Find the values of θ at which there are horizontal tangent lines on the graph of r = 1 + cos θ.

 Find The Values Of At Which There Are Horizontal Tangent Lines On The Graph Of R = 1 + Cos .

Answers

Answer 1

Answer:

θ = π/3 and 5π/3

Step-by-step explanation:

x = r cos θ and y = r sin θ.  Therefore:

dy/dx = (dy/dθ) / (dx/dθ)

dy/dx = (r cos θ + r' sin θ) / (-r sin θ + r' cos θ)

In this case, r = 1 + cos θ and r' = -sin θ.

dy/dx = ((1 + cos θ) cos θ + (-sin θ) sin θ) / (-(1 + cos θ) sin θ + (-sin θ) cos θ)

dy/dx = (cos θ + cos²θ − sin²θ) / (-sin θ − sin θ cos θ − sin θ cos θ)

dy/dx = (cos θ + cos²θ − (1 − cos²θ)) / (-sin θ (1 + 2 cos θ))

dy/dx = (cos θ + 2 cos²θ − 1) / (-sin θ (1 + 2 cos θ))

The tangent line is horizontal when the numerator of dy/dx is 0 and the denominator of dy/dx is not 0.

0 = 2 cos²θ + cos θ − 1

0 = (cos θ + 1) (2 cos θ − 1)

cos θ = -1 or 1/2

θ = π/3, π, 5π/3

0 = -sin θ (1 + 2 cos θ)

sin θ = 0 or cos θ = -1/2

θ = 0, 2π/3, π, 4π/3

Therefore, the answer is θ = π/3 and 5π/3.


Related Questions

A function g(x) has x-intercepts at (5,0) and (6, 0). Which could be g(x)?
g(x) = 2(x + 1)(x + 6)
g(x) = (x + 6) (2x - 1)
g(x) = 2(x-2)(x-6)
g(x) = (x + 6) (x + 2)​

Answers

Answer:

g(x) = a(x-5) ( x-6)

Step-by-step explanation:

A function g(x) has x-intercepts at (5,0) and (6, 0)

g(x) = a(x-5) ( x-6)

If you mistyped and the x intercepts are at 2 and 6

g(x) = 2(x-2)(x-6)

Solve the following inequality for hh. Write your answer in simplest form.
h-8>4h+5

Answers

Answer:

h<-13/3

Steps:

Move variable to the left side and change its sign

h-4h-8>5

Move constant to right and change its sign

h-4h>5+8

Collect like terms

-3h>5+8

Add the numbers

-3h>13

Divide both sides of the inequality by -3 and flip the inequality sign

h<-13/3

If I have 21 cone shaped hats with a diameter of 16in and height of 9in. How much paper do I need to cover the hats?

Answers

Answer:

10584 inches of paper

Step-by-step explanation:

to find the surface area of a cone the formula you use is A= πr(r+h2+r2)  *you need to divide 16 by two to find the radius 16 is the diameter*

and this subsituted is π8(8+9(2)+8(2)) and this equals 504 inches and you then multiply by 21 to get enough paper to cover all 21 cones and that equals 10584 inches

Mark wanted to find out which types of movies students in his grade at school preferred. He divided the list of student names in his grade into boys and girls and then from each group, drew names from a hat so that the number in each group was proportional to the total number of boys and girls in his grade. Will this sample represent the population, and if so, which type of sampling method did Mark use? A. Yes, simple random sampling. B. Yes, systematic random sampling. C. Yes, stratified random sampling. D. No, this was not a random sample.
NOT"C"

Answers

Answer:

the answer is not b

Step-by-step explanation:

The random sampling used is stratified random sampling

What is simple random sampling ( SRS )?

The most accurate type of probability sampling is random sampling. Each person in a population has an equal probability of getting selected for the sample thanks to this method of sampling.

This kind of sampling is perfect for highly controlled investigations where it is unacceptable to have human bias in the selection of the sample.

Each sampling unit in a population has an equal probability of being included in the sample when using simple random sampling (SRS). As a result, each potential sample has an equal probability of being chosen.

Given data ,

Mark used a stratified random sampling method by dividing the list of students into boys and girls and drawing from each group in proportion to their numbers in the population. This method helps ensure that the sample is representative of the population in terms of gender.

Hence , the sample is likely to represent the population. Answer: C. Yes, stratified random sampling.

To learn more about simple random sampling click :

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cha cha cha reallllllllllllll smooth what the answer?

Answers

Answer:

d=12

Step-by-step explanation:

7d-28=56

Add 28 to both sides

7d=84

Divide both sides by 7

d=12

Cha Cha Real Smooth indeed! :3

Answer:

d=12

Step-by-step explanation:

first step is to add 28 to both sides and get 7d=84

then divide 84 by 7

d=12

good luck :D

find the value of c that creates a perfect square trinomial

Answers

Answer:

C equals 64

Step-by-step explanation:

To find the missing value you have to take B divide it by two and then square it. So 16/2 ----> 8 ----->  8 squared =64. Hope this helped

A cylinder has a height of 14 inches and a radius of 5 inches. What is its volume? Use ​ ≈ 3.14 and round your answer to the nearest hundredth.

Answers

Answer:

1099 inches

Step-by-step explanation:

We know that formula for volume of a cylinder is:

[tex]V=\pi r^2h[/tex]

[tex]V=\pi 5^2*14[/tex]

[tex]V=\pi 25*14[/tex]

[tex]V=\pi 350[/tex]

[tex]V=3.14*350[/tex]

[tex]V=1099[/tex]

Answer:

1099

Step-by-step explanation:

1/4 (8x - 20) - 16 = 10x + 51

Answers

Answer:

The answer is -9 (If your looking for x)

Step-by-step explanation: I would suggest using the app Photomap for this kind of stuff. You take a picture of an equation and it gives you a step by step explanation and the answer. Glad I could help!


If the length of a rectangular television screen is 20 inches and its height is 15 inches, what is the length of its
diagonal

Answers

Answer:

25 inches

Step-by-step explanation:

Using the Pythagorean theorem;

20^2 + 15^2 = C^2

C= 25

Answer:

its 25

Step-by-step explanation:

there right

Amy has 2 crayons. Cameron gives her 5 more crayons. How many crayons does Amy have now?

Answers

Answer:

Amy now has 7 crayons.

Step-by-step explanation:

2 crayons + 5 crayons = 7 crayons.

Answer:

Amy has 7 crayons now

Step-by-step explanation:

2+5=7

2) On average, 75% of the planes leave on time. If 120 planes left on time
yesterday, how many total planes were there?​

Answers

Answer:

Step-by-step explanation:160

Use your ruler to draw a rectangle that measures
by 4 1/2 by 2 3/4 inches, and find its area.

Answers

Answer:

Area of rectangle, [tex]A=\dfrac{99}{8}\ \text{inch}^2[/tex]

Step-by-step explanation:

We have,

A rectangle that measures [tex]4\dfrac{1}{2}\times 2\dfrac{3}{4}[/tex].

It is required to find the area of the rectangle

The product of length and breadth of a rectangle is called its area. It can be calculated as :

[tex]A=4\dfrac{1}{2}\times 2\dfrac{3}{4}\\\\A=\dfrac{9}{2}\times \dfrac{11}{4}\\\\A=\dfrac{99}{8}\ \text{inch}^2[/tex]

So, the area of the rectangle is [tex]\dfrac{99}{8}\ \text{inch}^2[/tex].

See the question above.
A) 96 ft2
B) 24 ft2
C) 48 ft2
D) 38 ft2

Answers

Answer:

C. 48 [tex]ft^2[/tex]

Step-by-step explanation:

The formula for finding the area of a triangle is:

[tex]\frac{1}{2}bh[/tex]

The "[tex]b[/tex]," is the base (10), and the "[tex]h[/tex]," is the height (9.6), so plug it in:

[tex]\frac{1}{2} (10)(9.6)\\[/tex]

[tex]\frac{1}{2} (96)[/tex]

48 [tex]ft^2[/tex]

I need help immediately with middle school

Answers

Answer:

5*7=35 square units

Step-by-step explanation:

area of parrelogram =B×h

plzz mark me brainliest

Which set of angles can form a triangle?

A: 2 acute and 1 right

B: 1 acute and 2 obtuse

C: 1 acute and 2 right

D: 1 acute, 1 right and 1 obtuse

Answers

Answer:

A: 2 acute and 1 right

Step-by-step explanation:

A triangle is a polygon that has 3 sides as well as three angles

Triangles can be can be classified based on their three sides as well as their three angles.

Based on their sides, we have 3 types of triangles.

1. Equilateral triangle: This means they have 3 equal sides.

2. Isoceles triangle: They have only 2 equal sides.

3. Scalene triangle: The 3 sides are not equal

Based on their angles, we have 3 triangles.

a. Acute triangle

b. Obtuse triangle

c. Right angle triangle.

One of the way by which we can form a triangle, is by forming it with 2 acute angles and 1 right angle.

Answer:

The correct option is;

A: 2 acute and 1 right

Step-by-step explanation:

Here we have the definitions of the angles as follows;

The sum of angles in a triangle = 180°

An acute angle is an angle less than 90°

An obtuse angle is an angle more 90 ° but less 180°

A right angle triangle is an angle equal to 90°

∴ A right angle triangle is an angle = 90°

Therefore, we compare each of the options

A: 2 acute and 1 right = <90 +<90 +90 < 180 =180 or >180 Possible

B: 1 acute and 2 obtuse = 1<90 + > 90 +>90 > 180°

C: 1 acute and 2 right = <90 + 90 + 90 > 180

D: 1 acute + 1 right + 1 obtuse = <90 + 90 + >90 > 180

Therefore, the only possible correct option is A.

Use the graph to find the relationship between variable

Answers

Where is the picture at?

Jasmine and her brother Nat are saving to buy a digital camera that costs $135. Jasmine earns $15 per week for babysitting and spends $5 of it. Nat earns $18 per week for walking dogs and spends $7 of it. Which expression can be used to find how many weeks it will take to save for the digital camera?

Answers

Answer:

7 weeks

Step-by-step explanation:

Jasmine=$10

Nat=$11

10+11=$22 each week

135/22=6.14

at least 7 weeks



Find the area of the composite figure.

a.) 364 sq ft
b.) 796 sq ft
c.) 800 sq ft
d.) 980 sq ft​

Answers

Answer:

B. 796 sq ft

Step-by-step explanation:

All of these shapes are rectangles.

The formula for the area of a triangle is length x width.

Dining Room

26 x 14 = 364

Bedroom

14 x 8 = 112

Enclosed Porch

20 x 16 = 320

Then you add all of the products.

364 + 112 = 476

476 + 320 = 796

15 POINTS! will be selecting Brainliest!

How do I do this? Please give step-by-step answers + "work"- will be selecting Brainliest!

Answers

Just divide the bottom numbers then when you figure that out you multiply the top number by the number you divided at the bottom

This is the last one!!!

Answers

Answer:

B

Step-by-step explanation:

the mean goes up more than the median

How much water will it take to fill the aquarium?
Please anawer this question. Will put as brainliest if i get an answer

Answers

Answer:

4096

Step-by-step explanation:

You multiply 32 in. by 8 in. by 8 in.

Answer:

4096

Step-by-step explanation:

to find volume you can just multiply length, width, and height together :) Hope this helps!!! ❤

a system of equations is shown.

Answers

Answer:

x=4 , y=1

Step-by-step explanation:

y = 2x-7

y = -x +5

Set them equal to each  other

2x-7 = -x+5

add x to each side

2x+x-7 = -x+x+5

3x-7 = 5

Add 7 to each side

3x-7+7 = 5+7

3x = 12

Divide by 3

3x/3 = 12/3

x=4

Now find y

y = -x+5

y = -4 +5

y = 1

Mabel is the costume manager for a large theater company. She has 20 different hats,
including 4 pork pie hats.
What is the probability that a randomly chosen hat from Mabel's inventory will be a pork pie
hat?
Write your answer as a fraction or whole number.
P(pork pie hat) =
Submit​

Answers

Answer:

4/20 or 1/5 being simplified

Step-by-step explanation:

What is the solution to
2x-y=1
Зу + 3 = 6x
Using system of equations

Answers

Answer:

infinitely many solutions

Step-by-step explanation:

Reduce the second equation by dividing all three terms by 3:  y + 1 = 2x.  Then compare this result to the first equation 2x - y = 1.  Noe that these two equations are identical.  Thus, the two lines coincide.  There are "infinitely many solutions."

Find the sum: 4√3+11√12
15√15
15√3
26√3
48√3

Answers

Answer:

26√3

Step-by-step explanation:

4√3+11√12

We know that sqrt(ab) = sqrt(a) sqrt(b)

4√3+11[tex]\sqrt{4} \sqrt{3}[/tex]

4√3+11*[tex]2 \sqrt{3}[/tex]

4√3+22 [tex]\sqrt{3}[/tex]

Combine like terms

26√3

Answer:

[tex]26\sqrt{3}[/tex]

Step-by-step explanation:

Simplify them:

[tex]11\sqrt{12} = 22\sqrt{3}[/tex]

[tex]22\sqrt{3}+4\sqrt{3}=26\sqrt{3}[/tex]

25POINTS!!!

In ΔFGH, h = 7.1 cm, f = 6.7 cm and ∠G=103°. Find the length of g, to the nearest 10th of a centimeter.

Answers

Answer:

10.8 cm

Step-by-step explanation:

Using cosine law

g² = 7.1² + 6.7² - 2(7.1)(6.7)cos(103)

g² = 116.7018433

g = 10.80286274

Answer:

10.8

Step-by-step explanation:

deltamath

You and two friends go to eat pizza there are two deals 10% off of 1 pizza or 30% off of 3 pizzas. Do you want to each pay separately and get 10% off of your bill or do you want to spl it the bill 3 wa

Answers

Answer:

Step-by-step explanation:

Assuming the price of 1 pizza is $x. Total number of pizzas is 3.

Considering the first deal,

10% off your pizza is 10/100 × x = 0.1x. The amount that you will pay is x - 0.1x = 0.9x

The cost of the remaining two pizzas is 2 × x = 2x

The total amount to be paid is

2x + 0.9x = $2.9x

Considering the second deal, 30% off the bill would be

30/100 × 3x = 0.9x

The amount to be paid is

3x - 0.9x = $2.1x

Therefore, I prefer the second deal

You are jumping off the 12-foot diving board at the municipal pool. You bounce up at 6 feet per second and drop to the water. Your height (h) in feet above the water in terms of (t) seconds is given by h = -16t^2 + 6t + 12. What is the first step in determining the maximum height that you reach during your dive?

Answers

Answer:

h_max = 12.56 units

(the exercise does no specify the units)

Step-by-step explanation:

You have the following function:

[tex]h(t)=-16t^2+6t+12[/tex]    (1)

To calculate the maximum height reach by you, you first derivative the function h(t) respect to t:

[tex]\frac{dh(t)}{dt}=-32t+6[/tex]

Next, you equal the result of the derivative to zero and you obtain t:

[tex]-32t+6=0\\\\t=\frac{6}{32}=\frac{3}{16}[/tex]

Then, you replace this value of t in the equation (1) and you obtain the maximum height:

[tex]h(t)=-16(\frac{3}{16})^2+6(\frac{3}{16})+12[/tex]

[tex]h(t)=12.56[/tex]

hence, the maximum height is 12.56 units

What is the distance between (-3, 4) and (-3, 10)?

Answers

Answer:

6 is the the distance between the points

The required distance between coordinates (-3, 4) and (-3, 10) is 6 units.

Given that,
The distance between the coordinates (-3, 4) and (-3, 10) is to be determined.

What is coordinate?

Coordinate, is represented as the values on x - axis and y - axis of the graph. where the values of the x-coordinate are called abscissa and the values of y coordinate are called ordinate.

Let the distance between two points be C.
Tio determined the distance between two points [tex](x_1, y_1)[/tex] and ([tex]x_2,y_2[/tex]) is given by the distance formula
C = √(x_1-x_2)^2(y_1-y_2)^2}

Putting the given coordinate in the above formula.

C = (-3+3)^2+(4 -10)^2}
C = {(0)^2+(6)^2}
C = √36
C = 6

The required distance between points is 6 units.

Thus, the required distance between coordinates (-3, 4) and (-3, 10) is 6 units.

Learn more about coordinate here:

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Find the distance between point a and point b

Answers

Answer:

5 units

Since they both have the same y value, just subtract the a value's x from the b's x value

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Which is the following best states the main idea of this passage Explain why knowing parts of speech would be helpful to a reader? Ex. Adjective, articles, nouns, pronouns, interjection etc. Write the equation of the line in slope intercept form y=mx + b that would have the slope of two and crosses through the ordered pair (-3,5). Which is the correct way torewrite the following sentencefragment?Last night, or this morning truly,up on the housetop. Miguel stated that any monomial can be a cube root. Sylvia disagreed and said that a monomial cube root must have exponents divisible by 3. Who is correct, and why?Sylvia is correct. Any variable term, when it is cubed, always has an exponent divisible by 3.Miguel is correct. Any monomial can be a perfect cube root because, when it is cubed, the variables will have exponents divisible by 3.Sylvia is correct. In order for the cube to have exponents that are divisible by 3, the cube root has to be divisible by 3.Miguel is correct. Sylvia confused the perfect square root with the perfect cube root. 50. Express-44 in the form bi where b is a real number.TIME SENSITIVE In which year was Nelson Mandela released from prison? simplify the following equationx^5 * x^-3 =x^nA. 4B. -2C. 2D. 8 boston is the capital of what state? What principle was behind the young peoples' decision to join the Children's March of 1963? A) civil disobedience B) division of labor C) militant protest D) pacifism A grassland environment, called a savanna, would be able to support the energy needs of carnivores becauseAgrasslands have grasses and plants to eat for animals that are not carnivores, which then provides animals for the carnivores to eat.Bgrasslands provide grasses and plants for carnivores to eat, which then provides food for animals that are not carnivores.Canimals that are carnivores can survive by eating leaves and plants that grasslands have.Danimals that are carnivores are able to hide from predators in the high grass of the grasslands. Gregor Mendel crosses a true-breeding tall plants (TT) with a true-breeding short plant (tt). What are the predicted phenotypes for the offspring ?a. all tallb. all short c. ratio of 1 tall : 1 short d. ratio of 3 tall : 1 short Find the vertex of the parabola whose equation is y = x^2 - 4x + 6.(-2, 18)(2, 2)(2,6) Mr. Marchand cut a pizza into 8 equal slices. He ate 2 slices.Write 2 equivalent fractions to describe how much pizza Mr. Marchand ate.Write 2 equivalent fractions to describe how much pizza was left 8 cmAnswer:cmPart BFind circumference of the circle. Use= 3.14.A 25. 12 cmB 50.24 cmC 73.06 cmD200.96 cm Consider the system of equations. 5x + 2y = 6, 10x + 4y = 12 Which equation is equivalent to the first equation of the system and can be used to solve the system using the linear combination method Identify the adjectives in the following sentence: "Henry's dreariest days are the rainy ones: they mirror the teary disposition he has these days. What was the name of the book written by Muslim scholar Ibn Sina that summarized all known diseases and treatments?Compendious Book on CalculationCanon of MedicineHouse of WisdomBook of Kells Why did European powers decide to divide Africa between them and how did they do it?I Why does tim end up going to war?A.- he thinks that people who go to canada are cowards B.- he thinks the war is justified and will stop communismC.- he afraid of what people will think of him if he doesnt go