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uysha [10]
3 years ago
8

What many eighths in 0.25

Mathematics
1 answer:
Sergeu [11.5K]3 years ago
4 0
There are 3 eighths in 0.25
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Marine biologists have determined that when a shark detectsthe presence of blood in the water, it will swim in the directionin w
siniylev [52]

Solution :

a). The level curves of the function :

$C(x,y) = e^{-(x^2+2y^2)/10^4}$

are actually the curves

$e^{-(x^2+2y^2)/10^4}=k$

where k is a positive constant.

The equation is equivalent to

$x^2+2y^2=K$

$\Rightarrow \frac{x^2}{(\sqrt K)^2}+\frac{y^2}{(\sqrt {K/2})^2}=1, \text{ where}\ K = -10^4 \ln k$

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 :

$\triangledown C= \frac{\partial C}{\partial x}i+\frac{\partial C}{\partial y}j$

$\Rightarrow \triangledown C =-\frac{2}{10^4}e^{-(x^2+2y^2)/10^4}(xi+2yj),$ and

$\triangledown C$ points in the direction of most rapid increase in concentration, which means $\triangledown C$ is tangent to the most rapid increase curve.

$r(t)=x(t)i+y(t)j$  is a parametrization of the most $\text{rapid increase curve}$ , then

$\frac{dx}{dt}=\frac{dx}{dt}i+\frac{dy}{dt}j$ is a tangent to the curve.

So then we have that $\frac{dr}{dt}=\lambda \triangledown C$

$\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} $

∴ $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{2y}{x}$

Using separation of variables,

$\frac{dy}{y}=2\frac{dx}{x}$

$\int\frac{dy}{y}=2\int \frac{dx}{x}$

$\ln y=2 \ln x$

⇒ y = kx^2 for some constant k

but we know that $y(x_0)=y_0$

$\Rightarrow kx_0^2=y_0$

$\Rightarrow k =\frac{y_0}{x_0^2}$

∴ The path of the shark will follow is along the parabola

$y=\frac{y_0}{x_0^2}x^2$

$y=y_0\left(\frac{x}{x_0}\right)^2$

7 0
2 years ago
Find the solution of the system of equations.<br> 10x+10y=40 <br> 5x+3y=8
ddd [48]

Answer:

x = -2 & y = 6

Step-by-step explanation:

10x+10y=40

10x+10y+ -10y=40+-10y

10x=-10y+40

\frac{10x}{10} =\frac{-10y+40}{10}

x=-y+4

5x+3y=8

5(-y+4)+3y=8

-2y+20=8

-2y+20+-20=8+-20

-2y=-12

\div2

y=6

x=-y+4

x=-6+4

x=-2

Hope this helps.

7 0
2 years ago
HELP PLEASE WILL GIVE BRAINLIEST WORTH 50 POINTS
Alja [10]

Answer:

y = 8.13 x -16125.713


hope it helps

Step-by-step explanation:

Sum of X = 14004

Sum of Y = 977

Mean X = 2000.5714

Mean Y = 139.5714

Sum of squares (SSX) = 53.7143

Sum of products (SP) = 436.7143

Regression Equation = ŷ = bX + a

b = SP/SSX = 436.71/53.71 = 8.13032

a = MY - bMX = 139.57 - (8.13*2000.57) = -16125.71277

ŷ = 8.13032X - 16125.71277

8 0
3 years ago
Read 2 more answers
3f + 9 &lt; 21<br> Help please
fiasKO [112]

Answer:

f < 4

Step-by-step explanation:

3f + 9 < 21

21 - 9 = 12

12 / 3 = 4

4 0
3 years ago
Read 2 more answers
8 1/4 pounds of carrots to make 6 carrot cakes. how many pounds of carrots does frank need to make one carrot cake.
jasenka [17]
The answer would be 1.4


6 0
3 years ago
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