Mathematics Middle School

For what values of θ on the polar curve r=θ, with 0≤θ≤2π , are the tangent lines horizontal? Vertical?

Answers

Answer 1
Given that [tex]r=\theta[/tex], then [tex]r'=1[/tex]

The slope of a tangent line in the polar coordinate is given by:

[tex]m= \frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta} [/tex]

Thus, we have:

[tex]m= \frac{\sin\theta+\theta\cos\theta}{\cos\theta-\theta\sin\theta} [/tex]



Part A:

For horizontal tangent lines, m = 0.

Thus, we have:

[tex]\sin\theta+\theta\cos\theta=0 \\ \\ \theta\cos\theta=-\sin\theta \\ \\ \theta=- \frac{\sin\theta}{\cos\theta} =-\tan\theta[/tex]

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are horizontal are:

θ = 0

θ = 2.02875783811043

θ = 4.91318043943488



Part B:

For vertical tangent lines, [tex] \frac{1}{m} =0[/tex]

Thus, we have:

[tex]\cos\theta-\theta\sin\theta=0 \\ \\ \Rightarrow\theta\sin\theta=\cos\theta \\ \\ \Rightarrow\theta= \frac{\cos\theta}{\sin\theta} =\sec\theta[/tex]

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are vertical are:

θ = 4.91718592528713
Answer 2
Final answer:

For the polar curve r = θ, the tangent lines are horizontal when θ = 0 and θ = π, and the tangent lines are vertical when θ = π/2 and θ = 3π/2.

Explanation:

The question is asking us to find the values of θ for which the tangent lines to the polar curve r = θ are horizontal and vertical. To do this, we need to first derive the slope of the tangent line in polar coordinates.

The derivative dr/dθ represents the slope of the radius with respect to θ, or in other words, the slope of the curve. When r = θ, the derivative dr/dθ = 1. This slope is equal to the tangent of the angle that the tangent line makes with the positive x-axis.

Therefore, for the tangent lines to be horizontal, the angle needs to be 0 or π rad (0 or 180 degrees). For the tangent lines to be vertical, the angle needs to be π/2 or 3π/2 rad (90 or 270 degrees). Using the equation r = θ in rad, we find the horizontal tangents are when θ = 0 and θ = π, while the vertical tangents are when θ = π/2 and θ = 3π/2.

Learn more about Polar Coordinates here:

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Related Questions

I need more help....sorry haha

Answers

Number 7:   11/12-9/10 = 110/120-108/120=2/120= 1/60

number 8:     5/18-1/3=15/54-5/54=10/54 =5/27

Number 7: 11/12-9/10 = 110/120-108/120=2/120= 1/60

number 8: 5/18-1/3=15/54-5/54=10/54 =5/27

math question down below

Answers

for the first drain the equation you need would be: 2n = 108

for drain B your equation would be: 5n = 108

these are your equations because a number times the amount of water the drain drains per minute will equal 108

to solve use inverse operations and divide 108 by the amount of water the drain drains in one minutes 

Drain A- 108/2 = 54
it takes 54 minutes for drain a to empty a pool with 108 gallons of water

Drain B- 108/5 = 21.6
it takes 21.6 minutes for drain b to empty a pool with 108 gallons of water in it

How do you write the equation?

Answers

The number of hours she practices each day is unknown, so let's call it x.

She practices dancing x hours each day she goes to practice.
Since she practices 3 days per week, in a week she practices 3x hours.
In two weeks she practices 2(3x) hours, or 6x hours.

We are told she practices 12 hours in 2 weeks, so 6x must equal 12.

The equation is

6x = 12

We can solve it by dividing both sides by 6.

x = 2

She practices 2 hours per day.

Check:
2 hours per for 3 days = 2 * 3 hours = 6 hours in 1 week.
In 2 weeks, she practices 2 * 6 hours = 12 hours.
Our solution is correct.

What is 42% in simplest form?plz show me how to do it

Answers

Hey there!

42%= 42/100 


42/100=2/2×21/50


So 21/50 is the simplest form of 42%

if G varies inversely as r, and G = 18 when r= 4, find G when r= 6

Answers

The model to use here is   y = k/x.

If g =18 when r = 4, then 18 = k/4.  Solving for k, k = 72.

Then g = 72/r.  If r = 6, g = 72/6 = 12 (answer)

Six months after a divorce, the former wife and husband each take a test that measures divorce adjustment. The wife’s score is 63, and the husband’s score is 59. Overall, the mean score for divorced women on this test is 60 (SD = 6); the mean score for divorced men is 55 (SD = 4). Which of the two has adjusted better to the divorce in relation to other divorced people of their own gender? Explain your answer to a person who has never had a course in statistics

Answers

Note that
(63 – 60)/6 = 1/2
(59 – 55)/4 = 1
So, in relation to other divorced women, the score of 63 is only 1/2 SD above the average. But in relation to other divorced men, the score of 59 is 1 SD above the average. That means the husband has adjusted better to the divorce in the relation to other divorced people of his gender.

Did i choose the correct answer?

Answers

The correct answer is 
A) MN
You are correct. If you drew the triangles upright on a piece of paper you would be able to see you are correct

please may you help me or I will get detention

Answers

The range is the highest number minus the lowest number. In this case, the highest number is 10 and the lowest number is 6, so the range is 10-6=4

To find the students in the group, simply add up the frequency from each mark. I'll give you an example solution, but I challenge you to do this on your own!

For example, say that there are 4 students with 3 marks and 6 students with 4 marks. There are then 4+6=10 total students in the group

To find the mean, simply add up all up the marks with the amount of frequencies and divide it by the amount of students.

Take our example from earlier - we can write it as a list. Since there are 4 students with 3 marks, we have
3, 3, 3, 3 ( 3 four times)
Since there are 6 students with 4 marks, we have
4, 4, 4, 4, 4, 4 (4 six times)

Add them all up, and divide them by the total amount of students (10) to get 
(3+3+3+3+4+4+4+4+4+4)/10=(3*4+4*6)/10=36/10=3.6

Good luck, and feel free to ask questions if you're confused!

Answer:

a) 4

b) 40 students

c) 8

Step-by-step explanation:

10-6=4, so a=4

6+ 7+8+ 9+10= 40

6+7+8+9+10=40; 40/5 =8

Plz mark me brainliest

11/3 this is equivalent to
3.6

3.6-

3.67

3.666

Answers

3.666 since 3/11 is equivalent to dividing 3 by 11
Hello there!

To find the decimal of a fraction, all you need to do is take the number on top then divide it by the number at the bottom.

Or you can say: Take the numerator divide it by the denominator.

So;
11/3 is equal to 3.666667

In this case, the answer is option C. 3.67

I hope this helps!

Graph y= 12x–3 .
Where would I put the points ?

Answers

We need to graph y = 1/2x - 3.
y = 1/2x - 3 is from the general form of a linear function y = ax + b, in which a represents the slope, b the y-intercept, and x and y the coordinates of 2 points.
Since it's a linear function, the graphical representation of that equation is a straight line.

To graph that function, we need at least 2 points from that line. And we'll find their coordinates using the equation.

1- For x = 0, y = -3 (Since the y-intercept is b = -3).
2- For x = 2. y = 1/2x - 3 = 1/2*2 - 3 = -2

So we now have the 2 points A (0, -3) and B (2, -2).Place those points on the graph, and reunite them with a line and you'll get your graphical representation of the linear function y = 1/2x - 3 (You can see the graphical representation in the image alongside my answer).

Hope this helps! :)

Answer:

Look at my picture, it will show you where to plot the points, and you can also see the green checkmark above which verifies that this answer is correct. Hope this helps!

Step-by-step explanation:

This answer is in the picture. Also, it is not y = 12=-3, it's y = 1/2x-3

The person who asked this question copy pasted and the computer didn't put the symbol with the fraction.

Anyways, this is the correct answer for the problem, so make sure you look at the picture, and I hope I helped you! :)

Good luck on whatever you are doing. ;)

When John bought his new computer, he purchased an online computer help service. The help service has a yearly fee of $25.50 and a S10.50 charge for each help session a person uses. If John can only spend S170 for computer help this year, what is the maximum number of help sessions he can use this year?

Answers

The yearly fee can be subtracted, resulting in 170-25.50=144.5

So with 10.50 a session, you must divide, resulting in around 13.8 sessions.

13 sessions if you need a whole number.

Hope this helps

Maya's school held an aluminum collection program in which they collected aluminum and brought it to be recycled. In the first week, Maya collected 4/20 lb of aluminum and her friend Abigail collected 7/8 lb. a. which girl collected more aluminum? b. how much aluminum did the two girls collect?

Answers

If you put this into the calculator, 4/20 is .2 and 7/8 is .875 therefore Abigail collected more. If you use them as the same denominator, you have 8/40 for Maya and 35/40. And if you add these two numbers together, you get 1.075 which is 43/40 as a fraction

a)If Maya collected 4/20 lb of aluminum and her friend Abigail collected 7/8 lb Abigail collected more aluminum.

b) 43/40 lb of aluminum the two girls collect.

What is a fraction?

Fraction number consists of two parts, one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.

It is given that, in the first week, Maya collected 4/20 lb of aluminum, and her friend Abigail collected 7/8 lb.

We have to simplify the expression,

Maya collected 0.2 lb of aluminum, and her friend Abigail collected 0.875 lb.

From the given fraction it is obtained as Abigail collected more aluminum.

The amount  of aluminum did the two girls collect is,

=4/20+ 7/8

=43/40

Thus, If Maya collected 4/20 lb of aluminum and her friend Abigail collected 7/8 lb Abigail collected more aluminum and 43/4 lb of aluminum than the two girls collect.

Learn more about the fraction here:

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form an equation and solve it to find the dimensions of the rectangle
(perimeter 48cm)
(height 2x-2)
(width 5x+4)

Answers

To find x, use the equation 2l+2w

In this case, your equation would look like

2(2x-2)+2(5x+4)= 48 cm

Hope this helps!

14 boys and 21 girls will be equally divided into groups. Find the greatest number of groups that can be created if no one is left out.

Answers

I think is 0.6666........and so on
It is 5 because 14 and 21 are both divisable by 7

A company needs to take 330 sample sensor readings. If the sensor collects data at 5 1 2 samples per second, how long will it take the company to take all 330 samples?

Answers

Note: this answer was made with the assumption that you meant 5 1 2 as 5 1/2 / 5.5, though you may correct me if that is the wrong interpretation.

It will only take the company about 60 seconds, or a minute to take all 330 samples, for 330/5.5 = 60.

Answer:

hmmm...

Step-by-step explanation:

Which expression is NOT equivalent to 8 ∙ x ∙ 2 ∙ y?

1. 8x + 2y

2. 16yx

3. 16xy

Answers

8 * x  * 2 * y = 16xy
Wrong answer is option 1
The answer is:  "Answer choice: 1):  "8x + 2y".
__________________________________________
   Note:  " 8 * x * 2 * y = 16xy ".  

So, choice 3):   "16xy is correct". 

Choice 2):  "16yx" is also correct, since:  "16xy = 16yx".

Choice 1)": is incorrect, so this is the incorrect answer.

to  be sure.  let's plug in a values, "3" for "x" ; and "4" for "y"; 
______________________________
Choice 1) " 8x + 2y"  ;    8(3) + 2(4) = 24 +8 = 32 .

Now, try:  "16xy".  16(3)(5) = 48(5) = 240.   

Note that:  "32≠40" .
____________________________________
So, Choice 1):   "8x + 2y" ;  is the correct answer.
______________________________________

Jason has 7 puzzles. Quincy has the same number of puzzles as Jason. How many puzzles do they have altogether? Solve. Write or draw to explain

Answers

All together they have 14 puzzles. You add Jason's seven to Quincy's seven and you will end up with 14 puzzles.

Marcy split her income from last week equally between paying her student loans, rent, and savings. She put $650 in savings. How much did Marcy earn last week?

Answers

Final answer:

Marcy's total weekly income was calculated by multiplying the amount she put in savings ($650) by the number of equal parts (3), resulting in a total of $1950 earned last week.

Explanation:

To find out how much Marcy earned last week, we need to consider that she split her income equally into three parts: paying her student loans, rent, and savings. Given that she put $650 in savings, and each of the three parts is equal, we can calculate her total weekly income by multiplying $650 by 3 (the number of equal parts).

Total weekly income = $650 (amount put in savings) × 3

Total weekly income = $1950

Therefore, Marcy earned a total of $1950 last week.

A snow storm dumped 18 7/10 inches of snow in a 12 hour period. How many inches were falling per hour

Answers

All you have to do for this is take your inches (18 7/10 or 18.7) and divide it by the number of hours (12) which makes 1.56in per hour.
make 18 7/10 into a improper fraction and multiply that by 12.
Hope this helped.

PART ONE-----
Patel bought a model rocket kit from a catalog. The price of the kit was $124.95. The state sales tax of 7% was added, and then a $19 charge for shipping was added after the sales tax. What was the total amount Patel paid including tax and shipping costs?


PART TWO---
Patel received an allowance of $15 per week. How many weeks it will take him to purchase the kit?

Answers

Part one: $152.70

Part two: About 11 weeks

Hope this helps!!

What is the perimeter of a rectangle if the length of the short sides are 5", and the other sides are both y?

Answers

The equation would be set up as;

2(5) +2(y) =P
P-10=2y
(P/2) - 5 = y

Not sure how you wanted this to be set up but any of the answers above should work? 


Please mark brainiest and good luck :D.

which equation shows y+1/5=3x in standard form?

Answers

I believe the correct answer is 15x + 5y =-1 but im not positive cause i did the problem and got -1 but the equation i got looks different i got the equation -15x + 5y = -1
These are the steps i followed
You always begin by the Given equation, which is in this case:
y+ = 3x
To remove the fraction, you multiply the whole equation by 5:
5y + 1 = 15x
Then, you move the "15x" the left-side of the equation, changing it's sign
-15x + 5y +1 = 0
You move the "1" to the right side, changing it's sign
-15x + 5y = -1

The answer is 15x-5y=1.

Tyler delivers newspapers. He collected $10 from each customer. He has 5 customers on each street, and he delivers to 4 different streets. How much money did Tyler collect?

Answers

$200 is the correct answer.
He would earn $200 because if he collects $10 from each customer, then 5 times 4=20, 20 times 10= 200

(02.06)Ralph needs 6 cups of milk to make a chocolate shake for his family. His measuring device measures liquids by the pint. If 1 pint is equal to 2 cups, how many pints should Ralph measure?

3 pints
4 pints
8 pints
12 pints

Answers

3 pints will be your answer.
if 1 pint=2 cups and you need to get 6 cups of milk, you need to see which of your answers will equal 6 when timed by 2.
12x2=24. Wrong
8x2=16. Wrong
4x2=8 Wrong
3x2=6. Correct!

1 pint = 2 cups

 so 6 cups / 2 cups = 3 pints

Three boxes are stacked one on top of the other. One box is 3 feet 7 inches tall, one is 3 feet 10 inches tall, and one is 4 feet 9 inches tall. How high is the stack? Write your answer in feet and inches. Use a number less than 12 for inches.

Answers

21 ft. 7 in. that is the answer because you add it up then divide the in

Mr. Jones bought a block of fudge that weighed 1/4 of a pound. He divided the fudge into 2 equal pieces. What was the weight of each piece of fudge? *

Answers

You would do 1/4 x 1/2 to get 1/8 of a pound.

I need to know number 35 I don’t know how to do this promblem

Answers

Hello! Expanded form is when you write an addition problem by place value in order to get the sum for the answer. For example, expanded form of 123 is 100 + 20 + 3. In this case, 92.3 is out number we are going to expand in addition form. 92.3 in expanded form is 90 + 2 + 0.3. This is because 9 in the tens place has a value of 90, the 2 in the ones place has a value of 2, and the value of 3 in the tenths place has a value of 0.3.

what are the answers

Answers

11. 10x+10
13. 14x-14
15. 26x-50
11. 10x+10
13. 14x-14
15. -22x-50

Sam wants to find the best buy on cans of tuna. He reads these four ads in the newspaper. Place the ads in order from best to worst buy. [A better buy is one with a lower cost per unit.] ITEMBANK: Move to Bottom $12.53 for seven cans$5.97 for three cans$7.95 for five cans$8.34 for six cans Best 2nd Best 3rd Best Worst

Answers


Best to worst;

8.34/6 = $1.39

7.95/5 = $1.59

12.53/7 = $1.79

5.97/3 = $1.99

Answer: 5.97 for three cans >  $12.53 for seven cans > $7.95 for five cans > $8.34 for six cans

Step-by-step explanation:

Since, $12.53 for seven cans

The cost of one can = [tex]\frac{12.53}{7} = 1.79[/tex]

$5.97 for three cans

The cost of one can = [tex]\frac{5.97}{3} = 1.99[/tex]

s$7.95 for five cans

The cost of one can = [tex]\frac{7.95}{5} = 1.59[/tex]

$8.34 for six cans

The cost of one can =  [tex]\frac{8.34}{6} = 1.39[/tex]

Since, 1.99 > 1.79 > 1.59 > 1.39

Therefore, the required order of prices,

5.97 for three cans >  $12.53 for seven cans > $7.95 for five cans > $8.34 for six cans

The total cost for office expenses to run the organic strawberry farm varies directly with the number of months it is used. Which equation represents this scenario? https://coststudyfiles.ucdavis.edu//uploads/cs_public/c5/a1/c5a181af-8e1b-474a-9509-443c3cb223ed/strawberry2ndyrcc2011.pdf pages 15-16

Answers

Answer:

weed-hand

y=63x

$1,512.00

Step-by-step explanation:


Answer:

third option, second option, then first option

Step-by-step explanation:

just took the assignment

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