What is the measure for x

By the chain rule,

[tex]\dfrac{dy}{dx} = \dfrac{dy}{dt} \cdot \dfrac{dt}{dx} = \dfrac{\frac{dy}{dt}}{\frac{dx}{dt}}[/tex]

So first differentiate both x and y with respect to t.

• dx/dt :

[tex]x = \dfrac{\sin^3(t)}{\sqrt{\cos(2t)}} = \sin^3(t) \left(\cos(2t)\right)^{-\frac12}[/tex]

By the product rule,

[tex]\dfrac{dx}{dt} = \dfrac{d(\sin^3(t))}{dt} \left(\cos(2t)\right)^{-\frac12} + \sin^3(t) \dfrac{d\left(\cos(2t)\right)^{-\frac12}}{dt}[/tex]

By the power and chain rules,

[tex]\dfrac{dx}{dt} = \dfrac{d(\sin(t))}{dt} 3\sin^2(t) \left(\cos(2t)\right)^{-\frac12} -\dfrac12 \sin^3(t) \left(\cos(2t)\right)^{-\frac32} \dfrac{d(\cos(2t))}{dt}[/tex]

[tex]\dfrac{dx}{dt} = 3\sin^2(t) \cos(t) \left(\cos(2t)\right)^{-\frac12} -\dfrac12 \sin^3(t) \left(\cos(2t)\right)^{-\frac32} (-2 \sin(2t))[/tex]

We can clean this up a bit and write

[tex]\dfrac{dx}{dt} = \dfrac{3\sin^2(t) \cos(t)}{\sqrt{\cos(2t)}} + \dfrac{\sin^3(t) \sin(2t)}{\sqrt{\cos^3(2t)}}[/tex]

[tex]\dfrac{dx}{dt} = \dfrac{\sin^2(t)}{\sqrt{\cos^3(2t)}} \left(3 \cos(t) \cos(2t) + \sin(t) \sin(2t)\right)[/tex]

• dy/dt : (I'll skip some steps here, the derivative is nearly the same)

[tex]y = \cos^3(t) \left(\cos(2t)\right)^{-\frac12}[/tex]

[tex]\dfrac{dy}{dt} = 3\cos^2(t) (-\sin(t)) \left(\cos(2t)\right)^{-\frac12} + \cos^3(t) \left(-\dfrac12\right) \left(\cos(2t)\right)^{-\frac32} (-2 \sin(2t))[/tex]

[tex]\dfrac{dy}{dt} = -\dfrac{3\cos^2(t) \sin(t)}{\sqrt{\cos(2t)}} + \dfrac{\cos^3(t) \sin(2t)}{\sqrt{\cos^3(2t)}}[/tex]

[tex]\dfrac{dy}{dt} = -\dfrac{\cos^2(t)}{\sqrt{\cos^3(2t)}} \left(3 \sin(t) \cos(2t) - \cos(t) \sin(2t)\right)[/tex]

Finally, we get dy/dx by dividing the two derivatives above:

[tex]\dfrac{dy}{dx} = \dfrac{-\frac{\cos^2(t)}{\sqrt{\cos^3(2t)}} \left(3 \sin(t) \cos(2t) - \cos(t) \sin(2t)\right)}{\frac{\sin^2(t)}{\sqrt{\cos^3(2t)}} \left(3 \cos(t) \cos(2t) + \sin(t) \sin(2t)\right)}[/tex]

[tex]\dfrac{dy}{dx} = -\cot^2(t) \dfrac{3\sin(t)\cos(2t) - \cos(t)\sin(2t)}{3\cos(t)\cos(2t) + \sin(t)\sin(2t)}[/tex]

and by applying some trig identities,

[tex]\dfrac{dy}{dx} = -\cot^2(t) \dfrac{\sin(3t) - 2\sin(t)}{\cos(3t) + 2\cos(t)}[/tex]

[tex]\boxed{\dfrac{dy}{dx} = -\dfrac{\cos(3t)}{\sin(t) (2\cos(2t) + 1)}}[/tex]

Answer:

[tex]$\frac{dy}{dx}=-\cot (3 t)$[/tex]

Very important note!

Both answer are correct, the thing is: the expression can be more simplified  as

[tex]\sin(t)(2\cos(2t)+1) = \sin(3t)[/tex]

that will lead the the answer I just gave.

Step-by-step explanation:

The question is looking for [tex]\dfrac{dy}{dx}[/tex]

We have

[tex]x = \dfrac{\sin^3 (t)}{\sqrt{\cos(2t)}} \quad \text{ and } \quad y = \dfrac{\cos^3 (t)}{\sqrt{\cos(2t)}}[/tex]

One approach would be calculating [tex]\dot{x}[/tex] and [tex]\dot{y}[/tex] considering

[tex]\dfrac{dy}{dx} = \dfrac{\frac{dy}{dt}}{\frac{dx}{dt}}[/tex]

We would have

[tex]$\frac{dx}{dt}=\frac{3 \sin ^2(t) \cos(t)}{\sqrt{\cos (2 t)}} + \frac{\sin (2 t) \sin ^3(t)}{\cos ^{3/2}(2 t)} $[/tex]

[tex]$= \frac{\sin^{2} t}{\cos ^{3/2}(2 t)}(3\cos (t) \cos(2t)+2\sin^{2} (t) \cos(t))$[/tex]

[tex]$=\boxed{ \frac{\sin ^2(t) (2 \cos (t)+\cos (3 t))}{\cos ^{3/2}(2 t)}}$[/tex]

===============================================

[tex]$\frac{dy}{dt}=\frac{\sin (2 t) \cos ^3(t)}{\cos ^{\frac{3}{2}}(2 t)}-\frac{3 \sin (t) \cos ^2(t)}{\sqrt{\cos (2 t)}}$[/tex]

[tex]$= \frac{\cos^{2} t}{(\cos ^{3/2}(2 t)} (-3\sin (t) \cos (2t)+2\cos^{2} (t) \sin (t))$[/tex]

[tex]$=\boxed{ \frac{\sin (t)-\sin (5 t)}{4 \cos ^{3/2}(2 t)}}$[/tex]

Finally,

[tex]$\frac{dy}{dx}= \frac{\sin (t)-\sin (5 t)}{4 \cos ^{3/2}(2 t)} \cdot \frac{\cos ^{3/2}(2t) }{ \sin ^2(t) (2 \cos (t)+\cos (3 t))}$[/tex]

[tex]$=\frac{\cos(t)(2\cos^{2} t-3(2\cos^{2} t-1))} {\sin(t)(2\sin^{2} t+3(1-2\sin^{2} t))} $[/tex]

[tex]$=-\frac{4\cos^{3} (t)-3\cos (t)}{3\sin (t)-4\sin^{3} (t)} $[/tex]

[tex]$=-\frac{\cos (3t)}{\sin (3t)}$[/tex]

[tex]=\boxed{-\cot (3t)}[/tex]

Answer:

yes, congruent means the same. both parallelograms are just rotated, their side lengths are the same. so yes, the parallelograms are congruent.

Answer:

Yes

Step-by-step explanation:

They are equal. They are just different in positions. Try to make the second parallelogram horizontal then you'll see.

Answer:

it depends on how old the child is but theoretically yes because what if the child was ten the parent would be very young but would be thirty.

Step-by-step explanation:

[tex]\large\underline{\sf{Explanation-}}[/tex]

Given that

HOPE is a parallelogram.

It means,

Now, HE || PO & HP is transversal.

Now, In triangle HOP

We know, Exterior angle of a triangle is equals to sum of interior opposite angles.

It means, 40° + z = 70°

[tex]\longmapsto[/tex]"z" = 70° - 40° = 30°.

Now, OH || FE & HE is transversal.

So, we know, sum of Adjacent angles of a parallelogram is supplementary.

[tex]\rm \longmapsto\:z + 40\degree + x = 180 \degree[/tex]

[tex]\rm \longmapsto\:30\degree + 40\degree + x = 180 \degree[/tex]

[tex]\rm \longmapsto\:70\degree + x = 180 \degree[/tex]

[tex]\rm \longmapsto\:x = 180 \degree - 70\degree[/tex]

[tex]\longmapsto\:\bf{"x"} = 110 \degree [/tex]

Therefore, the measure of the following angle are:

Answer: "A 180° rotation about its center" is the one transformation among the following choices given in the question that will always map a parallelogram onto itself. The correct option among all the options that are given in the question is the third option or the penultimate option.

Step-by-step explanation: I hope it helps you!

Answer:

870.1

Step-by-step explanation:

When you round it you get 870.1 but without rounding the answer is 970.056..

Please mark as brainliest

Answer:

0.55 least            3

3/5= .6 greater    2

2/3= .66​ greatest 1

Step-by-step explanation:

give me brainliest pls

Answer:

y=1/3(x-5)-3

graph is in attachment I added

Step-by-step explanation:

So you would subtract 3 from both sides which makes the equation look like this y=1/3(x-5)-3

And the graph is in the picture.

Answer:

B

Step-by-step explanation:

3 in the expression is part of -3x called the coefficient of the variable. See the definitions below to distinguish the difference between each term:

A.

variable  - an unknown value represented with a letter

B.  

coefficient  - a number written with a variable denoting how many

C.  

constant  - a number without a variable denoting how much

D.  

exponent - a number written on a base denoting how many times to multiply the base by itself

Answer:

hi can I be your best friend

9514 1404 393

Answer:

  y -2 = 3/2(x -1)

Step-by-step explanation:

The slope formula tells you the slope:

  m = (y2 -y1)/(x2 -x1)

  m = (8 -2)/(5 -1) = 6/4 = 3/2

Using the first point in the point-slope form ...

  y -k = m(x -h) . . . . . line through (h, k) with slope m

  y -2 = 3/2(x -1)

Answer:

so simple division he can lift 44kg

one pound is 2.2kg

well you know what to do you divide the number to get the amount he can lift

so 44/2.2 and we get 20 pounds he can lift

Answer:

-2b

Step-by-step explanation:

since

|x| = x if x>0

and

|x| = -x if x<0

then it is -2b

The given modulus function at (b < 0) is written as (f(b) = -2b) and this can be determined by using the properties of the modulus function.

Given :

Expression  --  |2b| where b < 0

The following steps can be used in order to rewrite the given expression:

Step 1 - The generalized modulus function is given by:

f(x) = |x|

Step 2 - When x > 0 the above function becomes:

f(x) = x

Step 3 - When x < 0 the above function becomes:

f(x) = -x

Step 4 - So, the given expression is also a modulus function. The given modulus function at (b < 0) is written as:

f(b) = -2b

For more information, refer to the link given below:

https://brainly.com/question/14070556

Answer: 1 / 52

Step-by-step explanation: Relating to what given card at the moment.

Answer:

[tex]f(x)=\frac{7}{2}x+10[/tex]

Step-by-step explanation:

These relations can be rewritten as ordered pairs.

Insert these ordered pairs into the slope formula to find the slope.

[tex]m=\frac{y_2-y_1}{x_2-x_1}\\\\m=\frac{-11-3}{-6-(-2)}\\\\m=\frac{-14}{-4}\\\\m=\frac{7}{2}[/tex]

Use the information found and insert it into the point-slope formula.

[tex]y-y_1=m(x-x_1)\\\\y-3=\frac{7}{2}(x-(-2))\\\\y-3=\frac{7}{2}x+7\\\\y=\frac{7}{2}x+10[/tex]

Answer:

(1.5,-5)

Step-by-step explanation:

Answer:

yes it is a triangle.this is because three vertices or point will form a triangle

Answer:

-2

Step-by-step explanation:

If you want explanation lmk

Hi there!

The answer is: 49

Because if we square a negative number, we get a positive number.

But there's a difference between (-7)² and -(7)².

(-7)² equals 49

-(7)²=-49

Hope this helps you!

~Just a felicitous girlie

#HaveAnAmazingDay

[tex]SilentNature[/tex]

yes, because there would be an abnormality showing up in the mri scan

Answer:

5845.67147554

Step-by-step explanation:

(45)(150)cos(30degrees)

The work done by the lawn mower is 3375 ft-lb.

The work done is equal to the force applied times the distance moved, but only in the direction of the force. In this case, the force is not horizontal, so we need to find the component of the force that is parallel to the ground.

The force applied to the lawn mower is 45 lb, and the angle between the force and the horizontal is 30°. We can use the sine function to find the component of the force that is parallel to the ground:

[tex]F_{parallel[/tex] = F * sin(theta) = 45 lb * sin(30°) = 22.5 lb

The work done is then:

W = [tex]F_{parallel[/tex] * d = 22.5 lb * 150 ft = 3375 ft-lb

Therefore, the work done is 3375 ft-lb.

Here is the solution in mathematical notation:

Work = [tex]F_{parallel[/tex] * d = F * sin(theta) * d = 45 lb * sin(30°) * 150 ft = 3375 ft-lb

To learn more about work done here:

https://brainly.com/question/2750803

#SPJ2

(C)

Step-by-step explanation:

The first condition states that the graph has 5 real zeros. Based on the choices, only (b) and (c) satisfy this condition because the other two choices contain 6 real zeros.

Now the second condition states that its end behavior approaches negative infinity as x approaches positive infinity. As you can see, graph (b) approaches positive infinity as x approaches positive infinity. This is clearly NOT the correct answer, which leaves us with (c) because we can see that it approaches negative infinity as x approaches positive infinity.

Answer:

1.2 g/cm^3

Step-by-step explanation:

Assuming the volume of the solid has the units [tex]cm^{3}[/tex], density is calculated by dividing mass by volume.

[tex]density=\frac{180}{150}=1.2 g/cm^{3}[/tex]

Answer:

42 cups

Step-by-step explanation:

Answer:

What are the questions?

Step-by-step explanation:

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