An iron ball weighs 80 N on the surface

Find the area of the part of the plane 5x 3y z = 15 that lies in the first octant.

Lunna [17]

This part of the plane lies above a triangle with boundaries $x=0$ and $y=0$ along the coordinate axes, as well as the line

$z=0 \implies 5x + 3y = 15 \implies y = \dfrac{15 - 5x}3$

When $y=0$ , we have $15-5x=0\implies x=3$ . So this triangle is the set

$T = \left\{(x,y)\in\Bbb R^2 ~:~ 0\le x\le3 \text{ and } 0\le y\le\dfrac{15-5x}3\right\}$

Also, when $x=0$ , we have $y=\frac{15}3=5$ . So the triangle has length 3 and width 5, hence area 1/2•3•5 = 15/2.

Let $z=f(x,y) = 15 - 5x - 3y$ . Then the area of the plane over $T$ is

$\displaystyle \iint_T dA = \iint_T \sqrt{1 + \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2} \, dx \, dy$

We have

$\dfrac{\partial f}{\partial x} = -5 \implies \left(\dfrac{\partial f}{\partial x}\right)^2 = 25$

$\dfrac{\partial f}{\partial y} = -3 \implies \left(\dfrac{\partial f}{\partial y}\right)^2 = 9$

$\implies\displaystyle \iint_T dA = \sqrt{35} \iint_T dx\,dy = \boxed{\frac{15\sqrt{35}}2}$

since the integral

$\displaystyle \iint_TdA$

is exactly the area of $T$ , 15/2.

8 0

2 years ago

Red giants are smaller than main sequence stars, which are smaller than white dwarfs.

avanturin [10]

<span> False. A Red Giant evolves from a main sequence star after it commences Helium fusion and undergoes a great expansion; in the case of our Sun it would expand to encompass Earth's orbit. Thus, it would have approximately the same mass (a small amount being expelled in the helium flash), and vastly greater volume, equating to vastly lower density.

You're welcome :)
</span>

8 0

3 years ago

Describe succinctly the relationship between how far a galaxy is from us (its distance), versus how fast it is moving.

Eva8 [605]

Answer:

Distance is directly proportional to the velocity

Explanation:

In 1929, Edwin Hubble's wrote an article that talked about relationship between the distance and recession speed/velocity of galaxies which led to what is known as the Hubble Law. This law states that galaxies are moving away from the earth at velocities proportional to their distances.

Thus is written as;

v = H_o•d

Where;

v is velocity

d is distance

H_o is Hubble's constant rate of cosmic expansion.

He came to this conclusion by generating a graph known as Hubble's classic graph which was a graph of observed velocity vs distance for nearby galaxies.

7 0

3 years ago

the metal wire in an incandescent lightbulb glows when the light is switched on and stops glowing when it is switched off. the s

Travka [436]

When the metal wire in an incandescent lightbulb glows when the light is switched on and stops glowing when it is switched off, this is an example of resistance, which provides light and heat.

3 0

3 years ago

What planet has a visible mark referred to as the the Great Red Spot?

vivado [14]

Jupiter that is the answer

good luck

6 0

3 years ago