Solution :
a). The level curves of the function :
[tex]$C(x,y) = e^{-(x^2+2y^2)/10^4}$[/tex]
are actually the curves
[tex]$e^{-(x^2+2y^2)/10^4}=k$[/tex]
where k is a positive constant.
The equation is equivalent to
[tex]$x^2+2y^2=K$[/tex]
[tex]$\Rightarrow \frac{x^2}{(\sqrt K)^2}+\frac{y^2}{(\sqrt {K/2})^2}=1, \text{ where}\ K = -10^4 \ln k$[/tex]
which is a family of ellipses.
We sketch the level curves for K =1,2,3 and 4.
If the shark always swim in the direction of maximum increase of blood concentration, its direction at any point would coincide with the gradient vector.
Then we know the shark's path is perpendicular to the level curves it intersects.
b). We have :
[tex]$\triangledown C= \frac{\partial C}{\partial x}i+\frac{\partial C}{\partial y}j$[/tex]
[tex]$\Rightarrow \triangledown C =-\frac{2}{10^4}e^{-(x^2+2y^2)/10^4}(xi+2yj),$[/tex] and
[tex]$\triangledown C$[/tex] points in the direction of most rapid increase in concentration, which means [tex]$\triangledown C$[/tex] is tangent to the most rapid increase curve.
[tex]$r(t)=x(t)i+y(t)j$[/tex] is a parametrization of the most [tex]$\text{rapid increase curve}$[/tex] , then
[tex]$\frac{dx}{dt}=\frac{dx}{dt}i+\frac{dy}{dt}j$[/tex] is a tangent to the curve.
So then we have that [tex]$\frac{dr}{dt}=\lambda \triangledown C$[/tex]
[tex]$\Rightarrow \frac{dx}{dt}=-\frac{2\lambda x}{10^4}e^{-(x^2+2y^2)/10^4}, \frac{dy}{dt}=-\frac{4\lambda y}{10^4}e^{-(x^2+2y^2)/10^4} $[/tex]
∴ [tex]$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{2y}{x}$[/tex]
Using separation of variables,
[tex]$\frac{dy}{y}=2\frac{dx}{x}$[/tex]
[tex]$\int\frac{dy}{y}=2\int \frac{dx}{x}$[/tex]
[tex]$\ln y=2 \ln x$[/tex]
⇒ y = kx^2 for some constant k
but we know that [tex]$y(x_0)=y_0$[/tex]
[tex]$\Rightarrow kx_0^2=y_0$[/tex]
[tex]$\Rightarrow k =\frac{y_0}{x_0^2}$[/tex]
∴ The path of the shark will follow is along the parabola
[tex]$y=\frac{y_0}{x_0^2}x^2$[/tex]
[tex]$y=y_0\left(\frac{x}{x_0}\right)^2$[/tex]
What is the sum?
46 mins left please help!!!!!
Erin has 4 cards, one with the letter A, one with the letter B, one with the letter C, and one with the letter D. She places the cards in a row in random order.
What is the probability that either the words BAD or CAB will appear as part of the row?
Answer:
P = 1/6
Step-by-step explanation:
We have 4 places where we can put the cards.
Let's count the number of possible outcomes for each place.
In the first place, we can put one of 4 cards.
in the second place, we can put one of 3 cards (because there is already one card in the first place)
In the third place, we can put one of 2 cards
In the last place, we can put one of one cards.
The total number of different combinations is the product between the number of options for each place, then we have:
C = 4*3*2*1 = 24
Now we need to count the number of combinations such that the words BAD or CAB appear.
Those are:
BADC
CBAD
CABD
DCAB
4 outcomes out of 24.
Then the probability will be equal to the quotient between the number of outcomes where one of these words appears (4) and the total number of possible outcomes (24)
The probability is:
P = 4/24 = 1/6
The probability would be 16.7%
If the volume of two similar spheres are 128m^3 and 1458m^3.
1. What is their scale factor? (in a ratio)
2. What is the ratio of their surface areas?
Answer:
ZOOM PLZ COME WE R BORED
MEETING ID 798 4170 2552
PASSWORD:XCNeV3
Step-by-step explanation:
Scale factor is 11.39 and ratio is 729:64
what is scale factor?The scale factor is a measure for similar figures, who look the same but have different scales or measures.
volume of two similar spheres are 128m³ and 1458m³.
Scale factor= volume of sphere 1/ volume of sphere 2
= 1458/128
= 729/64
= 11.39
and the ratio = 729 : 64
learn more about scale factor here:
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How do you solve this
9514 1404 393
Answer:
x = 8
Step-by-step explanation:
The measure of the inscribed angle is half the measure of the intercepted arc.
9x -25 = 94/2
9x = 72 . . . . . . . add 25
x = 8 . . . . . . . . . divide by 9
Victor drew trapezoid PQRS on a
coordinate plane. The coordinates of each
vertex are:
P(8,4) Q(10, 4) R(13, -1) S(8, -1)
tion
What is the length, in units, of side PS?
A. 2
B. 3
C. 4
D. 5
Answer:
Step-by-step explanation:
b
Answer:
b hope it helps make brainlliest ty
Step-by-step explanation:
can you make me brainlliest
Someone please help me. I will mark brainliest. I really need help ASAP.
Answer: 38.8
Step-by-step explanation:
use formula C = π * D
Diameter = 6
C = π * 6
C = 18.8495559215 = 18.8
18.8 + 10 + 10 = 38.8
What is sec 0 when cot 0 = -4/2?
Answer:
[tex]sec 0=\frac{\sqrt{66} }{8}[/tex]
Step-by-step explanation:
[tex]Cot0=-4\sqrt{2}[/tex]
[tex]sec0=\frac{\sqrt{33} }{4\sqrt{2} }[/tex]
[tex]=\frac{\sqrt{33} }{4\sqrt{4} } *\frac{4\sqrt{2} }{4\sqrt{2} }[/tex]
[tex]=\frac{4\sqrt{66} }{32}[/tex]
[tex]=\frac{\sqrt{66} }{8}[/tex]
--------------------
hope it helps...
have a great day!!
Suppose y varies directly with x. Write an equation relating x and y.
y = -22.5 when x = 15
a.
b.
y = -0.75x
2
y=-3
c. y=-1.5x
d. y = 1.5x
Please help me!! I don’t understand!!!
Answer:
Ask your John Brown teacher!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Step-by-step explanation:
If you place a 37-foot ladder against the top of a building and the bottom of the ladder is 22 feet from the bottom of the building, how tall is the building? Round to the nearest tenth of a foot
Answer:
29.7 ft tall
Step-by-step explanation:
29.75ft. tall is the building.
What is Pythagorean theorem?Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.
Here, we have,
from the figure , we get,
Hypotenuse = 37 ft.
Base= 22 ft.
Let, the building = height = x ft.
Now , by Pythagorean theorem, we have,
x=√37²-22²
=√885
=29.75
Hence, 29.75ft. tall is the building.
To learn more on Pythagorean theorem click:
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During the summer, Dwayne Works on his uncles dairy farm. One of Dwaynes Many chores is to load bales of hay onto a wagon and haul them from the field into the barn.
There is a proportional relationship between the number of wagon loads, x and the number of bales of hay Dwayne can haul, y.
What is the constant of proportionality? Write your answer as a whole number or decimal.
The constant of proportionality of the proportional relationship between the number of wagon loads, x and the number of bales of hay Dwayne can haul, y, is given as follows:
4.
How to define a proportional relationship?The standard definition of a proportional relationship is given as follows:
y = kx.
Hence the solution for the constant is given as follows:
k = y/x.
As the table represents a proportional relationship, the constant can be calculated from any pair, hence:
k = 20/5 = 4.
More can be learned about a proportional relationship at https://brainly.com/question/10424180
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Is (1,2) (1,3) (1,4) a function?
Answer:
yes
Step-by-step explanation:
What is 8 minus a number
8-n
8/n
8n
N/8
Which one
If p and q vary inversely and p is 8 when q is 8, determine q when p is equal to 16.
Answer:
25
Step-by-step explanation:
i smarte
This right triangular prism has a base with an area of 15 square feet. The height of the prism is 7 feet. What is V, the volume of the triangular prism?
Answer:
[tex]area = {15}^{2} \\ v = \: l \times b \times h \\ area = l \times b = \\ x \times x = {15}^{2} \\ volume = 15 \times h \\ = 15 \times 7 \\ = {105}^{3} feet[/tex]
Answer:
105
Step-by-step explanation:
its right trust me
but make sure this is the question that you are doing
Find the volume.
Pls help!
Answer:
72 in^3
Step-by-step explanation:
Volume: lwh
V = 6 · 3 · 4
V = 18 · 4
V = 72 in^3
Select all the equations where r 3 is a solution.
Answer:
The right answers are 1,4 and 6
Answer:
x-3 =0
6=2x
x^2 = 9
Step-by-step explanation:
substitute the values, and you will get the answer!
have a great day!!
A round stained-glass window has a diameter of 12 inches. What is the circumference to the nearest inch? No links.
Answer:
37.68 inches OR about 38 inches
Step-by-step explanation:
To find the circumference from the diameter, you need to multiply the diameter by pie (3.14)
1÷-½ aswer please!!
Answer:
[tex]( - 2)[/tex]
Step-by-step explanation:
[tex]1 \div - \frac{1}{2} \\ 1 \times - \frac{2}{1} \\( - 2)[/tex]
Hope this helps you.
have a nice day!
Need help fast please,
Answer:
with?
Step-by-step explanation:
Answer:
I can help you really fast, if you give me something to work on
Step-by-step explanation:
also, my life is blank like this question
prove that the points A(1 , -1 ) B(5, 2) C(9, 5) are collinear.
Answer:
AB= [tex]\sqrt(5-1)^2 +(2+1)^2 = \sqrt16+9 =5[/tex]
BC= [tex]\sqrt(5-9)^2 + (2-5)^2 = \sqrt16+9=5[/tex]
AC = [tex]\sqrt(1-9)^2 + (-1-5)^2 = \sqrt64+36 = 10[/tex]
So, AC = AB + BC
A, B, C are collinear points
Step-by-step explanation:
I hope this helps!
The difference between two numbers is 3. What is the larger number if the smaller number is -4? *
Answer:-1
Step-by-step explanation:
Subtract plus 3
Answer:
-1
Step-by-step explanation:
x-y=3
x—Larger number
y—smaller number
x-(-4)=3
x+4=3
x=–1
A tour helicopter travels at a constant rate of 80 mph. If the tour takes 2 hours, how far does the helicopter travel?
A. 40 mi.
B. 80 mi.
C. 120 mi.
D. 160 mi.
Answer:
D
Step-by-step explanation:
80 miles per hour, each hour it will travel 80 miles so for two hours tou do
80 x 2 = 160
Answer:
D
Step-by-step explanation:
80x2=40
it's just simple multiplecation but then again I cant spell multiplication so I mean
plz help i give 50 points
Answer:
area of trapezium = 20 in^2
Step-by-step explanation:
here's your solution
==> area of trapezium = 1/2*(sum of parrel side)*distance between them
==> length of parrel sides = 6 in and 2 in
==> height = 5 in.
==> area of trapezium = 1/2(6 + 2) * 5
==> area of trapezium = 20 in^2
hope it helps
Step-by-step explanation:
sorry for this
"i try so can you rate"
Dr. Victory is a psychologist studying the effects of high-level classes, bullying, and music classes on the anxiety levels of middle school students. He has previously determined that the effects from class levels, bullying experiences, and music classes are independent. It has been determined that the probability that a student is taking a high-level English class is 0.23, the probability that a student has been bullied is 0.36, and 1/3 of students are in music classes.
(a) What is the probability that a randomly selected student is in a high-level English class and has been bullied?
(b) What is the probability that a randomly selected student is in a music class and has been bullied?
(c) What is the probability that a randomly selected student is in a music class, has been bullied, and is in a high-level English class?
Answer:
a) 0.0828 = 8.28% probability that a randomly selected student is in a high-level English class and has been bullied
b) 0.12 = 12% probability that a randomly selected student is in a music class and has been bullied
c) 0.0276 = 2.76% probability that a randomly selected student is in a music class, has been bullied, and is in a high-level English class
Step-by-step explanation:
Probabilities of independent events happening together:
The probabilities of independent events happening together is the multiplication of the probabilities of each event happening.
In this question:
0.23 probability that a student is taking a high-level English class.
0.36 probability that a student has been bullied.
1/3 = 0.3333 probability that a student is in music class.
(a) What is the probability that a randomly selected student is in a high-level English class and has been bullied?
0.23*0.36 = 0.0828
0.0828 = 8.28% probability that a randomly selected student is in a high-level English class and has been bullied
(b) What is the probability that a randomly selected student is in a music class and has been bullied?
0.36*1/3 = 0.12
0.12 = 12% probability that a randomly selected student is in a music class and has been bullied
(c) What is the probability that a randomly selected student is in a music class, has been bullied, and is in a high-level English class?
0.23*0.36*1/3 = 0.23*0.12 = 0.0276
0.0276 = 2.76% probability that a randomly selected student is in a music class, has been bullied, and is in a high-level English class
HELP!
Which expression is equivalent to -6(-2/3+2x) ?
Answer:
-6(-2/3+2x)
=12/3+12x
=4+12x
4(1+3x)
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A $55 sweater is on sale for 20% off. Today only, you can take an additional 10% off the sale price. How much does the sweater cost today?
Answer:$39.60
Step-by-step explanation:
$55 x 0.80= $44
$44 x 0.90=$39.60
Or
$55x 0.20=$11
55-11=44
$44x 0.10=4.40
$44 - $4.40=$39.60
20 POINTS!!!!!!!The monthly salary of an employee with 33 hours of training can be written as
D-Salary(33) or S(33)
D is correct hope this helps
If a dog eats 1/4 pound of food each day, how many days will it take the dog to eat a 5 pound bag of food?
Answer:
20 days
Step-by-step explanation:
0.25 pound of food is eaten in 1 day
1 pound of food is eaten in 1 / 0.25
5 pound of food is eaten in 5/0.25 = 20 days
the sum of 4 and 6 times a number z
Answer:
(4 + 6) × Z = 10 × Z
Step-by-step explanation:
The equation of a line is typically written as y=mx+b where m is the slope and b is the y-intercept. If you know two points that a line passes through, this page will show you how to find the equation of the line.