Find the equation (in terms of xx and yy) of the tangent line to the curve r=3sin2θr=3sin⁡2θ at θ=π/6θ=π/6

Answers

Answer 1

the slope of the tangent line to a curve is the first derivative, dy/dx, of the curve at the point of tangency.

In this problem the point of tangency has theta = pi/6, so the corresponding r = 5 sin[(3)(pi/6)] = 5.
Converting these (r, theta) coordinates to (x, y) coordinates you use the polar to cartesian coordinate conversion eqns.:
x = r cos(theta) and y = r sin(theta).
So x = 5 cos(pi/6) = 4.33; y = 5 sin(pi/6) = 2.5
So the point of tangency where the tangent line intersects the curve is (x,y) = (4.33, 2,5)

Using the formula from the attached website, which I assume your teacher derived in class:
dy/dx = [[dr/d(theta)] sin(theta) + r cos(theta)] / [[dr/d(theta)] cos(theta) - r sin(theta)]
From r = 5sin(3 theta) , dr/d(theta) = 15 cos(3 theta)
dy/dx = [ [15 cos(3 theta)] sin(theta) + r cos(theta)] / [ [15 cos(3 theta)] cos(theta) - r sin(theta)]
dy/dx evaluated at theta = pi/6 and r = 5 is:
dy/dx = [ [15 cos(3 pi/6] sin(pi/6) + 5 cos(pi/6)] / [ [15 cos(3 pi/6)] cos(pi/6) - 5 sin(pi/6)]
dy/dx = - cot(pi/6) = -1.732

So we have the tangent line of the form y = (-1.732)x + b where the point (x,y) = (4.33, 2.5) is on the line.
b = 2.5 + (1.732)(4.33) = 10

So the tangent line is y = -1.732 x + 10

Related Questions

Find the difference.

5wx2 − 2wx2

Answers

5wx2 − 2wx2
=3wx2

hope it helps

Which inequality describes all the solutions to 5(3 - x) < -2x + 6?

A) x < -9
B) x > 3
C) x > 7

Answers

The inequality describes the solution to 5(3 - x) < -2x + 6 will be x > 3. The correct option is B.

What is inequality?

Inequality is the relationship between two expressions that are not equal, employing a sign such as ≠ “not equal to,” > “greater than,” or < “less than.”.

The given expression 5(3 - x) < -2x + 6 will be solved as below:-

5(3 - x) < -2x + 6

15 - 5x < -2x + 6

-5x + 2x < -15 + 6

-3x < -9

x > 3

Therefore, the inequality describes the solution to 5(3 - x) < -2x + 6 will be x > 3. The correct option is B.

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If 5(x+7)=-3, then 5x+35=-3

Answers

5(x+7)=-3
1st step: Distributive Property
5x+35=-3
So, yes this is true.

A regular hexagon is shown.

What is the length of the apothem, rounded to the nearest inch? Recall that in a regular hexagon, the length of the radius is equal to the length of each side of the hexagon.
4 in.
5 in.
9 in.
11 in.

Answers

Since the triangle will be a 30-60-90, because 360°/6 = 60° for each central angle, then that is bisected to find the height of each of the 6 equilateral triangles. This height is the same as the apothem.
In a 30-60-90 triangle, the sides are is the ratio of:
[tex]x \: \\ x \sqrt{3} \\ and \: 2x[/tex]
where x is opposite the 30° and 2x is opposite the 90°
Since the base is 10 and the hypotenuse is 10, 1/2 that is 5 (the x), and so:
[tex]x \sqrt{3} = 5 \sqrt{3} = 8.66[/tex]
So rounded up, you apothem then is closest to D) 9 in.

The length of the apothem of a regular polygon that has a radius of 10 inches would be 9 inches .

What is the apothem of a regular polygon?

The apothem of a regular polygon is defined as a line segment from the center to the midpoint of one of its sides.

It is Given that the hexagon has a radius of 10 inches and the length of the radius is equal to the length of each side of the hexagon.

The apothem AB divides the side of the hexagon into two equal parts of 5 Inches.

Using Pythagoras Theorem

[tex]10^2 = 5^2 + AB^2\\AB^2 = 10^2 -5^2\\AB^2 = 100 - 25\\AB^2 = 75\\AB = \sqrt{75}[/tex]

AB = 8.66

So, rounded up, we get the apothem closest to D that is 9 in.

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What do they call the guy who invented the first blue jeans

Answers

Answer:

Levi Strauss!

Step-by-step explanation:

By the 1860s, Levi Strauss's blue pants were daily wear for miners and farmers and cattlemen throughout the West. In 1873 he bought, for $69—the price of the patent application — an idea from a Russian immigrant tailor in Reno for making miner's pants stronger by riveting the critical seams.

Hope this helped!

On which number line do the points represent negative 4 and 1 over 2 and +3?

Answers

A because if you look t is on +3 and r is on -41/2

Suppose you divide a 52 card deck into its four suits, and draw one card from each suit. what is the probability that you draw two red face cards and the two black aces?

Answers

Diamonds(red) 3 face cards out of 13
clover - only 1 black ace
hearts(red) 3 face cards out of 13
spade - only 1 black ace

3/13 * 3/13 *1/13 *1/13 = 9/28561

please help. !!!!!!!!

Answers

Basically you find the data points and fill in the values. For Point J it is (2,4). They want you to do this: (2-4,4-6)...it then becomes (-2,-2).
So
J: (-2,-2)
K: (-1, -4)
I: (-2, -5)
Those are your new points. I hope this helps love! :)

Solve 12g=12(2/3g-1)+11

Answers

Solve for g:12 g = 12 ((2 g)/3 - 1) + 11
Put each term in (2 g)/3 - 1 over the common denominator 3: (2 g)/3 - 1 = (2 g)/3 - (3)/3:12 g = 12 (2 g)/3 - (3)/3 + 11
(2 g)/3 - (3)/3 = (2 g - 3)/3:12 g = 12 (2 g - 3)/3 + 11
12/3 = (3×4)/3 = 4:12 g = 4 (2 g - 3) + 11
4 (2 g - 3) = 8 g - 12:12 g = 8 g - 12 + 11
Grouping like terms, 8 g - 12 + 11 = 8 g + (11 - 12):12 g = 8 g + (11 - 12)
11 - 12 = -1:12 g = 8 g + -1
Subtract 8 g from both sides:12 g - 8 g = (8 g - 8 g) - 1
12 g - 8 g = 4 g:4 g = (8 g - 8 g) - 1
8 g - 8 g = 0:4 g = -1
Divide both sides of 4 g = -1 by 4:(4 g)/4 = (-1)/4
4/4 = 1:Answer: g = (-1)/4

The sum of two numbers is 35 . the smaller number is 19 less than the larger number. what are the numbers?

Answers

let the smaller number be x and the larger be (x+19)
x+x+19=35
2x=16
x=8
x+19=27

the numbers are 8 and 27.

Which of these ratios is NOT in simplest form? 12/5 8/5 7/10 9/15

Answers

9/15, it can be simplified to 3/5

9/15
9 / 3 = 3
15 / 3 = 5

F. determine if (x + 3) is a factor of (x) = 2x3 + x2 –8x + 21 by using synthetic division. if so, find the other factors.

Answers

since its x + 3 we use -3

-3| 2 1 -8 21 (these are the # without the x)
___________________

-6 15 -21
-------------------------------
2 -5 7 0

Drag the 2 down, multiple it by -3 to get -6
1+(-6) = -5. -5 * -3 =15 . 15+(-8) = 7
7*-3 = -21. -21 +21 = 0
Since last # is 0 we have a factor.

factor is 2x^2 -5x -7 (we start with a power 1 less the original.)

Question 25 the sum of three numbers is 79 . the second number is 9 less than the first. the third number is 2 times the first. what are the numbers?

Answers

I believe the numbers are 22, 13, and 44.

He length of a rectangle is six times its width. if the perimeter of the rectangle is 70 in , find its area.

Answers

The width is 150 inches squared

Help please
Solve for r.

s=2πrh

Answers

s/2πh=r

hope this helps u

Here it is
I hope that will help you

A stop sign is a regular octagon formed by cutting triangles off the corners of a square. if we have a square with sides of length square root 2, what is the side length of the resulting octagon

Answers

the answer is 3.5147 inches

Final answer:

The side length of the resulting octagon is 2√2.

Explanation:

A stop sign is a regular octagon formed by cutting triangles off the corners of a square. To find the side length of the resulting octagon, we can use the Pythagorean theorem. Since the square has sides of length √2, its diagonal can be found by multiplying its side length by √2. This gives us a diagonal length of 2√2. The diagonal of each triangle cut off the corners of the square will be the side length of the resulting octagon. So, the side length of the octagon is 2√2.

Nathan and Steven stacked boxes on a shelf. Nathan lifted 13 boxes and Steven lifted 15 boxes. The boxes that Nathan lifted each weighed 12 lb more than the boxes Steven lifted.

Let s represent the weight of the boxes that Steven lifted.

Which expression represents the total number of pounds Nathan lifted?

s + 12

12(s+13)

15(s+12)

13(s+12)

Answers

13(s+12) is your answer
13 boxes that each weigh 12 more pounds than steven's boxes.

Answer:

13(s+12) is your answer

Step-by-step explanation:

Which of the following equations represents the perpendicular bisector of WX graphed below?

Thank you! :)

Answers

The coordinates of the 2 given points are W(-5, 2), and X(5, -4).

First, we find the midpoint M using the midpoint formula:

[tex]\displaystyle{ M_{WX}= (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} )= (\frac{-5+5}{2}, \frac{2+(-4)}{2} )=(0, -1).[/tex]

Nex, we find the slope of the line containing M, perpendicular to WX. We know that if m and n are the slopes of 2 parallel lines, then mn=-1.

The slope of WX is [tex]\displaystyle{ m= \frac{y_2-y_1}{x_2-x_1}= \frac{2-(-4)}{-5-5}= \frac{6}{-10}= -\frac{3}{5} [/tex].

Thus, the slope n of the perpendicular line is [tex]\displaystyle{ \frac{5}{3} [/tex].

The equation of the line with slope [tex]\displaystyle{ n= \frac{5}{3} [/tex] containing the point M(0, -1) is given by:

[tex]\displaystyle{ y-(-1)=\frac{5}{3}(x-0)[/tex]

[tex]\displaystyle{ y+1= \frac{5}{3}x[/tex]

[tex]\displaystyle{ 3y+3=5x[/tex]

[tex]\displaystyle{ 5x-3y-3=0 [/tex]

Answer: 5x-3y-3=0

A can of paint is 15 centimeters high and has a ddiameter of 13.6 cm. what is the volue of the can? Round to the nearest tenth.

Answers

Hello! A can is the shape of a cylinder. The formula for the volume of a cylinder is this: pi * r^2 * h. The radius is 1/2 of the diameter. Half of 13.6 is 6.8. 6.8^2 is 46.24. 46.24 * 15 is 693.6. Now, we can multiply by pi as 3.14. 693.6 * 3.14 is 2,177.904 or 2,177.9 when rounded to the nearest tenth. The volume of the cam is approximately 2,177.9 cubed centimeters.

Answer:

V=πr2h=π·6.8^2·15≈2179.00866

2179 cm^3

Step-by-step explanation:

Can you help me solve this proof please?

Answers

Okay, so I haven't done proofs in a bit, so when it comes to proving that lines are congruent (or angles), you can try to prove that the triangles are congruent (CPCTC- Corresponding Parts of Congruent Triangles are Congruent). To prove that triangles are congruent, use SSS (Side side side- meaning all sides are congruent), ASA (angle side angle- meaning corresponding angle, then side, then angle are congruent), SSA (2 corresponding sides are congruent and an angle is also congruent) or SAS (you get it). We should use ASA, since obviously it says that ABD and CBD are congruent and obviously BD and BD are congruent and ADB aND CDB are congruent (both are 90degrees). So, we have proven that these traingles are congruent. Now use CPCTC as the last step, since AD and CD are corresponding parts of the triangles.

When constructing parallel and perpendicular lines, how are the steps similar

Answers

The construction of parallel lines includes copying an angle by creating arcs with a compass.
When constructing parallel and perpendicular lines the steps are similar because they both include creating and connecting intersecting arcs.

The coordinates of the vertices of quadrilateral JKLM are J(-3,2), K(4,-1), L(2,-5) and M(-5,-2).

Find the slope of each side of the quadrilateral and determine if the quadrilateral is a parallelogram.

Answers

Slope of JK=((-1)-2)/(4-(-3)=-3/7 Slope of KL=((-5)-(-1)/(2-4=2 Slope of LM=((-2)-(-5))/(-5-2)=-3/7 Slope of MJ=(2-(-2))/((-3)-(-5))= 2 JK is parallel to LM and KL is parallel to MJ. Therefore JKLM is a parallelogram.

A one celled organism measures 32 millimeters in a length in a photograph. If the photo has been enlarged by a factor of 100, what is the actual length of the organism? Show your work

Answers

Final answer:

The actual length of the organism is 0.32 millimeters, which is obtained by dividing the photograph length (32 mm) by the magnification factor (100).

Explanation:

To determine the actual length of the one-celled organism when the photograph has been enlarged by a factor of 100, we use the formula:

Actual length = Photograph length ÷ Magnification factor

Substituting the given values:

Actual length = 32 mm ÷ 100

Actual length = 0.32 mm

This calculation shows that the actual length of the organism is 0.32 millimeters, which is equivalent to 320 micrometers (320 µm). It's interesting to note that this is well within the size range for many microorganisms, which may require magnification to be seen clearly, such as with a microscope.

As a student you are able to earn extra money by assisting your neighbors with odd jobs. If you charged $10.25 an hour for your assistance about how many hours would you need to work to earn $8,425

Answers

8425/10.25
821.9
It would take about 822 hours

For an hour of assistance if a student charged $10.25 than total number hours a student need to work to earn $8,425 is 821.95 hours and this can be determine by the help of unitary method.

Given :

Charge for 1 hour assistance = $10.25

Total earning through assistance = $8425

In this problem unitary method can be use to determine the total number of hours need to work to earn $8425. In unitary method, single unit value is determine first to evaluate the necessary value by multiplying the single unit value.

For an hour of assistance if a student charged $10.25 than total number hours a student need to work to earn $8,425 can be:

[tex]= \dfrac{8425}{10.25} \times 1[/tex]

= 821.95 hours

So a student need to work a total of 821.95 hours to earn $8425.

For more information, refer the link given below:

https://brainly.com/question/4725220

In isosceles triangle the length of the base is twice shorter than the length of a leg. The perimeter of this triangle is 50cm. Find the lengths of all sides.

Answers

Perimeter is sum of all 3 sides. Isosceles triangle has two same sides and one short side.
If the short side is X, the other two sides would be 2X and 2X (since the short side is twice shorter)
P = x + 2x + 2x
50 = 5X
10 = X
Short side is 10, other two sides are 2(10) or 20.

find the difference without using a calculator 201-72.35

Answers

the answer is 128.65

Here is your answer:

Since its decimal ma!e your equation look like this:

201.00
72.35-
Its going to make 99 because its zeros but add the 1
10-5=5
Then make another 9:
And another use the 2
10-3=7
10-2=8

Which will equals:

201-72.35=128.65
(Since there's 100. Decimal)

To make punch, Sophia mixes 2 liters of clear soda with 70 centiliters of cranberry juice and 236 milliliters of pineapple juice. How much punch does she have in total?

Answers

2.936L
0.236 + 0.7= 0.936

The length of each side of a square prism is ten ft

Answers

What is the question?

HOW many times can times 6 go into 59

Answers

6 can go into 59, nine times. Therefore 9 times 6 equals 54

All you have to do is divide 59/6, you get 9.83 hopefully this helps

is it possible to draw three lines in two planes such that all three lines are skew?

Answers

Answer with explanation:

→Two lines in three dimensional plane are said to be Skew if they are neither parallel nor Intersecting.

→Skew lines can be drawn in three dimensional plane only.

→Drawn two skew lines in a Plane , and then drawn another line in a plane and found that , they are skew.

⇒Yes, it is possible to draw three lines in two planes such that all three lines are skew.

Skew lines are non-coplanar lines that are not parallel, and they do not intersect.

In a 2D plane, it is not possible to draw three skew linesIn a 3D plane, it is possible to draw three skew lines

The question does not state the type of plane. So, I will answer this question in 2 ways.

(a) 2D plane

Lines in 2D planes can only be drawn in a flat direction.

This means that, when three lines are drawn in the two planes, one of the following would happen.

The three lines would intersectTwo lines would intersect

See attachment for illustration

So, 3 lines in a 2D plane cannot be skewed lines

(b) 3D plane

Lines in 3D planes can only be drawn in several directions.

The above statement imply that:

Several lines can be drawn where none of the lines would be parallel and none of the lines would intersect.

This means that, when three lines are drawn in the two planes, one of the following would happen.

The three lines would intersectTwo lines would intersectNone of the lines would intersect

The third highlight means that:

It is possible to have skew lines in a 3D plane