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Varvara68 [4.7K]
3 years ago
9

7. When dy/dx = (x -6)1/2, what does y equal?

Mathematics
1 answer:
Sergio039 [100]3 years ago
3 0
The answer is below
\frac{dy}{dx} = \frac{1}{2} (x - 6)
multiply both sides by dx

dy = \frac{1}{2} (x - 6)dx
integrate both sides of the diffrential

integral \: dy = integral \: \frac{1}{2} (x - 6)dx
simplify and integrate to get y alone don't forget to tack on c with your indefinite integral

y = \frac{x {}^{2} }{4} - 3x + c
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Use lagrange multipliers to find the shortest distance, d, from the point (4, 0, −5 to the plane x y z = 1
Varvara68 [4.7K]
I assume there are some plus signs that aren't rendering for some reason, so that the plane should be x+y+z=1.

You're minimizing d(x,y,z)=\sqrt{(x-4)^2+y^2+(z+5)^2} subject to the constraint f(x,y,z)=x+y+z=1. Note that d(x,y,z) and d(x,y,z)^2 attain their extrema at the same values of x,y,z, so we'll be working with the squared distance to avoid working out some slightly more complicated partial derivatives later.

The Lagrangian is

L(x,y,z,\lambda)=(x-4)^2+y^2+(z+5)^2+\lambda(x+y+z-1)

Take your partial derivatives and set them equal to 0:

\begin{cases}\dfrac{\partial L}{\partial x}=2(x-4)+\lambda=0\\\\\dfrac{\partial L}{\partial y}=2y+\lambda=0\\\\\dfrac{\partial L}{\partial z}=2(z+5)+\lambda=0\\\\\dfrac{\partial L}{\partial\lambda}=x+y+z-1=0\end{cases}\implies\begin{cases}2x+\lambda=8\\2y+\lambda=0\\2z+\lambda=-10\\x+y+z=1\end{cases}

Adding the first three equations together yields

2x+2y+2z+3\lambda=2(x+y+z)+3\lambda=2+3\lambda=-2\implies \lambda=-\dfrac43

and plugging this into the first three equations, you find a critical point at (x,y,z)=\left(\dfrac{14}3,\dfrac23,-\dfrac{13}3\right).

The squared distance is then d\left(\dfrac{14}3,\dfrac23,-\dfrac{13}3\right)^2=\dfrac43, which means the shortest distance must be \sqrt{\dfrac43}=\dfrac2{\sqrt3}.
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3 years ago
In a neighborhood 60% of the houses have a garage and a fenced in backyard. Given that 80% of the houses in the neighborhood hav
Tpy6a [65]
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8 0
3 years ago
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The mass of the Earth is 5.972x10^24 kg. the mass of the moon is 7.3 x 10^22kg. find the total mass, in kg, of the Earth and the
torisob [31]

Answer:

6045000000000000000000000 kg.

Step-by-step explanation:

We have been given that the mass of Earth is 5.972\times10^{24} kg. The mass of the Moon is 7.3\times10^{22} kg.

To find the total mass we will add mass of Earth and Moon.

First of all let us convert the given masses in standard form.

5.972\times10^{24}=5972000000000000000000000  

7.3\times10^{22}=73000000000000000000000

5.972\times10^{24}+7.3\times10^{22}  

5972000000000000000000000+73000000000000000000000

6045000000000000000000000

Therefore, the mass of Earth and Moon is 6045000000000000000000000 kg.

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Constant of Proportionality
Slav-nsk [51]
Hmmm.....................
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2 years ago
An ice cream factory makes 29 quarts of ice cream in 5 hours. How many quarts could be made in 48 hours ? What was that rate per
Vadim26 [7]

We can easily get the quarts per hour rate by dividing the number of quarts by the number of hours:

\text{29 quarts in 5 hours} \implies \dfrac{29}{5} = 5.8 \text{ quarts in 1 hour}

Now that we have the quarts per hour rate, we can easily address the question: the factory could make

5.8\cdot 48 = 278.4

quarts in 48 hours, with a daily rate of

5.8\cdot 24 = 139.2

quarts per day

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3 years ago
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