The minimum value of the function [tex]f(x,y)[/tex] is [tex]-4[/tex] at the point [tex](1,0)[/tex] and the maximum value of the function [tex]f(x,y)[/tex] is [tex]50[/tex] at the points [tex](-2, -2\sqrt{3}), (-2, 2\sqrt{3})[/tex].
[tex]f(x,y)=2x^2+3y^2-4x-2, g(x,y)=x^2+y^2-16.[/tex]
[tex]f_x (x,y)=4x+0-4-0=4(x-1)[/tex]
[tex]f_y (x,y)=0+6y-0-0=6y[/tex]
[tex]4(x-1)=0, x=1[/tex]
[tex]6y=0, y=0.[/tex]
So, critical points of [tex]f(x,y)[/tex] is [tex](1,0)[/tex].
Put this in [tex]g(x,y)=x^2+y^2-16[/tex].
[tex]g(1,0)=1^2+0^2-16=-15<0[/tex].
So, critical point is in the interior point of the given region.
[tex]g_x (x,y)=2x+0-0=2x[/tex]
[tex]g_y (x,y)=0+2y-0=2y[/tex]
Let there is a parameter [tex]\lambda[/tex] such that [tex]f_x = \lambda g_x[/tex] and [tex]f_y = \lambda g_y[/tex].
[tex]4(x-1)=\lambda (2x)[/tex] and [tex]6y=\lambda (2y)[/tex]
So, [tex]2(x-1)=x\lambda[/tex] and [tex]\lambda=3[/tex]
So, [tex]2(x-1)=3x[/tex]
So, [tex]-2=x[/tex]
Put [tex]x=-2[/tex] in [tex]x^2+y^2=16[/tex], to determine the value of [tex]y[/tex].
[tex](-2)^2+y^2=16[/tex]
[tex]y^2=12[/tex]
[tex]y=\pm 2\sqrt{3}[/tex]
So, the obtained possible extrema are [tex](-2, -2\sqrt{3}), (-2, 2\sqrt{3})[/tex].
Now, find [tex]f(x,y)[/tex] at [tex](1,0), (-2, -2\sqrt{3}),[/tex] and [tex](-2, 2\sqrt{3})[/tex].
[tex]f(1,0)=2(1)^2+3(0)^2-4(1)-2[/tex]
[tex]=2-4-2[/tex]
[tex]=-4[/tex]
[tex]f(-2,-2\sqrt{3})=2(-2)^2+3(-2\sqrt{3})^2-4(-2)-2[/tex]
[tex]=8+36+8-2[/tex]
[tex]=50[/tex]
[tex]f(-2,2\sqrt{3})=2(-2)^2+3(2\sqrt{3})^2-4(-2)-2[/tex]
[tex]=8+36+8-2[/tex]
[tex]=50[/tex]
So, the minimum value of the function [tex]f(x,y)[/tex] is [tex]-4[/tex] at the point [tex](1,0)[/tex] and the maximum value of the function [tex]f(x,y)[/tex] is [tex]50[/tex] at the points [tex](-2, -2\sqrt{3}), (-2, 2\sqrt{3})[/tex].
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If (3,6) is a point on the graph of y=f(x) , what point must be on the graph of y=f(-x)? Explain.
The function y = f(-x) has the points (-3,-6).
When we substitute -x into a function f(x) to get f(-x), we are essentially reflecting the graph of f(x) across the y-axis. This is because replacing x with -x negates the x-values, effectively flipping the function horizontally.
Given that (3, 6) is a point on the graph of y = f(x), if we substitute -3 for x in f(-x), we get:
[tex]\[ f(-(-3)) = f(3) \][/tex]
So, the corresponding point on the graph of [tex]\(y = f(-x)\)[/tex] is [tex]\((3, 6)\)[/tex] since the function values stay the same when we reflect across the y-axis. Therefore, the point [tex](-3, -6)\)[/tex] is on the graph of [tex]\(y = f(-x)\)[/tex].
solve each equation over [0,2pi)
4cos^4x-13cos^2x+3=0
Ms. Rios buys 453 grams of strawberries she has 23 grams left after making smoothies how many grams of strawberries did she use
Ms. Rios used 430 grams of strawberries to make her smoothies. This is calculated by subtracting the amount left (23 grams) from the total amount purchased (453 grams).
Explanation:To determine how many grams of strawberries Ms. Rios used for making smoothies, we can subtract the quantity of strawberries left unprocessed from the total quantity she originally purchased. In this case, Ms. Rios bought 453 grams of strawberries and had 23 grams left after making smoothies.
The formula to determine the solution would be: Total amassed quantity - Remaining quantity = Used quantity
By filling the above formula with our values, the solution will be as follows: 453 grams (total) - 23 grams (remaining) = 430 grams (used).
Thus, Ms. Rios used 430 grams of strawberries for making smoothies.
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[tex]F(x)=(6 \sqrt{x} -2)(5 \sqrt{x} +7)[/tex]
The intelligence quotient (iq) test scores are normally distributed with a mean of 100 and a standard deviation of 15. what is the probability that a person would score 130 or more on the test?
The length of a rectangle is 4 meters less than twice the width. If the area of the rectangle is 286 square meters, find the dimensions.
The dimensions of the rectangle are length = 22 meters and width = 13 meters
What is area of rectangle?[tex]A=l\times w[/tex], where 'l' is the length and 'w' is the width of the rectangle.
For given question,
Suppose 'l' is the length and 'w' is the width of the rectangle.
The length of a rectangle is 4 meters less than twice the width.
So, we get an equation,
⇒ l = 2w - 4
The area of the rectangle is 286 square meters.
⇒ A = 286 sq. m.
Using the formula for the area of the rectangle,
[tex]\Rightarrow A=l\times w\\\\\Rightarrow 286=(2w-4)\times w\\\\\Rightarrow 286=2w^2-4w\\\\\Rightarrow 2w^2-4w-286=0[/tex]
Now, we solve the quadratic equation [tex]2w^2-4w-286=0[/tex]
[tex]\Rightarrow 2w^2-4w-286=0\\\\\Rightarrow w^2-2w-143=0\\\\\Rightarrow (w-13)(w+11)=0\\\\\Rightarrow w-13=0~~~or~~~w+11=0\\\\\Rightarrow w=13~~~or~~~w=-11[/tex]
w = -11 is not possible.
So, the width of the rectangle is 13 meters.
And the length of the rectangle would be,
[tex]\Rightarrow l \\= 2w - 4\\=(2\times 13)-4\\=22[/tex]
Therefore, the dimensions of the rectangle are length = 22 meters and width = 13 meters
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In one year, Michael earned $6300 as a work study in college. He invested part of the money at 9% and the rest at 7%. If he received a total of $493 in interest at the end of the year, how much was invested at 7%? How much was invested 9%?
The sat scores have an average of 1200 with a standard deviation of 60. a sample of 36 scores is selected. what is the probability that the sample mean will be larger than 1224? round your answer to three decimal places.
Final answer:
The probability that the sample mean of SAT scores will be larger than 1224 is 0.008. This is calculated using the z-score, which in this case is 2.4 after determining the standard error of the sampling distribution.
Explanation:
To find the probability that the sample mean will be larger than 1224, we will use the concept of the sampling distribution of the sample mean. Given the population mean (μ) is 1200 and standard deviation (σ) is 60, and that the sample size (n) is 36, the standard deviation of the sampling distribution, known as the standard error (SE), is
σ/√n = 60/√36 = 10.
We calculate the z-score for the sample mean of 1224 using the formula
z = (X - μ)/SE = (1224 - 1200)/10 = 2.4
A z-score of 2.4 indicates that the sample mean is 2.4 standard errors above the population mean.
To find the probability associated with this z-score, we refer to the normal distribution table or use a calculator. The probability to the left of z = 2.4 is 0.9918. Therefore, the probability that the sample mean is greater than 1224 is
1 - 0.9918 = 0.0082
which can be rounded to three decimal places as 0.008.
which of the fallowing functions has a slope 3/2 and contains the midpoint segment between (6, 3) and (-2, 11)?
Out of 6 women would consider themselves baseball fans, with a standard deviation of
if the smaller of two consecutive even intergers is subtracted from 3 times the larger the result is 42
Yearbook sales this year increased 120% over last years yearbook sales. If 465 yearbooks were sold last year, how many were sold this year?
Answer:
This year 1,023 books were sold.
Step-by-step explanation:
Last year the number of yearbooks were sold = 465
This year sales are increased 120% over last years sales.
this year sale = 465 + (120% of 465)
= 465 + ([tex]\frac{120}{100}[/tex] × 465)
= 465 + (1.2 × 465)
= 465 + 558
= 1,023 books
This year 1,023 books were sold.
Write the following comparison as ratio reduced to lowest terms 194 inches to 17 feet
For each of the following functions, find the maximum and minimum values of the function on the circular disk: x^2+y^2≤1. Do this by looking at the level curves and gradients.
f(x,y)=x+y+4
maximum value =
To find the maximum and minimum values of the function f(x, y) = x + y + 4 on the circular disk x^2 + y^2 ≤ 1, evaluate the function at the boundary of the disk.
Explanation:To find the maximum and minimum values of the function f(x, y) = x + y + 4 on the circular disk x^2 + y^2 ≤ 1, we can use the method of level curves and gradients.
First, find the gradient of the function f(x, y) using partial derivatives.Next, find the critical points of the function by setting the gradient equal to zero and solving for x and y.Finally, evaluate the function at the critical points and the boundary of the circular disk to find the maximum and minimum values.In this case, since the function f(x, y) = x + y + 4 is linear, it does not have any critical points. Therefore, the maximum and minimum values of the function on the circular disk x^2 + y^2 ≤ 1 are obtained by evaluating the function at the boundary of the disk.
When x^2 + y^2 = 1, the function value is largest at the point (x, y) = (-1, 0), giving a maximum value of -1 + 0 + 4 = 3. The function value is smallest at the point (x, y) = (1, 0), giving a minimum value of 1 + 0 + 4 = 5.
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if it is a square is it a quadrilateral
A 250-kVA, three-phase, 480-volt transformer requires 75 gallons of insulating oil to keep it properly cooled. How many liters is this?
This problem can be directly solved by using a conversion factor. Simple research will tell us that 1 gallon contains about 3.78 Liter. Therefore the volume in Liter is:
volume = 75 gallons * (3.78 Liter / gallon)
volume = 283.5 Liters
Find the unit rate by using WKU. David drove 135 miles in 3 hours.
Simplify the expression where possible. (r 3) -2
Which numbers are a distance of 4 units from 7 on the number line? A number line ranging from negative 3 to 15. Select each correct answer.
3
11
7
15
4
−3
Answer: 3, 11
Step-by-step explanation:
As u see in this screen shot there is wrongs and rights
A farmer has 260 feet of fencing to make a rectangular corral. What dimensions will make a corral with the maximum area? What is the maximum area possible?
Beyond Euclidean Geometry.
Many airlines use maps to show the travel paths of all their flights, which are called route maps. For instance, K12Air has a route map that describes all the possible routes to and from Samsville, Shiloh, Camden, Chelsea, Jamestown, and Lorretta.
You have been provided a route map for K12Air. Write a question about this map that involves Hamiltonian or Euler circuits or paths.
Help me come up with a question?
A suitable question to ask about the K12Air route map in the context of Hamiltonian or Euler circuits or paths could be:
Is it possible to find a Hamiltonian circuit on the K12Air route map that allows a plane to travel through each city exactly once before returning to the starting city?
To formulate a question involving Hamiltonian or Euler circuits or paths, one must understand the difference between these concepts:
- A Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once. If this path returns to the starting vertex, it is called a Hamiltonian circuit.
- An Euler path is a path in a graph that visits every edge exactly once. If this path starts and ends at the same vertex, it is called an Euler circuit.
Given the context of the K12Air route map, which describes all the possible routes to and from various cities, the question should focus on whether it's possible to traverse the graph representing the route map in a way that satisfies the conditions of either a Hamiltonian or an Euler circuit/
For the Hamiltonian circuit, the question is whether there exists a sequence of flights that allows a plane to start at one city, visit every other city exactly once, and return to the starting city without repeating any city. This would require the route map to have a Hamiltonian circuit, which is a more stringent condition than an Euler circuit because it involves visiting all vertices exactly once.
For an Euler circuit, the question would be whether there exists a sequence of flights that allows a plane to traverse every possible route exactly once before returning to the starting point. This would require the route map to have an Euler circuit, meaning every edge (route) is used exactly once.
In the case of K12Air, the question about the Hamiltonian circuit is particularly interesting because it tests the connectivity of the route map and the possibility of a round trip that covers all cities without repetition. This could be relevant for planning efficient travel itineraries or for optimizing the use of airline resources. If the route map does not allow for a Hamiltonian circuit, one might then ask if a Hamiltonian path exists, which would not require returning to the starting city.
To answer such a question, one would need to analyze the connectivity of the graph represented by the route map, possibly using theorems related to Hamiltonian graphs, such as Dirac's theorem or Ore's theorem, which provide sufficient conditions for a graph to contain a Hamiltonian circuit.
A school bus uses 3/4 of a tank of gas to drive back and forth to school in a week with 5 school days. How much gas does the bus use to drive back and forth to school in a week with 4 school days? Write your answer in simplest form.
Suppose there is a strong positive correlation between v and w. Which of the following must be true?
The length of a rectangle is twice its width. The perimeter is 60 ft. Find its area.
Answer:
200 feet squared
Step-by-step explanation:
W=Width
2W=Length
Perimeter = 2*Length + 2*Width
Now use substitution for the Length
60 = 2(2W) + 2(W)
60=4W + 2W = 6W divide both sides by 6
60/6 = 6W/6
10 = W
Width = 10 and Length is twice as long so it is 20. 10+10+20+20=60
The area is Length * Width = 20*10=200
Graph the line for y+1=−35(x−4) on the coordinate plane. What are the coordinates that's all I need to know
If the apy of a savings account is 3.7%, and if the principal in the savings account were $3600 for an entire year, what will be the balance of the savings account after all the interest is paid for the year?
Find the value of kk for which the constant function x(t)=kx(t)=k is a solution of the differential equation 3t3dxdt+5x−3=03t3dxdt+5x−3=0.
How many ones in 800?
Two trains arrived at a station at 2:55 P.M., with one arriving on Track A, and the other arriving on Track B. Trains arrive on Track A every 16 minutes, and they arrive on Track B every 18 minutes. At what time will trains next arrive at the same time on both tracks? A) 4:07 P.M. B) 5:19 P.M. C) 6:31 P.M. D) 7:43 P.M.
Zeno has to go a distance, d, to get to his destination. He claims he can never get there because he has to travel half that distance d, then half of half of distance d, and so on. He says that he has to do an infinite number of tasks and that it is impossible. Use an infinite geometric series to help Zeno. Identify a1 and r to form the infinite geometric series that represents the problem.
a1=1/2d
r=1/2
sup it needs twenty character apparently
The [tex]a_{1} =\frac{d}{2}[/tex] and [tex]r=\frac{1 }{2 }[/tex] are for the infinite Geometric sequence.
What is geometric sequence ?
Geometric sequence a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
i.e. [tex]a+ar+ar^2+ar^3+......[/tex]
Here, [tex]a=[/tex] First term and [tex]r=[/tex] common ratio
Next term formula, [tex]a_{n} =a_{1}\ *\ r^{n-1}[/tex]
We have,
[tex]d=[/tex] distance to her destination,
Now,
According to the question;
Zeno has to travel half that distance i.e. [tex]\frac{d}{2}[/tex],
Then half of half of distance i.e. [tex]\frac{d}{4}[/tex],
Then half of [tex]\frac{d}{4}[/tex] distance i.e. [tex]\frac{d}{8}[/tex] and so on.
So,
Here, we have,
[tex]\frac{d}{2},\frac{d}{4},\frac{d}{8}, .........[/tex]
So,
We have Geometric Sequence,
Here,
[tex]a_{1} =\frac{d}{2}[/tex] and
Now,
[tex]r=[/tex] common ratio,
[tex]r=\frac{a_{2} }{a_{1} }[/tex]
[tex]r=\frac{\frac{d}{4} }{\frac{d}{2} }[/tex]
We get,
[tex]r=\frac{1 }{2 }[/tex]
So, These are [tex]a_{1} =\frac{d}{2}[/tex] and [tex]r=\frac{1 }{2 }[/tex] of the Geometric sequence.
Hence, we can say that he [tex]a_{1} =\frac{d}{2}[/tex] and [tex]r=\frac{1 }{2 }[/tex] are for the infinite Geometric sequence.
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