Find the maximum and minimum values of f(x,y) = 2+2x+4y-x^2-y^2 on the triangular region in the first quadrant bounded by the lines x=0, y0, y=9x.
Answers
I think your question seems like :-
Find the maximum and minimum values of f(x,y) = 2+2x+4y-x²-y² on the triangular region in the first quadrant bounded by the lines x = 0, y = 0, y = 9-x. The reason is below :
If you see carefully, the given lines were x = 0, y = 0, y = 9x , also these lines forms a Triangle in first quadrant (Given) , but these all lines doesn't form a triangle in 1st quadrant , we can't change the other two lines (As those two lines will work as permanent sides and support for the triangle) but third one can be , also neither 9 + x nor (9/x) can't form triangle in 1st quadrant with the given other two lines .Only with y = 9 - x the given situation can be achieved
But before starting the question let's recall some of the concepts :-
For a given function let it be f(x,y) ,
f(x,y) can have maximum or minimum if AC - B² > 0 , as per A < 0 and A > 0 , if A < 0 then maxima at critical points and if A > 0 , then minima at critical points f(x,y) have saddle points if AC - B² < 0 From AC - B² = 0 , nothing can be concludedWhere ,
[tex]{\boxed{\bf{A=\dfrac{\partial^{2}f}{\partial x^{2}}}}}[/tex] [tex]{\boxed{\bf{B=\dfrac{\partial^{2}f}{\partial x\partial y}}}}[/tex] [tex]{\boxed{\bf{C=\dfrac{\partial^{2}f}{\partial y^{2}}}}}[/tex]Now , consider the function ;
[tex]{:\implies \quad \sf f(x,y)=2+2x+4y-x^{2}-y^{2}}[/tex]
Partial Differentiating both sides w.r.t.x
[tex]{:\implies \quad \sf \dfrac{\partial f}{\partial x}=2-2x}[/tex]
Now , partial differentiating both sides w.r.t.y
[tex]{:\implies \quad \sf \dfrac{\partial^{2}f}{\partial x\partial y}=0}[/tex]
Now , consider ;
[tex]{:\implies \quad \sf \dfrac{\partial f}{\partial x}=2-2x}[/tex]
Partial differentiating both sides w.r.t.x
[tex]{:\implies \quad \sf \dfrac{\partial^{2}f}{\partial x^{2}}=-2}[/tex]
Now , consider ;
[tex]{:\implies \quad \sf f(x,y)=2+2x+4y-x^{2}-y^{2}}[/tex]
Partial differentiating both sides w.r.t.y
[tex]{:\implies \quad \sf \dfrac{\partial f}{\partial y}=4-2y}[/tex]
Now , partial differentiating both sides w.r.t.y
[tex]{:\implies \quad \sf \dfrac{\partial^{2}f}{\partial y^{2}}=-2}[/tex]
Now to find the critical points , first order partial derivative of f(x,y) both w.r.t.x and w.r.t.y must be 0. So , if you equate them to 0 , you will get x = 1 and y = 2. So critical point is only one i.e (1,2)
Now , from the above conditions , let calculate AC - B² first , from which we will get maxima or minima
[tex]{:\implies \quad \sf (-2)(-2)-(0)^{2}}[/tex]
[tex]{:\implies \quad \sf 4\> 0}[/tex]
So , AC - B² > 0 and also A < 0 , so maxima at (1,2) , to find maxima we need to put x = 1 , y = 2 in f(x,y) i.e f(1,2) = 2 + 2(1) + 4(2) - 1² - 2² = 7 . Now , for minima , draw the triangle see attachment 1 . Now , in the attachment critical points are (0,9) , (0,0) and (9,0) . Now , we need to find f(x,y) at these all points too . If you calculate then you will get f(0,9) = -43 , f(0,0) = 2 and f(9,0) = -61 , out of which -61 is minimum
So , the required maxima and minima are 7 and -61
Note :- Also , refer to the attachments no. 2,3,4 as well for why I choosed the third line to be 9 - x rather than others.
Related Questions
1/3 of 3 feet = how many inches
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Answer:
12 Inches
Step-by-step explanation:
3 ÷ 3 = 1
1 feet is 12 inches
The local cinema sells student tickets for $7.25. A school paid a total of $203 for a class of students to attend the cinema. Which equation can be used to determine the number of students, s that are in this class?
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y = 5 and y = -1
Is it parallel,perpendicular or neither
Answers
Answer:
Parallel
Step-by-step explanation:
The graphs of these two equations will never intersect. Hope this helps.
suppose there is a normally distributed population with a mean of 250 and a standard deviation of 50. if xbar is the average of a sample of 36, find P(246
Answers
Using the normal distribution and the central limit theorem, it is found that:
[tex]P(246 \leq \bar{x} \leq 260) = 0.5693[/tex]
Normal Probability DistributionIn a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].In this problem:
The mean is of [tex]\mu = 250[/tex].The standard deviation is of [tex]\sigma = 50[/tex].A sample of 36 is taken, hence [tex]n = 36, s = \frac{50}{\sqrt{36}} = 8.3333[/tex].The probability of a sample mean between 246 and 260 is the p-value of Z when X = 260 subtracted by the p-value of Z when X = 246, hence:
X = 260:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{260 - 250}{8.3333}[/tex]
[tex]Z = 1.2[/tex]
[tex]Z = 1.2[/tex] has a p-value of 0.8849.
X = 246:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{246 - 250}{8.3333}[/tex]
[tex]Z = -0.48[/tex]
[tex]Z = -0.48[/tex] has a p-value of 0.3156.
0.8849 - 0.3156 = 0.5693, hence:
[tex]P(246 \leq \bar{x} \leq 260) = 0.5693[/tex]
To learn more about the normal distribution and the central limit theorem, you can check https://brainly.com/question/24663213
Factor the greatest common factor (GCF) from each polynomial. - 12y + 48x
Answers
Answer:
-12(y-4x)
Step-by-step explanation:
-12y+48x=-12(y-4x)
Need help please with this math question
Answers
Answer:
1/20=5%
Step-by-step explanation:
Janet is 33.3% finished with
her math homework. If she
has 24 questions, how many
does she have left to finish?
Answers
What is the volume of the right rectangular prism below?
Answers
Answer:
C
Step-by-step explanation:
2 in. x 1 1/4 in. x 1 1/2 in. = 2/1 in. x 5/4 in. x 3/2 in. = 30/8 cubic inches = 3 6/8 = 3 3/4 cubic inches
2.
Show your work
24*
V
2x
Angle =
Answers
Answer:
90-24=76°
Step-by-step explanation:
____________________
what is the sum of -4y+3 and 11y-5, i will give brainliest
Answers
Answer:
so the answer is 7y - 2 (i think)
Step-by-step explanation:
you just have to minus 11y and 4y then add 3 - 5
please mark me brainliest
Answer:
7y - 2
Step-by-step explanation:
solving variable and constant terms separately,
variable term:
-4y + 11y = 7y
constant term :
-5 + 3 = -2
together:
7y - 2
A person rolls a standard six-sided die 6 times. In how many ways can he get 3 fours, 2 ones, and I two?
Answers
Answer:
Good afternoon! no. of ways = 10! / [ 6! * 3! * 1!]
= 840
So the answer is 10.
_AnnieTheDreamGirl, 3/8/2022
If the scale factor of sides is 3, then the scale factor of area is ___ and the scale factor of
volume is __
Answers
Answer:
9
27
Step-by-step explanation:
the scale factor of area is 3² = 9
the scale factor of volume is 3³ = 27
Answer:
1/2 then 4Ts
Step-by-step explanationexplanation: good luck
evaluate the expression 6n^2-15 when n=5
Answers
Hey there!
6n^2 - 15
= 6(5)^2 - 15
= 6(5^2) - 15
= 6(5 * 5) - 15
= 6(25) - 15
= 150 - 15
= 135
Therefore, your answer is: 135
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
The price of an all-access ticket to the fair is $42.99 before tax. Roland bought the all-access ticket for 40% off. He then paid 7% sales tax on the discounted price. Part A: How much did Roland pay in sales tax? Show all work and steps in your solution. (5 points) Part B: What is the total amount that Roland paid for the ticket? Show all work and steps in your solution.
Answers
Answer:
See below
Step by step explanation:
Here it is given that , the price of a ticket is $42.99 before tax . And he got a discount of 40% . So the cost after discount will be ,
[tex]\longrightarrow Cost = \$ 42.99 - \$42.99\times \dfrac{40}{100} \\[/tex]
Simplify ,
[tex]\longrightarrow Cost = \$ 42.99 - \$17.196 \\[/tex]
[tex]\longrightarrow Cost = \$ 25.79 [/tex]
Again it is given that he paid a tax of 7% on the discounted price . So the tax paid in dollars will be ,
[tex]\longrightarrow Tax = 7\% \ of \ \$25.79\\ [/tex]
Simplify ,
[tex]\longrightarrow Tax = \dfrac{7}{100}\times \$ 25.79 [/tex]
[tex]\longrightarrow \underline{\underline{ Tax =\$ 1.80 }} [/tex]
Answer of Part A : He paid a sales tax of $1.80 .
Now the total amount paid can be calculated by adding sales tax to the cost after discount .
[tex]\longrightarrow Total\ Amount = \$ 25.79 +\$ 1.80\\ [/tex]
Add ,
[tex]\longrightarrow \underline{\underline{ Total \ Amount = \$ 27.59}} [/tex]
Answer of Part B : The total amount paid by Ronald for ticket is $27.59 .
There are 145 students at a small school and 41 of them are freshman. What fraction of the students are freshman
Answers
Answer: 41/145
Step-by-step explanation:
Freshmen/Total students = 41/145
ayuda por favor
Resta: 4805 - 1568
Answers
Answer:
4805 - 1568 = 3237
Please mark as brainliest;)
Do not understand it??
Answers
Question 4 of 5
Find the height of a cone with a diameter of 12 m whose volume is 340 m2
Use 3.14 for tr and round your answer to the nearest meter.
12 m
A. 28 m
O B.2 m
O C. 12 m
O D. 9m
Answers
Answer:
h = 9 m
Step-by-step explanation:
[tex]V_{cone}=340\: m^3[/tex]
diameter (d) = 12 m
[tex]\implies radius\:(r)= \frac{12}{2}=6\:m[/tex]
[tex] \because \: V_{cone}= \frac{1}{3} \pi {r}^{2} h \\ \\ \implies \: h = \frac{3V_{cone}}{\pi {r}^{2} } \\ \\ \implies \: h = \frac{3(340)}{3.14 {(6)}^{2} } \\ \\ \implies \: h = \frac{1020}{113.04} \\ \\ \implies \: h = 9.02335456 \\ \\ \implies \: h \approx 9 \: m[/tex]
Helppp!
a. box 1: always, sometimes , never / box 2: remain ,do not remain
b. box 1: always, sometimes , never/ box 2: always, sometimes , never
c. box 1 : always, sometimes , never/ box 2: remain, not remain/ box 3: remain, not remain
d. box 1 : awlays, sometimes, never/ box 2 : an equal distance, different distance/ box 3 : some, all, no
e. box 1 : awlays, sometimes, never/ box 2 : would, would not / box 3 : all, no.
Answers
Answer:
They are all correct i think
Step-by-step explanation:
Find the factors pairs of 78
Answers
Answer:
Therefore, the pair factors are (1, 78), (2, 39), (3, 26), and (6, 13).
Give me brainliest answer
Answer:
1 x 78, 2 x 39, 3 x 26, 6 x 13
Step-by-step explanation:
Factors of 78 are 1, 2, 3, 6, 13, 26, 39, and 78.
I just need the answers
Answers
Answer:
A=102 and B=0
Step-by-step explanation:
Sub values for X and Y into equation, put into your calculator and you will get the correct answer.
Answer:
a) 102
b) 0
Step-by-step explanation:
A:
[tex]10(10)-(2(5)-12)[/tex] [tex]100-(10-12)[/tex] [tex]100-(-2)[/tex] [tex]100+2[/tex] [tex]102[/tex]B:
[tex]2\left(10\right)-35+4\left(10\right)-\left(3\left(10\right)-5\right)[/tex] [tex]20-35+40-(30-5)[/tex] [tex]25-(25)[/tex] [tex]0[/tex]
Prism A is a dilation of Prism B. The height of Prism A is 612 ft, and the volume of Prism A is 8734 ft³. The height of Prism B is 314 ft.
What is the volume of Prism B?
Enter your answer as a mixed number in simplest form by filling in the boxes.
Answers
Answer:
volume of prism B=4481.16993463 ft^3
Step-by-step explanation:
Prism B's height was multiplied by the scale factor to get prism A's height.
So x is the scale factor and 314x=612
x=1.949
volume of prism A is volume of prism B multiplied by x
8734=1.949(volume of prism B)
Volume of prism B=4481.16993463 ft^3
Answer:
10 31/32
Step-by-step explanation:
I took the k12 test
Find all points on the x-axis that are 16 units from the point (4,-8)
Answers
Step-by-step explanation:
all points on the x axis have a y value of 0.
so, we need to find all points (x, 0) with a distance of 16 to (4, -8).
the distance to a point is the Hypotenuse of the right-angled triangle with the coordinate differences as legs.
and so Pythagoras applies :
c² = a² + b²
with c being the Hypotenuse (side opposite of the 90° angle).
so, in our case we have
16² = (4-x)² + (-8 - 0)²
256 = 16 - 8x + x² + 64 = x² - 8x + 80
x² - 8x - 176 = 0
if we have no other idea of factorization the general solution of such a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = -8
c = -176
x = (8 ± sqrt((-8)² - 4×1×-176))/(2×1) =
= (8 ± sqrt(64 + 704))/2 = (8 ± sqrt(768))/2 =
= (8 ± sqrt(256×3))/2 = (8 ± 16×sqrt(3))/2 =
= 4 ± 8×sqrt(3)
x1 = 4 + 8×sqrt(3) = 17.85640646...
x2 = 4 - 8×sqrt(3) = -9.856406461...
so, the points on the x axis that are 16 units apart from (4, -8) are
(x1, 0) and (x2, 0)
I am not sure in what format you need the answer.
please fill in for x1 and x2 from above what you need.
What quadratic function bests fits this data?
(refer to the picture)
A) y= -9.38x^2 + 74.80x + 186.1
B) y= -9.38x^2 -74.80x+186.1
C) y= 9.38x^2 - 74.80x + 186.1
D) y= 9.38x^2 + 74.8x + 186.1
Answers
Brainliest if correct
Answers
Answer:
44% is shaded
Step-by-step explanation:
Having carefully counted, I find 44 of the squares shaded. I'm sure of it.
Answer:
Yes, the other is correct. The answer is 44%
Step-by-step explanation:
In a certain class, 25% of students wear
glasses. If 6 students wear glasses, how many
students are in the class?
Answers
choose the sentence that complete the fact family like 2×9=18 18÷9=2 9×2=18 is it a 36÷2=18 b 18÷6=3 c 36÷18=2 d 18÷2=9 is it abc or d
Answers
2 * 9 = 18
18 * 9 = 2
2 * 9 = 18
18/2 = 9
The numbers we are working with are 2, 9, and 18.
a is wrong (there is a 36)
b is wrong (there is a 6)
c is wrong (there is a 36)
d is correct (there is a 2, 9, and 18)
Therefore, our final answer is d.
Hope this helps!
Solve both Question related to Volume of Solids. 50 Points
Answers
Given for the cone:
base radius: 3aperpendicular height: 4agiven for the spheres:
radius: aFind volume of cone:
[tex]\sf \hookrightarrow \dfrac{1}{3} \pi r^2h[/tex]
[tex]\sf \hookrightarrow \dfrac{1}{3} \pi (3a)^2(4a)[/tex]
[tex]\sf \hookrightarrow 12\pi a^3[/tex]
Find volume of sphere:
[tex]\sf \hookrightarrow \dfrac{4}{3} \pi r^3[/tex]
[tex]\sf \hookrightarrow \dfrac{4}{3} \pi a^3[/tex]
Divide volume of cone by volume of sphere:
[tex]\sf \hookrightarrow \dfrac{12\pi a^3}{ \dfrac{4}{3} \pi a^3}[/tex]
[tex]\rightarrow 9[/tex]
Therefore proved 9 such spheres can be made using the metal cone.
We know that volume of cone is [tex]12\pi a^3[/tex]
[tex]\sf \rightarrow 12\pi a^3 = 12936[/tex]
[tex]\sf \rightarrow a^3 = \dfrac{12936}{12\pi}[/tex]
[tex]\sf \rightarrow a = \sqrt[3]{\dfrac{12936}{12\pi}}[/tex]
[tex]\sf \rightarrow a =7[/tex]
Therefore the radius of sphere is 7 cm
yall I need help with Algebraic Experrion could yall like help me with a & b & c with the work it would really help me
Thank you! <3
Answers
Answer:
A) is 9b
B) is 2
C) is 3y+6
Step-by-step explanation:
A) (b)(2+7)
=(b)(2) + (b)(7)
= b2+b7
=b9
B) (2)(4-3)
= (2)(1)
= 2
C) 3(2+y) =(3)(2+y)
=(3)(2)+(3)(y)
=6+3y
=3y+6
A dilation produces a smaller figure. Which is a possible scale factor? –2 One-third 1 4
Answers
Answer:
1/3 (one-third)
Step-by-step explanation:
A scale factor that produces an image smaller than the original figure is less than 1. A scale factor that produces an image bigger than the original figure is more than 1.
Thus, the only option that's less than 1 is one-third.