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1 year ago

Make sure your answer is in

Mathematics

1 answer:

romanna [79]1 year ago

6 0

Answer:

210° , 330°

Step-by-step explanation:

using the sides of the 30- 60- 90 triangle

with legs 1, $\sqrt{3}$ and hypotenuse 2 , then

$sin^{-1}$ $\frac{1}{2}$ = 30° ← related acute angle

since $sin^{-1}$ - $\frac{1}{2}$

then angle is in third / fourth quadrants , then required angles are

180° + 30° = 210° ← in third quadrant

360° - 30° = 330° ← in fourth quadrant

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Use the Chain Rule (Calculus 2)

atroni [7]

1. By the chain rule,

$\dfrac{\mathrm dz}{\mathrm dt}=\dfrac{\partial z}{\partial x}\dfrac{\mathrm dx}{\mathrm dt}+\dfrac{\partial z}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}$

I'm going to switch up the notation to save space, so for example, $z_x$ is shorthand for $\frac{\partial z}{\partial x}$ .

$z_t=z_xx_t+z_yy_t$

We have

$x=e^{-t}\implies x_t=-e^{-t}$

$y=e^t\implies y_t=e^t$

$z=\tan(xy)\implies\begin{cases}z_x=y\sec^2(xy)=e^t\sec^2(1)\\z_y=x\sec^2(xy)=e^{-t}\sec^2(1)\end{cases}$

$\implies z_t=e^t\sec^2(1)(-e^{-t})+e^{-t}\sec^2(1)e^t=0$

Similarly,

$w_t=w_xx_t+w_yy_t+w_zz_t$

where

$x=\cosh^2t\implies x_t=2\cosh t\sinh t$

$y=\sinh^2t\implies y_t=2\cosh t\sinh t$

$z=t\implies z_t=1$

To capture all the partial derivatives of $w$ , compute its gradient:

$\nabla w=\langle w_x,w_y,w_z\rangle=\dfrac{\langle1,-1,1\rangle}{\sqrt{1-(x-y+z)^2}}}=\dfrac{\langle1,-1,1\rangle}{\sqrt{-2t-t^2}}$

$\implies w_t=\dfrac1{\sqrt{-2t-t^2}}$

2. The problem is asking for $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ . But $z$ is already a function of $x,y$ , so the chain rule isn't needed here. I suspect it's supposed to say "find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$ " instead.

If that's the case, then

$z_s=z_xx_s+z_yy_s$

$z_t=z_xx_t+z_yy_t$

as the hint suggests. We have

$z=\sin x\cos y\implies\begin{cases}z_x=\cos x\cos y=\cos(s+t)\cos(s^2t)\\z_y=-\sin x\sin y=-\sin(s+t)\sin(s^2t)\end{cases}$

$x=s+t\implies x_s=x_t=1$

$y=s^2t\implies\begin{cases}y_s=2st\\y_t=s^2\end{cases}$

Putting everything together, we get

$z_s=\cos(s+t)\cos(s^2t)-2st\sin(s+t)\sin(s^2t)$

$z_t=\cos(s+t)\cos(s^2t)-s^2\sin(s+t)\sin(s^2t)$

8 0

2 years ago

A carnival tent is set up using a 32-foot pole as the center. The Support wires each have an angle of elevation of 38 degrees fr

BARSIC [14]

Answer:

about 52 feet

Step-by-step explanation:

The figure is omitted--you may draw it to confirm my answer.

sin (38°) = 32/x

x sin (38°) = 32

x = 32 / sin (38°) = about 52 feet

5 0

1 year ago

The pollutant PCB (polychlorinated biphenyl) can affect the thickness of pelican eggshells. Thinking of the thickness, T, of the

gayaneshka [121]

Answer:

The expression f(200) depicts the value of the concentration of the pollutant, which when substituted in the expression T = f(P) will give the value of the thickness of the pelican eggs.

Step-by-step explanation:

Since we are given that T = f(P)

Where T = thickness of the eggs

P = concentration in ppm ((parts per million))

Considering f(200) comparing to f(P)

The expression f(200) depicts the value of the concentration of the pollutant, which when substiuted in the expression T = f(P) will give the value of the thickness of the pelican eggs.

i.e substituted the value of the concentration in ppm to get the thickness of the pelican eggs in mm. since the expression shows thickness of pelican eggs as a function of the concentration. for every 200ppm of concentration, the numerical value of the thickness of pelican eggs can be gotten.

6 0

2 years ago

What is <br><br> 88x3-7+198-34x15+76-126=

PilotLPTM [1.2K]

-105

Multiply first, then subtract, then add.

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2 years ago

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Ilya [14]

Answer: 9 remainder 2

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