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3 years ago

6

I need someone to do this pls! Will give brainiest!

1 answer:

lyudmila [28]3 years ago

4 0

Answer: i'm sorry i have no idea how to do this

Step-by-step explanation:

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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

Help please <br> x=6<br> x= square root 80<br> x=12<br> x= square root 164

AURORKA [14]

Answer:

x = 12

Step-by-step explanation:

8 0

3 years ago

Ben is repackaging a 45-pound bag of rice into 8 smaller bags. How much rice is going to be in each smaller bags?

Anarel [89]

Each bag of rice is going to have 1/8 of the original amount, or 1/8 of 45.
45/8 = 5.625.
Each rice bag will have 5.625 pounds of rice.

4 0

2 years ago

Read 2 more answers

When x=8 and y=20 find y when x=42

ella [17]

When x=42 that means it is 8x6 so that means that the y = 120 becuase its 20x6
answer= y=120

7 0

3 years ago

Please help, would be greatful :)

IrinaVladis [17]

Answer:

x ≈ 31.0°

Step-by-step explanation:

Using the tangent ratio in the right triangle

tanx = $\frac{opposite}{adjacent}$ = $\frac{9}{15}$ , then

x = $tan^{-1}$ ( $\frac{9}{15}$ ) ≈ 31.0° ( to 1 dec. place )

7 0

3 years ago