation:
y = mx + b
We are given the slope -3/4.
y = -3/4x + b To find the b we will use the point (-3,8) We will use -3 for x and 8 for y and then solve for b
8 = -3/4 (-3) + b
8 = -9/4 + b Add 9/4 to both sides
8 + 9/4 = -9/4 + 9/4 + b
41/4 = b or 10 1/4 = b
Helping in the name of Jesus.
please help me with 4 math questions
Using linear negative association, According to the all four parts correct options are D ;A ;D ;D respectively
What is linear negative association?The slope of a line expresses a great deal about the linear relationship between two variables. If the slope is negative, there is a negative linear relationship, which means that as one variable increases, the other variable decreases. If the slope is zero, one increases while the other remains constant.
The first answer to the question is option D
The second answer to the question must be option A
Option D must be chosen for the third question.
Option D must be selected for Question 4.
Solution:
1.
square of 3 is 9
3 to the power of negative 2 is 1/ 9
cube of 3 is 27
3 to the negative power 3 is 1/27
2.
cylinder volume =πr²h
Given value
pi =3.14
r=5
h=10
Volume=3.14×5²×10
cylinder volume =785m³
3.
When a point is rotated 90 degrees anticlockwise about the origin, it becomes the point (x,y) (-y,x).
The coordinates of Point N are (4, 3)
N' will be the new coordinates (-3, 4)
As a result, the y-coordinate of N' is 4.
4.
Option D must be selected for Question 4.
To know more about linear nonlinear visit:
brainly.com/question/15830007
#SPJ1
explain using words, rather than equations, why if , the total energy eigen- functions cannot be written in the form . x(x)y(y)z(z) c(x, y, z)
The reason why total energy eigen-functions cannot be written in form x(x)y(y)z(z) c(x, y, z) is that the energy eigenfunctions depend on all three spatial coordinates, x, y, and z, simultaneously.
In other words, the probability density of finding the electron at any point in space is not just a product of separate functions that only depend on one of the coordinates at a time.
Instead, the probability density of finding the electron is determined by the complex wave function, which depends on all three spatial coordinates at the same time.
The wave function gives us information about the energy and the behavior of the electron in the system, and it cannot be factorized into separate functions that depend on only one coordinate at a time.
Therefore, we cannot write the total energy eigenfunction as a product of separate functions of x, y, and z. The total energy eigenfunction is a complex function that depends on all three coordinates simultaneously, and it cannot be separated into individual functions that depend only on one coordinate at a time.
Learn more about eigen at
https://brainly.com/question/30357013
#SPJ4
Justin purchased his dream car worth 18500 on a finance for 4 years. He was offered 6% interest rate.assuming no other chargers and no tax,we want to find his monthly installment.
Answer:
To calculate Justin's monthly installment, we need to use the formula for calculating the monthly payment for a loan:
P = (r * A) / (1 - (1 + r)^(-n))
where P is the monthly payment, r is the monthly interest rate, A is the loan amount, and n is the total number of payments (in months).
In this case, A = 18500, r = 0.06/12 = 0.005 (since the annual interest rate is 6%, we divide by 12 to get the monthly rate), and n = 4*12 = 48 (since the loan is for 4 years, or 48 months).
Plugging in these values, we get:
P = (0.005 * 18500) / (1 - (1 + 0.005)^(-48))
P ≈ $432.85
Therefore, Justin's monthly installment would be approximately $432.85
PLEASE HELPPP ME MATH
The coordinates of the image of point D that ensures that the triangles are congruent is (-2, 2)
How to determine the coordinates of point DGiven the triangle ABC and the incomplete triangle DEF
From the triangle transformation of points B and C, we can see that
The points of the triangle ABC are translated to the left by 6 units and downward by 4 units
Mathematically, this can be represented as
(x, y) = (x - 6, y - 4)
Given that
A = (4, 6)
So, we have
D = (4 - 6, 6 - 4)
Evaluate the difference
D = (-2, 2)
Hence, the position is (-2, 2)
Read more about transformation at
https://brainly.com/question/27224272
#SPJ1
Write the equation for a parabola with a focus at (2,2) and a directrix at x=8
Answer:
(y - 2)² = -12(x - 5)
Step-by-step explanation:
A parabola is a locus of points, which are equidistant from the focus and directrix;
Generic cartesian equation of a parabola:
y² = 4ax, where the:
Focus, S, is: (a, 0)
Directrix, d, is: x = -a
a > 0
Put simply, a is the horinzontal difference between the directrix and the vertex or between the vertex and focus;
Always a good idea to do a quick drawing of the graph;
We are the told the focus, F, is: (2, 2) and directrix, d, is: x = 8;
First thing to note, the vertex, or turning point will be in line with the focus vertically, i.e. they will share the same y-coordinate;
Horizonatally, it will be halfway between the focus and the directrix, i.e. halfway between 8 and 2;
Therefore, the vertex will be will be (5, 2);
We can also work out a:
a = 8 - 5 = 5 - 2
a = 3
Substituting this value of a into the generic cartesian equation:
y² = 4(3)x
y² = 12x
The focus and directrix will be:
S: (3, 0)
d: x = -3
Next thing to note, a parabola curves away from the directrix;
In this case, the directrix is x = 8, so the vertex will be the right-most point on the parabola, it will curve off to the left and the focus will also be to the left;
What we want to do is compare with y² = 12x;
This parabola, has a vertex (0, 0), which is the left-most point that curves off to the right and a focus also to the right;
Since we know the formula of this parabola, if we figure out how to transform it into the one in the question, we can find out it's equation;
What we should recognise first is that the parabola in the question is reflected in the y-axis, compared to y² = 12x;
So we apply the transformation that corresponds to this, i.e. use the f(-x) rule:
y² = 12(-x)
y² = -12x
Now the two graphs will have the same shape and orientation;
The focus and directrix will also be affected:
S: (-3, 0)
d: x = 3
Now, the only remaining difference would be the coordinates of the focus and directrix of the two graphs;
The focus of the graph in the question is 5 units to the right and 2 units upwards compared to the focus of y² = -12x;
The directrix is 5 units to the right of that of y² = -12x;
So we apply a translation transformation of 5 units right and 2 units up, like so:
(y - 2)² = -12(x - 5)
Replace y with (y - 2) to translate up 2 units;
Replace x with (x - 5) to translate 5 units right.
We know have a parabola with focus, (2, 2), directrix, x = 8 and vertex, (5, 2), i.e. the parabola in the question;
Hence, the equation of the parabola in the question is:
(y - 2)² = -12(x - 5)
It might seem a bit long and complicated to begin with, but can be done very quickly if you can get used to it.
label the parts of an atom
Answer:
1. Neutron
2. Nucleus shell
3. Proton
4. Electron
3
Find the gradient of the curve x³ + 3xy²-y³ = 1 at the point with coordinates (1, 3).
The gradient of the curve x³ + 3xy²- y³ = 1 at the point with coordinates (1, 3) is -29/9
What is the gradient of a curve?The gradient of a curve at a point is the value of its derivative at that point.
To find the gradient of the curve x³ + 3xy²- y³ = 1 at the point with coordinates (1, 3). we differentiate implicitly with respect to x.
So, we have that
x³ + 3xy²- y³ = 1
d(x³ + 3xy²- y³)/dx = d1/dx
dx³/dx + d3xy²/dx - dy³/dx = d1/dx
dx³/dx + 3y²dx/dx + 3xdy²/dx - dy³/dx = d1/dx
2x² + 3y² × 1 + 3x × dy²/dy × dy/dx - dy³/dy × dy/dx = d1/dx
2x² + 3y² × 1 + 3x × 2y × dy/dx - 3y² × dy/dx = 0
2x² + 3y² + 6xydy/dx - 3y²dy/dx = 0
6xydy/dx - 3y²dy/dx = -(2x² + 3y²)
Factorizing out dy/dx, we have that
6xydy/dx - 3y²dy/dx = -(2x² + 3y²)
(6xy - 3y²)dy/dx = -(2x² + 3y²)
dy/dx = -(2x² + 3y²)/(6xy - 3y²)
At (1,3) dy/dx = -(2x² + 3y²)/(6xy - 3y²)
= -(2(1)² + 3(3)²)/(6(1)(3) - 3(3)²)
= -(2(1) + 3(9))/(6(1)(3) - 3(9))
= -(2 + 27)/(18 - 27)
= -29/-9
= 29/9
So, dy/dx = 29/9
Learn more about gradient of a curve here:
https://brainly.com/question/30684208
#SPJ1
Find the product of 3√20 and √5 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.
Answer:
30, rational
Step-by-step explanation:
[tex]3\sqrt{20}\cdot\sqrt{5}=3\sqrt{4}\sqrt{5}\cdot\sqrt{5}=(3\cdot2)\cdot5=6\cdot5=30[/tex]
The result is rational because it can be written as a fraction of integers.
Barry spent 1/4 of his monthly salary for rent and 1/7 of his monthly salary for his utility bill. If $1411 was left, what was his monthly salary?
Answer: $2,324
Step-by-step explanation:
Let his salary be x.
He spent 1/4 of his salary, or x/4, and 1/7 of his salary, or x/7.
His salary was x. He spent x/4 and x/7, so we subtract those two amounts from his salary.
x - x/4 - x/7 is the amount he still has. He still has $1411. We equate the two and have an equation.
x - x/4 - x/7 = 1411
x/1 - x/4 - x/7 = 1411
We need to combine the three fractions on the left side, so we need to use a common denominator. The least common multiple of 1, 4, and 7 is 28, so 28 is the LCD.
28x/28 - 7x/28- 4x/28= 1411
17x/28= 1411
Multiply both sides by 28.
17x = 39,508
Divide both sides by 17.
x = 2,324
Check:
1/4 of his salary is 2,324/4 = 581
1/7 of his salary is 2,324/7 = 332
Now we subtract 581 and 332 from 2,324
2324 - 581 - 332 = 1411 which is what the problem stated.
Our answer $2,324 is correct.
Change this mixed number to an improper fraction. Use the / key to enter a fraction e.g. half = 1/2
No spam links, please.
Answer:
35/8
Step-by-step explanation:
A mixed fraction in the form [tex]a \dfrac{b}{c}[/tex] can be converted to an improper fraction using the following calculation:
[tex]a \dfrac{b}{c} = \dfrac{(a \times b) + b}{c}[/tex]
Here we have the improper fraction [tex]4 \dfrac{3}{8}[/tex]
Using the technique described
[tex]4 \dfrac{3}{8} = \dfrac{4 \times 8 + 3}{8} = \dfrac{32+ 3}{8} = \dfrac{35}{8}[/tex]
Ans: 35/8
What gravitational force does the moon produce on the Earth if their centers are 3.88x108 m apart and the moon has a mass of 7.34x1022 kg?
The gravitational force that the moon produces on the Earth is approximately 1.99x10²⁰ N.
What is Newton's law of gravitation?Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers, according to Newton's rule of universal gravitation.
We can use Newton's law of gravitation to solve this problem:
[tex]$F = G \frac{m_1 m_2}{r^2}$[/tex]
where F is the gravitational force, G is the gravitational constant, [tex]$m_1$[/tex] and [tex]$m_2$[/tex] are the masses of the two objects, and r is the distance between their centers.
Plugging in the given values, we get:
[tex]$F = (6.674\times10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2) \frac{(7.34\times10^{22} \text{ kg})(5.97\times10^{24} \text{ kg})}{(3.88\times10^8 \text{ m})^2}$[/tex]
Simplifying the expression, we get:
[tex]$F \approx 1.99\times10^{20} \text{ N}$[/tex]
Therefore, the gravitational force that the moon produces on the Earth is approximately 1.99x10²⁰ N.
To know more about Fraction visit:
brainly.com/question/8969674
#SPJ1