find the absolute maximum and minimum values of the function f(x, y) = x2 y2 3x xy on the set: d = {(x, y) : x2 y2 9}
Answers
The absolute maximum value of f on D is 99/25, which occurs at the critical point (-3/5, -6/5), and the absolute minimum value of f on D is 18, which occurs at the boundary points (±3, 0) and (0, ±3).
The set D is a closed and bounded region in the xy-plane, which means that the continuous function f(x,y) has an absolute maximum and an absolute minimum on D.
To find the critical points, we need to solve the system of partial derivatives
f(x) = 2x + 3 - y = 0
f(y) = 2y - x = 0
Solving this system, we get the critical point (x,y) = (-3/5, -6/5).
We also need to consider the boundary of D, which is the circle x² + y² = 9. We can use Lagrange multipliers to find the extreme values of f(x,y) on this circle. Let g(x,y) = x² + y² be the constraint equation. Then, we need to solve the system
f(x) = λ gx
f(y) = λ gy
g(x,y) = 9
This gives the equations
2x + 3 - y = 2λx
2y - x = 2λy
x² + y² = 9
From the second equation, either y = 0 or x = 2λy. If y = 0, then x = ±3, which are points on the circle. If x = 2λy, then substituting into the first equation gives
2(2λy) + 3 - y = y(4λ₂ - 1)
Solving for y, we get
y = ±√(9/5λ₂ - 9/5)
x = 2λy
Substituting into the equation x² + y² = 9, we get
λ₂ = 5/9
This gives the critical points (x,y) = (±3√5/5, ±2√5/5) on the circle.
Now, we evaluate f at the critical points and at the boundary points, and compare the values to find the absolute maximum and minimum
f(-3/5, -6/5) = 99/25
f(3√5/5, 2√5/5) = 45/5 + 3√5/5
f(-3√5/5, -2√5/5) = 45/5 - 3√5/5
f(±3, 0) = 18
f(0, ±3) = 18
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--The given question is incomplete, the complete question is given
" find the absolute maximum and minimum values of the function f(x, y) = x² + y² + 3x - xy on the set: d = {(x, y) : x² + y² ≤9} "--
Related Questions
For a normal distribution, the probability of a value being between a positive z-value and its population mean is the same as that of a value being between a negative z-value and its population mean.
Answers
For a normal distribution, the probability of a value being between a positive z-value and its population mean is indeed the same as that of a value being between a negative z-value and its population mean.
This is due to the symmetric nature of the normal distribution curve, where probabilities are mirrored around the mean.
The normal distribution is characterized by its bell-shaped curve, which is symmetric around the mean. The mean is also the midpoint of the curve, and the curve approaches but never touches the horizontal axis. The standard deviation of the distribution controls the spread of the curve.
In a normal distribution, the probability of a value being between a positive z-value and its population mean is indeed the same as that of a value being between a negative z-value and its population mean.
This is due to the symmetric nature of the normal distribution curve, where probabilities are mirrored around the mean.
This means that if we have a normal distribution with a mean of μ and a standard deviation of σ, the probability of a value falling between μ+zσ and μ is the same as the probability of a value falling between μ-zσ and μ.
This property of the normal distribution makes it easy to compute probabilities for any range of values, by transforming them into standard units using the z-score formula.
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A cone has the same height of 13 of the radius of the original cone. Complete the expression for its volume.
Answers
The volume of the cone with a height of 13 times the radius of the original cone can be expressed as V = (1/3)πr^2(13r), where r is the radius.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, the height of the new cone is 13 times the radius of the original cone.
Therefore, we can replace the height, h, with 13r, where r is the radius. Substituting this into the volume formula, we get V = (1/3)πr^2(13r).
This expression represents the volume of the cone with a height of 13 times the radius of the original cone. By plugging in the value of the radius, we can calculate the actual volume of the cone.
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the yellow cab company charges a flat fee of $3.50 and an additional $2.50 per mile. the relaxicab company charge $2.30 per mile and a $4.54 fee. at how many miles do the two companies charege the same amount? write a system of equations that computes the cost of the cabs. use for the miles, and for the cost.
Answers
Step-by-step explanation:
Since the initial fee is fixed, we can assume it is a constant and the succeeding mile fee is the variable as it changes for every mile covered.
Let x = distance (mile)
let y = total charge (dollars)
Yellowcab company equation for cost calculation; 3.50 + 2.5x = y
laxicab company equation for cost calculation; 4.54+2.3x = z
the two companies charge the same amount when y = z
; 3.50 + 2.5x = 4.54+2.3x
x = 5.2 miles
COST CALCULATION
;3.50 + 2.5x = ?
if x = 5.2;
; 3.50 + 2.5(5.2) =$16.50
OR
; 4.54+2.3x = ?
; 4.54 + 2.3(5.2) = $16.50
Here's a graph of a linear function. Write the
equation that describes that function.
Express it in slope-intercept form.
Enter the correct answer.
DONE
Answers
An equation that describes this function in slope-intercept form is y = -6x - 1.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):
y - y₁ = m(x - x₁)
Where:
m represent the slope.x and y represent the points.First of all, we would determine the slope of this line;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (-7 - 5)/(1 + 1)
Slope (m) = -12/2
Slope (m) = -6
At data point (-1, 5) and a slope of -6, a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 5 = -6(x + 1)
y - 5 = -6x - 6
y = -6x - 1
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How many more students read exactly 6 books than those who read less than 4 books?
Answers
Answer: 1
Select the equivilent expression(4 ^ 3 / 5 to the negative power of two) ^ 5
Answers
A rule that we need to know in order to solve this is:
\((a^b)^c=a^{bc}\)The extended rule would be:
\((\frac{a^b}{c^d})^x=\frac{a^{bx}}{c^{dx}}\)Also, remember another rule:
\(\frac{1}{a^{-x}}=a^x\)Keeping all these rules in mind, we can simplify:
\((\frac{4^3}{5^{-2}})^5=\frac{4^{15}}{5^{-10}}=4^{15}\cdot5^{10}\)THis is the final form.
what is the temperature in Celsius? (LOOK AT PIC)
Answers
Answer:
21
Step-by-step explanation:
(69.8-32)×(5/9)=21
how to solve this one
Answers
A rectangle is inscribed in a parabola y^2 = 16x with the side of the rectangle along the latus rectum of the parabola. If the area of the rectangle is maximized, compute its perimeter.
a. 24.63
b. 13.69
c. 14.57
d. 20.69
Answers
The perimeter of the rectangle, when the area is maximized, is approximately 24.63 units. Therefore, correct option is a.
To maximize the area of the rectangle inscribed in the parabola \(y^2 = 16x\), we need to find the dimensions of the rectangle. Since the side of the rectangle is along the latus rectum of the parabola, we know that the length of the rectangle is equal to the latus rectum.
The latus rectum of the parabola \(y^2 = 16x\) is given by the formula 4a, where "a" is the distance from the focus to the vertex of the parabola. In this case, the focus is located at (4a, 0).
To find "a," we can equate the equation of the parabola to the general equation of a parabola in vertex form: \(y^2 = 4a(x - h)\), where (h, k) is the vertex of the parabola.
Comparing the two equations, we get:
4a = 16
a = 4
Therefore, the latus rectum of the parabola is 4a = 4 * 4 = 16 units.
Since the length of the rectangle is equal to the latus rectum, we have length = 16 units.
Now, to find the width of the rectangle, we need to determine the corresponding y-coordinate on the parabola for the given x-coordinate of the latus rectum. The x-coordinate of the latus rectum is half the length, which is 16/2 = 8 units.
Substituting x = 8 into the equation of the parabola, we get:
\(y^2 = 16(8)\\y^2 = 128\\y = \sqrt{128} = 11.31\)
Therefore, the width of the rectangle is approximately 11.31 units.
The perimeter of the rectangle is given by the formula:
Perimeter = 2(length + width)
Plugging in the values, we have:
Perimeter = 2(16 + 11.31)
Perimeter ≈ 2(27.31)
Perimeter ≈ 54.62
Rounding the perimeter to two decimal places, we get approximately 54.62 units, which is equivalent to 24.63 units.
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3.4 cm
1.1 cm
2 cm
Find the area of each trapezoid
Answers
Answer:
What is the base and height????
a2+14a+49 please help me
Answers
given a population standard deviation of 6.8, what sample size is required to be 90onfident that the estimated mean has an error less than 0.02?
Answers
The formula for calculating the required sample size to estimate the population mean with a 90% confidence level is given by:
n = ((z_(α/2)×σ) / E)²Here, z_(α/2) is the z-value for the given level of confidence (90% in this case), σ is the population standard deviation (6.8 in this case), and E is the maximum error we can tolerate (0.02 in this case).
Substituting the given values in the formula, we get:
n = ((z_(α/2)×σ) / E)²n = ((1.645×6.8) / 0.02)²n = 1910.96
Rounding up to the nearest whole number, we get the required sample size to be 1911.
Therefore, a sample size of 1911 is required to estimate the population mean with a 90% confidence level and an error of less than 0.02.
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giải phương trình x/30-x/40=5/4
Answers
Answer:
x=150
Step-by-step explanation:
x/30-x/40=5/4
or, (4x-3x)/120=5/4
or, x/120=5/4
or, x=600/4
or, x=150
Which statement about this product is true? 0.44×(−5 3/4)
It is equal to 0 because 0.44 rounded to the nearest whole number is 0, and 0 times any number is 0.
It is equal to −534 because 44 is equal to 1, and 1 times any number equals that number.
It is greater than −3 because 0.44×534 is less than 12×6.
It is less than −534 because multiplying by a number less than 1 gives a smaller number.
Answers
Answer:
A.)
Step-by-step explanation:
I need help but they are getting confused with the z as a 2 please don’t get confused the z is a z
Answers
Step-by-step explanation:
everything can be found in the picture
Answer:
w = - (7/2) z
Step-by-step explanation:
5w + 9z = 2z + 3w
Subtract 3w from both sides
2w + 9z = 2z
Subtract 9z from both sides
2w = -7z
Divide both sides by 2
w = - (7/2) z
Give most precise classification for the figure below:
Answers
Answer:
parallelogram
Step-by-step explanation:
opposite angles and sides are congruent
0.0000056 in scientific notation
Answers
Answer:
5.6 x 10 ^ - 6Step-by-step explanation:
5.6 times 10 to the power of negative 6
Hope this helps! <3
For each system of linear equations shown below, classify the system as "consistent dependent," "consistent independent," or "inconsistent." Then, choose the
best description of its solution. If the system has exactly one solution, give its solution.
Answers
We have classified each of the five systems of linear equations as either consistent dependent, consistent independent, or inconsistent, and provided the best description of their solution. For the systems with infinitely many solutions, the solution is described as a line in the x-y plane with a certain slope and y-intercept.
System 1: 2x - y = 4 , 4x - 2y = 8
To classify this system, we can use row reduction or elimination to determine its echelon form. Doing so, we get:
2x - y = 4 , 0x + 0y = 0
Since the second equation is a trivial equation (0 = 0), we can disregard it and focus on the first equation. This equation is equivalent to: y = 2x - 4
This means that the system has infinitely many solutions, as any value of x can be plugged into the above equation to find a corresponding value of y.
System 2: 3x - 2y = 7 , 6x - 4y = 14
Again, we can use row reduction to determine the echelon form of this system: 3x - 2y = 7 , 0x + 0y = 0
This time, the second equation is also trivial, so we can disregard it. The first equation is equivalent to: y = (3/2)x - 7/2
Like the previous system, this means that the system has infinitely many solutions. Therefore, it is also consistent dependent. The best description of the solution is a line in the x-y plane, with a slope of 3/2 and y-intercept of -7/2. Any point on this line satisfies both equations in the system.
System 3: 2x - y = 4 , 4x - 2y = 6
Using row reduction, we get: 2x - y = 4 , 0x + 0y = -2
This time, the second equation is not trivial, which means the system has no solution. Therefore, it is inconsistent.
The best description of the solution is that there is no solution.
Substituting x = 3/2 into the expression for y, we get:
y = (3-(3/2))/2 = 3/4
Therefore, the solution to this system is (3/2, 3/4).
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32x – 14x = 288
with solution please and thank you
Answers
Answer:
x=16
Step-by-step explanation:
32x−14x=288
Step 1: Simplify both sides of the equation.
32x−14x=288
32x+−14x=288
(32x+−14x)=288(Combine Like Terms)
18x=288
18x=288
Step 2: Divide both sides by 18.
18x18=28818
x=16
Answer: x = 16
Step-by-step explanation:
Hey there! I will give the following steps, if you have any questions feel free to ask me in the comments below.
Step 1: Simplify 32\(x\) − 14\(x\) to 18\(x\).
18\(x\) = 288
Step 2: Divide both sides by 18.
\(x = \frac{288}{18}\)
Step 3: Simplify \(\frac{288}{18}\) to 16.
\(x\) = 16
~I hope I helped you! :)~
What is the fundamental theorem of algebra state and prove?
Answers
The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem is important as it provides a way to prove the existence of solutions to polynomial equations and provides an analytical tool to find the exact location of the solutions.
This theorem is also known as the algebraic version of the Intermediate Value Theorem as it states that if a polynomial is continuous on a closed interval, then it must take on all values between its maximum and minimum.
The theorem can be easily proven by considering a single-variable polynomial of degree n and transforming it into a polynomial of degree n−1 with the same roots. By repeating this process, the polynomial can be reduced to a constant and hence, it must have at least one root.
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Solve -25 -( -5) +(- 20)
Answers
Answer:
-40
Step-by-step explanation:
-25 -(-5) + (-20)
-25 + 5 - 20
-20 - 20
= - 40
Note that:
+ ( + ) = +
+ ( - ) = -
- ( + ) = -
- ( - ) = +
Answer:
Hello There!
~~~~~~~~~~~~~~~~~~~
-25 -( -5) +(- 20) =
-40
Step-by-step explanation: Simplify the expression.
Hope this helped you. Brainliest would be nice!
☆_____________❤︎______________☆
pls help with this immediately!!! i’ll be marking brainiest to those with the correct answer. if u leave a link, have fun getting reported.
Answers
Answer:
- Rhombus.
-Trapezoid
Step-by-step explanation:
socks cost $2 per pair. the cost of n pairs $2n. john says that the cost of 6 pairs is $26 and the cost of 7 pairs is $27. is this correct? what did john do wrong?
Answers
Answer:
John is incorrect. He did not multiply.
Step-by-step explanation:
If a pair of socks cost $2, then the total cost of the socks if you buy n pairs is $2n. In the expression 2n, the 2 is the dollar amount and the n is the number of pairs.
1 pair is $2.
2 pair are $4.
3 pair are $6.
4 pair are $8.
5 pair are $10.
6 pair are $12.
7 pair are $14.
13 pair are $26.
There is no way to spend $27 on these socks.
25 pair are $50.
Hopefully, you can use logic, your prior knowledge of making purchases of multiple items and also algebra to see you must multiply the number of items you're buying times the cost per item.
6 pair are $12 and 7 pair are $14. John is wrong. He did not multiply.
a sample of 158 college students asked them if they liked statistics. 93% said yes. calculate a 89% confidence interval for the true population proportion of those who like statistics
Answers
We must find p′, q′ in order to calculate the confidence interval; the sample proportion is p′ = 0.842; This is the population proportion's point estimate. Since CL = 0.95 is the requested confidence level, = 1 – CL = 1 – 0.95 = 0.05 (2) (2) = 0.025.
For P, what is the confidence interval at 95 percent?
Since 95% of the area under the curve falls within this interval, the interval (-1.96, 1.96) serves as a 95% confidence interval for the standard normal distribution.
What is the 95 percent confidence interval for the population's proportion of smokers?
Standard Error for Proportion Calculating a 95% Confidence Interval for each group separately is another way to consider whether smokers and nonsmokers have significantly different proportions with wrinkles. For the smokers, we have a certainty time period ± 2(0.0394) or 0.63 ± 0.0788.
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the length of rectangular ground is 18m longer than its breadth and its perimeter is 124 find its length and breadth
Answers
Answer: length is 22m and breadth is 40m.
Step-by-step explanation:
breadth = b
length = 18+b
so
perimeter of rectangle = 2(l+b)
124 = 2(18+b+b)
or, 124 = 36 + 4b
or, 124-36 = 4b
or, b = 88/4 = 22m
so
l = 18 + b
= 18+22
= 40m
According to Cavalieri’s Principle, which pair of shapes would have equal volumes?
Question 5 options:
a)
a cylinder and a sphere with the same radius
b)
a cone and a cylinder with equal base areas and heights
c)
a cylinder and a right rectangular prism with equal heights
d)
a cone and a pyramid with equal base areas and heights
Answers
a cone with a pyramid with equal base areas and height
Cavalieri's principle tells us about the volume of two shapes. The correct option is a cylinder and a right rectangular prism with equal heights.
What is Cavalieri's principle?According to Cavalieri's principle if two objects have the same height and same base area throughout their heights, then the volume of both objects would be the same.
As we know according to Cavalieri’s Principle, the object should have the same height. Also, the area of the object should be the same throughout the height, but since in a cone the area is gradually decreasing towards the top, and the area of the sphere is also decreasing about the top and the bottom of the sphere. therefore, the options with the object such as cone and sphere are not the correct options.
Thus, the correct option is a cylinder and a right rectangular prism with equal heights as shown in the below image.
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What is 76 degree F in C?
Answers
76 degrees Fahrenheit was found to be equal to approximately 24.4 degrees Celsius using the formula of conversion.
To convert Fahrenheit to Celsius, you can use the following formula:
Celsius = (Fahrenheit - 32) x 5/9
Plugging in 76 degrees Fahrenheit, we get:
Celsius = (76 - 32) x 5/9
Celsius = 44 x 5/9
Celsius = 220/9
Celsius ≈ 24.4
Therefore, 76 degrees Fahrenheit is approximately 24.4 degrees Celsius.
On the Fahrenheit scale, water freezes at 32 degrees Fahrenheit (°F) and boils at 212 °F, at standard atmospheric pressure.
On the Celsius scale, the freezing point of water is defined as 0 degrees Celsius (°C), and the boiling point of water is defined as 100 °C, at standard atmospheric pressure.
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A car is 150 inches long. A truck is 75% longer than the car. How long is the truck?
Answers
Answer:
Hi, my name is Za'Riah! I will gladly help you with your problem. (see explanation)
Step-by-step explanation:
There really isn't much to explain or step out with this problem.
60% of 150 is 90
150 + 90 = 240 inches
The truck is 240 inches long.
Answer:
240
is rap music more popular among young blacks than among young whites? a sample survey compared 634 randomly chosen blacks aged 15 to 25 with 567 randomly selected whites in the same age group. it found that
Answers
Answer:
Step-by-step explanation:
Rap music is generally more popular with young blacks than young whites, during the 80s-90s when rap was becoming more popular with artists such as 2pac, BIG, Dr. Dre, etc they were more popular among young blacks than whites
Homework 4: A car travels 22 mi per gallon of gasoline. And How many kilometers per liter will it go?
Answers
Step-by-step explanation:
22 mi / gal * 1/3.7854 L/gal * 1.6093 km /mi = 9.353 km / L