Mathematics High School

A curve is represented by x = 3sec θ, y = 3tan θ where 0 < θ < /2.
Which of the following is the y intercept of the tangent to the curve at (2√3, √3)?

Answers

Answer 1

Applying derivative concepts, the y-intercept of the line tangent to the curve at (2√3, √3) is: [tex]y = \sqrt{3}[/tex]

The equation of the line is:

[tex]y - y_1 = m(x - x_1)[/tex]

In which:

There is a point [tex](x_1, y_1)[/tex].The slope is [tex]m = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}[/tex]

In this problem, point [tex](2\sqrt{3}, \sqrt{3})[/tex], hence [tex]x_1 = 2\sqrt{3}, y_1 = \sqrt{3}[/tex]

To find the value of [tex]\theta[/tex], we have that:

[tex]3\tan{\theta} = \sqrt{3}[/tex]

[tex]\tan{\theta} = \frac{\sqrt{3}}{3}[/tex]

[tex]\theta = \frac{\pi}{6}[/tex]

Then, for the slope:

[tex]\frac{dy}{d\theta} = 3\sec^{2}{\theta}[/tex]

[tex]\frac{dx}{d\theta} = 3\sec{\theta}\tan{\theta}[/tex]

Considering [tex]\theta = \frac{\pi}{6}[/tex]:

[tex]\frac{dy}{d\theta} = 3\left(\frac{2}{\sqrt{3}}\right)^2 = 2[/tex]

[tex]\frac{dx}{d\theta} = 3\left(\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{3}\right) = 2[/tex]

[tex]m = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = 1[/tex]

Then, the equation of the line is:

[tex]y - y_1 = m(x - x_1)[/tex]

[tex]y - \sqrt{3} = x - 2\sqrt{3}[/tex]

[tex]y = x - \sqrt{3}[/tex]

The y-intercept is the value of y when x = 0, hence:

[tex]y = \sqrt{3}[/tex]

A similar problem is given at https://brainly.com/question/22426360

Related Questions

how much does 1/3 package weigh

Answers

Answer:

1/3 package of what

Step-by-step explanation:

brainlist
Which of the following situations could be modeled by an linear function?

Question 17 options:

The number of people in a stadium is decreasing at a rate of 45% every 10 minutes


The water level of a tank doubles every day


The population of a small town grows at a rate of 2.4% each year


A tree was 10 feet tall when purchased and grows at a rate of 2 feet per year

Answers

A tree was 10 feet tall when purchased and grows at a rate of 2 feet per year.

Let’s remember what a linear function is. A linear function has a set amount of growth. In this case, it will grow two feet. The other answers are exponential functions. They aren’t growing the exact same amount each time, but a percentage.

Hope this helps!

Answer: D

Step-by-step explanation:

x = {(2, 4), (3, 4), (4,4), (5, 4)) Is x a function and why?

Answers

Because they all have a different x number. Even though the y numbers are the same you can tell wether a set of numbers are a function by the x numbers. So since there are different x numbers it is a function. I hope this helps!

Answer:

x is a function

since it forms a horizontal line it passes the vertical line test and is a function

Just a quick math problem that I forgot how to do.

Answers

Answer:

16

Step-by-step explanation:

If they are parallel then the 2 angles are congruent, so

8x + 14 = 12x - 50

64 = 4x

16 = x

Answer:

x = 16

Step-by-step explanation:

For the 2 lines to be parallel, then

12x - 50 and 8x + 14 would be corresponding angles and congruent , so

12x - 50 = 8x + 14 ( subtract 8x from both sides )

4x - 50 = 14 ( add 50 to both sides )

4x = 64 ( divide both sides by 4 )

x = 16

Nia needs 15 pieces of string each ½ of a inch long. She cut a 6-inch piece of string into pieces that are each ½ of an inch long. How many more pieces of string does she need?

Answers

2 times plus 1 would be the answer I think

Help please (MATH)......................

Answers

Answer:

ft hu hu mmmmmmmmmuaaaaaaaaaaaaaah

I’m not sure but I think it’s 4 square feet

125 divided by 100 I have to make corrections

Answers

Answer:

1.25

Step-by-step explanation:

125/100 = 1.25

Answer:

it's 1.25

Step-by-step explanation:

125 ÷100 = 1.25

I NEED ANSWER NOW PLEASE TELL ME THE ANSWER AND HOW YOU GOT IT I WILL MARK U AS BRAINLIEST IT DUE IN 5 MINS
(GIVE ME ALL THE ANSWERS)

Answers

Answer:          

1 is 42,256 =              

3 × 105

+ 0 × 104

+ 5 × 103

+ 4 × 102

+ 4 × 101

+ 4 × 100

       

2 is

4 × 104

+ 2 × 103

+ 2 × 102

+ 5 × 101

+ 6 × 100

3 is   9,905,482 =

9 × 106

+ 9 × 105

+ 0 × 104

+ 5 × 103

+ 4 × 102

+ 8 × 101

+ 2 × 100

Step-by-step explanation:

What is 5 x 3/4 in fraction

Answers

Answer:

3 and 3/4

Step-by-step explanation:

Just ask.

(✿◡‿◡) HAPPY HOLIDAYS!!!

Answer:

15/4 in fraction and 3 3/4 in mixed number.

Step-by-step explanation:

Can someone help me please​

Answers

Answer:

D

Step-by-step explanation:

Graph: Is it a function? (Yes / no) why or why not?

Answers

short answer: yes.

the definition of a function is a graphical relation in which each x value (input) has only one y value (output). if we use the vertical line test, we can see that each x value on this graph corresponds to one y value and one only. if a certain x value had more than one y value, then it wouldn’t be a function.

hope this helped, i tried to keep this concise but if you need a better explanation let me know :)

Can someone help me out?

Answers

Answer:

d

Step-by-step explanation:

If I swim 575 yards for three days how many yards I swim

Answers

The answer is 1,725 yards

Step-by-step explanation:

If you swim 575 yards in 3 days, you swim 575 yards.

If you swim 575 yards per day for 3 days, then you swim 3 × 575 yd = 1725 yd

HELP!!!

‐6x – 1 = 23

Please do not just post the answer please explain how you got the answer. Whoever explains best gets brainliest!

Answers

1. Add the 1 to the 23 because if it's a negative you add and if it's positive you subtract and because you need x by itself
2. divide -6 to 24
3. x=-4

Answer:

-4

Step-by-step explanation:

23+1=24

24/-6=-4

Adam eats 2 apples in 5 minutes, at this rate how many apples will Adam eat in a hour?
6
10
24
120

Answers

Answer:

24 apples

Step-by-step explanation:

60 divided by 5 is 12 then you multiply 2 which is 24:)

The statement below is a different way to say that the sides of these similar parallelograms are proportional.

Complete the equations that go with the statement.

Answers

b/c = s/r

_________

o0o0o0o0o0

c/d = r/q

________

o0o0o0o0

The department of education decides to require an additional 2 tests per year of standardized test for all 10th graders. Would this equation be linear or exponential?
Group of answer choices

Exponential

Linear

Answers

It would be linear because it goes up each year by two.
If this is right may I have brainliest?

3,415.72 - 1,838.28 = 1577.44
4. Janet earns $6.25 an hour for guarding paintings. If she works 24.3 hours a week for a period of six
weeks, how much does she earn from working six weeks?

Answers

$911.25.

6.25x24.3= 151.875
151.875x6=911.25

Who ever can answer these three questions first I will give them a brainliest! Or two
Need help ASAP. Heeellllppppp ppplllllsss!!!


5. 6. And 7.

Answers

Answer:

mathemiatical porbs is is  it itroon poickg niucer g tnt g hayabusa sanjo          karaneteooo  break the code

Step-by-step explanation:

Select the graph of the equation y + x + 3 = 0.

Answers

Answer:

C.

Step-by-step explanation:

The picture I took is underneath.

help please???????????​

Answers

B. 6x9=9x6

Commutative property of multiplication: Changing the order of the numbers does not change the product.

The circumference of a circle, C, is directly proportional to the length of its radius, r. Write down a formula,using k, connecting the circumference of the circle and the length of its radius.

Answers

The proportional relationship connecting the circumference of the circle and the length of its radius is:

[tex]C = kr[/tex]

What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:

[tex]y = kx[/tex]

In which k is the constant of proportionality.

In this problem:

The output is the circumference of the circle.The input is the radius of the circle.

Hence, the formula is:

[tex]C = kr[/tex]

You can learn more about proportional relationships at https://brainly.com/question/25815502

Annie, Brianna & Carly are selling fruit boxes to raise money for their soccer team. All together the girls have sold 105 boxes of fruit. Carly has sold ten more than three times Brianna's sales. Brianna sold five more than Annie. How many boxes of fruit did Carly sell?

Answers

Step-by-step explanation:

Annie = x

Brianna = y

Carly = z

Now,

x + y + z = 105 ...(1)

Carly has sold ten more than three times Brianna's sales.

z = 10 + 3y ...(2)

Brianna sold five more than Annie.

y = 5 + x ...(3)

Now,

z = 10 + 3y

z = 10 + 3(5 + x)

z = 10 + 15 + 3x

z = 25 + 3x

Now,

x + y + z = 105

x + (5 + x) + 25 + 3x = 105

5x + 30 = 105

5x = 105 - 30

5x = 75

5x/5 = 75/5

x = 15

So,

y = 5 + x

y = 5 + 15

y = 20

Now,

z = 25 + 3x

z = 25 + 3(15)

z = 25 + 45

z = 70

Thus,

Annie Sold 15 boxes

Brianna Sold 20 boxes

Carly Sold 70 boxes

NO LINKS HELP (question is in picture)

Answers

Step-by-step explanation:

area of ∆ is 10in².

hope this helps you.

(PLS HELP FAST FOR 20 points) one step equation 18 + x = -9

Answers

Answer:

-27

Step-by-step explanation:

18 + x = -9

subtract by 18 on both sides

x = -27

You can check this by...

18 - 27 = -9

I hope this helps!!

this is your answer hope it's helpful for you

Pls help will give brainly

Answers

Answer:

y = - 2x - 4

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (- 4, 4) and (x₂, y₂ ) = (2, - 8) ← 2 points on the line

m = [tex]\frac{-8-4}{2-(-4)}[/tex] = [tex]\frac{-12}{2+4}[/tex] = [tex]\frac{-12}{6}[/tex] = - 2

The line crosses the y- axis at (0, - 4 ) ⇒ c = - 4

y = - 2x - 4 ← equation of line

ANSWER ALLL PLEASEEEEEEEEE ILL GIVE BRAINLIST !!!!!!!!!

Answers

Answer:

question 3 is $9.99

question 4 I believe is the third option

Graph the linear equation.

3x−4y=12
school: k 12

Answers

Answer: slope : 3/4 points: (0,-3) (4,0)

Step-by-step explanation:

i looked it up

Answer:

look at graph

Step-by-step explanation:

If these numbers represent temperature in degrees Celsius, which is the coldest ​

Answers

Answer: I don’t know

Step-by-step explanation: I don’t know because there is no numbers but the coldest Celsius I know is -273 degrees Celsius

Simplify and leave your answer in index form

Answers

Answer:8a^7

Step-by-step explanation:

Answer:

8a⁷ is the simplified form of 4a² x 2a⁵.

Step-by-step explanation:

4a² x 2a⁵=> 4 x a² x 2 x a⁵=> 8 x a²⁺⁵=> 8 x a⁷=> 8a⁷

Conclusion:

Therefore, 8a⁷ is the simplified form of 4a² x 2a⁵.

Hoped this helped.

[tex]BrainiacUser1357[/tex]

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