1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Luba_88 [7]
3 years ago
6

More Calculus! (I'm so sorry)

Mathematics
1 answer:
Olenka [21]3 years ago
6 0
Recall that converting from Cartesian to polar coordinates involves the identities

\begin{cases}y(r,\phi)=r\sin\phi\\x(r,\phi)=r\cos\phi\end{cases}

As a function in polar coordinates, r depends on \phi, so you can write r=r(\phi).

Differentiating the identities with respect to \phi gives

\begin{cases}\dfrac{\mathrm dy}{\mathrm d\phi}=\dfrac{\mathrm dr}{\mathrm d\phi}\sin\phi+r\cos\phi\\\\\dfrac{\mathrm dx}{\mathrm d\phi}=\dfrac{\mathrm dr}{\mathrm d\phi}\cos\phi-r\sin\phi\end{cases}

The slope of the tangent line to r(\phi) is given by

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm d\phi}}{\frac{\mathrm dx}{\mathrm d\phi}}=\dfrac{\frac{\mathrm dr}{\mathrm d\phi}\sin\phi+r\cos\phi}{\frac{\mathrm dr}{\mathrm d\phi}\cos\phi-r\sin\phi}

Given r(\phi)=3\cos\phi, you have \dfrac{\mathrm dr}{\mathrm d\phi}=-3\sin\phi. So the tangent line to r(\phi) has a slope of

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{-3\sin^2\phi+3\cos^2\phi}{-3\sin\phi\cos\phi-3\cos\phi\sin\phi}=\dfrac{3\cos2\phi}{-3\sin2\phi}=-\cot2\phi

When \phi=120^\circ=\dfrac{2\pi}3\text{ rad}, the tangent line has slope

\dfrac{\mathrm dy}{\mathrm dx}=-\cot\dfrac{4\pi}3=-\dfrac1{\sqrt3}

This line is tangent to the point (r,\phi)=\left(-\dfrac32,\dfrac{2\pi}3\right) which in Cartesian coordinates is equivalent to (x,y)=\left(\dfrac34,-\dfrac{3\sqrt3}4\right), so the equation of the tangent line is

y+\dfrac{3\sqrt3}4=-\dfrac1{\sqrt3}\left(x-\dfrac34\right)

In polar coordinates, this line has equation

r\sin\phi+\dfrac{3\sqrt3}4=-\dfrac1{\sqrt3}\left(r\cos\phi-\dfrac34\right)
\implies r=-\dfrac{3\sqrt3}{2\sqrt3\cos\phi+6\sin\phi}

The tangent line passes through the y-axis when x=0, so the y-intercept is \left(0,-\dfrac{\sqrt3}2\right).

The vector from this point to the point of tangency on r(\phi) is given by the difference of the vector from the origin to the y-intercept (which I'll denote \mathbf a) and the vector from the origin to the point of tangency (denoted by \mathbf b). In the attached graphic, this corresponds to the green arrow.

\mathbf b-\mathbf a=\left(\dfrac34,-\dfrac{3\sqrt3}4\right)-\left(0,-\dfrac{\sqrt3}2\right)=\left(\dfrac34,-\dfrac{\sqrt3}4\right)

The angle between this vector and the vector pointing to the point of tangency is what you're looking for. This is given by

\mathbf b\cdot(\mathbf b-\mathbf a)=\|\mathbf b\|\|\mathbf b-\mathbf a\|\cos\theta
\dfrac98=\dfrac{3\sqrt3}4\cos\theta
\implies\theta=\dfrac\pi6\text{ rad}=30^\circ

The second problem is just a matter of computing the second derivative of \phi with respect to t and plugging in t=2.

\phi(t)=2t^3-6t
\phi'(t)=6t^2-6
\phi''(t)=12t
\implies\phi''(2)=24

You might be interested in
What is 70% of 50? explain how you got this answer.
eimsori [14]
First, convert the percent into a decimal. You can make any percent into a decimal by moving the % sign two places to the left and making it into a decimal point.

70% = 0.70

In mathematics, if can also mean multiply.

0.70 * 50 = 35

So, 35 is the answer to this question.


7 0
3 years ago
Read 2 more answers
A blender runs at 120 volts at 4.16 amps. Calculate power
andrey2020 [161]

Answer:

P = 499.2 Watt

Step-by-step explanation:

p = iv \\ p = 4.16 \times 120 \\ p = 499.2 \: watt

6 0
3 years ago
Solve the inequality.
zysi [14]

Answer:

?= 2

X=1

Step-by-step explanation:

3(1+1)<9

3×2<9

6<9

Pls mark brainliest.

8 0
1 year ago
Read 2 more answers
B<br> с<br> 6x + 1<br> E<br> X + 11<br> D
Mnenie [13.5K]

Answer:

E

X + 11 i think that is the answer of this question

8 0
3 years ago
I can't find the surface area, is this confusing me.
Fed [463]
<span>Find the area of two sides (Length*Height)*2 sidesFind the area of adjacent sides (Width*Height)*2 sidesFind the area of ends (Length*Width)*2 ends<span>Add the three areas together to find the surface area</span></span>
3 0
3 years ago
Read 2 more answers
Other questions:
  • ivy bought 10 packs of cups for her holiday party. a pack of medium cups costs $1.80 and a pack of large cups costs $2.40. she p
    9·2 answers
  • A garden has four sides that are all the same length. Each side measures x+4 units. The garden's perimeter is 112 units. What is
    14·1 answer
  • Whats a simple meaning for polynomials?
    14·1 answer
  • The amounts of nicotine in a certain brand of cigarette are normally distributed with a mean of 0.972 g and a standard deviation
    5·1 answer
  • To rationalize the denominator of StartFraction 2 StartRoot 10 EndRoot Over 3 StartRoot 11 EndRoot EndFraction, you should multi
    10·2 answers
  • Plzzzzzzz help me with #1,2,3
    8·1 answer
  • PLEASE HELP!! DUE SOON :(
    12·1 answer
  • Simplify the expression using the order of operations. 3(7 + 4) – 8.2
    12·2 answers
  • loly beought a bag of hummy bears and ate one third of them the next say she ate 16 more hummy bears if she ate a total of 37 gu
    11·1 answer
  • Which graph represents a function? <br> Please help ASAP-thanks
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!