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
skad [1K]
3 years ago
7

15. If x=a Sin2t (I+Cos2t) and y=b Cos 2t (1-Cos2t) then finddy/dx at =22/7*4​

Mathematics
1 answer:
Sav [38]3 years ago
8 0

By the chain rule,

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}

It looks like we're given

\begin{cases}x=a\sin(2t)(1+\cos(2t))\\y=b\cos(2t)(1-\cos(2t))\end{cases}

where <em>a</em> and <em>b</em> are presumably constant.

Recall that

\cos^2t=\dfrac{1+\cos(2t)}2

\sin^2t=\dfrac{1-\cos(2t)}2

so that

\begin{cases}x=2a\sin(2t)\cos^2t\\y=2b\cos(2t)\sin^2t\end{cases}

Then we have

\dfrac{\mathrm dx}{\mathrm dt}=4a\cos(2t)\cos^2t-4a\sin(2t)\cos t\sin t

\dfrac{\mathrm dy}{\mathrm dt}=-4b\sin(2t)\sin^2t+4b\cos(2t)\sin t\cos t

\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{4b\cos(2t)\sin t\cos t-4b\sin(2t)\sin^2}{4a\cos(2t)\cos^2t-4a\sin(2t)\cos t\sin t}

\implies\boxed{\dfrac{\mathrm dy}{\mathrm dx}=\dfrac ba\tan t}

where the last reduction follows from dividing through everything by \cos(2t)\cos^2t and simplifying.

I'm not sure at which point you're supposed to evaluate the derivative (22/7*4, as in 88/7? or something else?), so I'll leave that to you.

You might be interested in
Find the area of the region between two concentric circular paths. If the radii of the circular paths are 210 m. and 490 m. resp
taurus [48]

Given :

  • The radii of the circular paths are 210 m. and 490 m.

To Find :

  • The area of the region between two concentric circular paths.

Solution :

Area of the outer circle :

\: \qquad  \dashrightarrow \sf{{ \pi \: \times  {(490)}^{2}  {m}^{2}}}

\: \qquad  \dashrightarrow \sf{{ \ \dfrac{22}{7} \times   \:490 \times 490 \:  {m}^{2}}}

\: \qquad  \dashrightarrow \sf{{ \ \dfrac{22}{ \cancel{7}} \times   \cancel{490} \times 490 \:  {m}^{2}}}

\: \qquad  \dashrightarrow \sf{{ \ {22} \times   70 \times 490 \:  {m}^{2}}}

\: \qquad  \dashrightarrow \bf{{ \ 754600 \:  {m}^{2}}}

⠀

Area of the inner circle :

\: \qquad  \dashrightarrow \sf{{ \pi \: \times  {(210)}^{2}  {m}^{2}}}

\: \qquad  \dashrightarrow \sf{{ \ \dfrac{22}{7} \times   \:210 \times 210 \:  {m}^{2}}}

\: \qquad  \dashrightarrow \sf{{ \ \dfrac{22}{ \cancel{7}} \times   \cancel{210} \times 210 \:  {m}^{2}}}

\: \qquad  \dashrightarrow \sf{{ \ {22} \times   30 \times 210 \:  {m}^{2}}}

\: \qquad  \dashrightarrow \bf{{ \ 138600 \:  {m}^{2}}}

⠀

Therefore,

Area of the region inside the circular paths :

\: \qquad  \dashrightarrow \sf{{ \ (754600 - 138600  )  \: {m}^{2}}}

\: \qquad  \dashrightarrow \bf{{ \ 616000 \:  {m}^{2}}}

6 0
2 years ago
the original price of a skateboard is 50% the sale price includes 20% discount what is the sale price
ss7ja [257]

Answer:

50 × 0.8 = $40 is the sale price

Step-by-step explanation:

7 0
3 years ago
A student says that the graph of the equation y = 3(x + 8) is the same as the graph of y = 3x, only translated upwards by 8 unit
Wittaler [7]

Given:

A student says that the graph of the equation y = 3(x + 8) is the same as the graph of y = 3x, only translated upwards by 8 units.

To find:

Whether the student is correct or not.

Solution:

Initial equation is

y=3x

f(x)=3x

Equation of after transformation is

y=3(x+8)

g(x)=3(x+8)

Now,

g(x)=f(x+8)      ...(i)

The translation is defined as

g(x)=f(x+a)+b        ...(ii)

Where, a is horizontal shift and b is vertical shift.

If a>0, then the graph shifts a units left and if a<0, then the graph shifts a units right.

If b>0, then the graph shifts b units up and if b<0, then the graph shifts b units down.

From (i) and (ii), we get

a=8\text{ and }b=0

Therefore, the graph of y=3x translated left by 8 units. Hence, the student is wrong.

8 0
2 years ago
Identify the cross section of the solid shown in the diagram
Ainat [17]

Answer:

B. Quadrilateral

Step-by-step explanation:

Just took the test

6 0
3 years ago
Sarah and Joe are seeking to join a gym. Sarah saw on television that Great Gym is offering membership at $19.50 per month, plus
xenn [34]
Sarah because if you add up the one time registration fee and the amount per month for one month it would be less than Joe's
8 0
3 years ago
Read 2 more answers
Other questions:
  • Daniel has 402 pieces in building set.He use 186 pieces to build a house.How many pieces does he have left?
    8·1 answer
  • With an 18% discount, Joan was able to save $13.23 for a coat. What was the original price of the coat?
    12·1 answer
  • What is 8.78 divided by 8
    15·2 answers
  • Solve for y in the equation below: -6x-2y=10​
    12·1 answer
  • PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!!
    9·1 answer
  • The graph below represents the distance of a dog from a trainer after a command is given.
    14·1 answer
  • Write the following quadratic expression in standard form: ( 2 x − 1 ) ( 3 x − 1 )
    9·1 answer
  • How do you slove 6x-=-24 ​
    10·2 answers
  • Classify each number below as a rational number or an irrational number.
    11·1 answer
  • ITZZZ EAZY PLZZZ<br> What is a system of equations?<br> What is a system of inequalities?
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!