Mathematics College

Find the points on x^2+3(y-2)^2 = 27
which are closest to and farthest from the origin

Answers

Answer 1

[tex](-4.89, 3)[/tex] and [tex](4.89, 3)[/tex] are the farthest from the origin and [tex](0, -1)[/tex] is the closest from the origin.

Let be a curve of the form [tex]x^{2}+3\cdot (y-2)^{2} = 27[/tex]. The distance with respect to origin is found by using the following Pythagorean identity:

[tex]r = x^{2}+y^{2}[/tex] (1)

Where [tex]r[/tex] is the square distance function.

We can modify (1) as following:

[tex]r = [27-3\cdot (y-2)^{2}]+y^{2}[/tex]

[tex]r = 27-3\cdot (y^{2}-4\cdot y +4)+y^{2}[/tex]

[tex]r = 27-3\cdot y^{2}+12\cdot y -12 + y^{2}[/tex]

[tex]r = -2\cdot y^{2}+12\cdot y+27[/tex] (1b)

Now we apply the First and second derivative tests to determine the minimum and maximum distances from the origin:

First derivative test

[tex]-4\cdot y +12 = 0[/tex]

[tex]y = 3[/tex]

Second derivative test

[tex]r'' = -4[/tex]

The y-component represents a maximum.

Now we graph the function with the ressource of a graphing tool, we find the following points:

Farthest points: [tex](-4.89, 3)[/tex], [tex](4.89, 3)[/tex].

Closest points: [tex](0, -1)[/tex].

[tex](-4.89, 3)[/tex] and [tex](4.89, 3)[/tex] are the farthest from the origin and [tex](0, -1)[/tex] is the closest from the origin.

To learn more on ellipses, we kindly invite to check this verified question: https://brainly.com/question/19507943

Find The Points On X^2+3(y-2)^2 = 27 Which Are Closest To And Farthest From The Origin

Related Questions

Help help math math math

Answers

Answer:

[tex]\displaystyle m = 3[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Algebra I

Coordinate Planes

Coordinates (x, y)

Slope Formula: [tex]\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

Step-by-step explanation:

Step 1: Define

Identify points.

Point (8, 1)

Point (11, 10)

Step 2: Find slope m

Simply plug in the 2 coordinates into the slope formula to find slope m.

Substitute in points [Slope Formula]:                                                              [tex]\displaystyle m = \frac{10 - 1}{11 - 8}[/tex][Order of Operations] Evaluate:                                                                      [tex]\displaystyle m = \frac{9}{3}[/tex]Simplify:                                                                                                             [tex]\displaystyle m = 3[/tex]

Answer:

m=3

Step-by-step explanation:

Mrs. Baca uses a phone card to call a relative in Colombia, It cost her 45 cents to talk for 15 minutes. Choose True or False for each statement.
A. For 45 cents, Mrs. Baca can talk for 3 minutes. True or False.
B. The call rate is 3 cents per minute. True or False.
C. The call rate can represent by the ratio 445:15. True or False.
D. Divide 75 by 15 to find the number of minutes Mrs. Baca can talk. True or False.

Answers

Answer:

A: False

B: True

C: False

D: False

Hope This Helps!!!

Marshall wants to buy a picture frame. The original price is $15,
How much will Marshall pay if he buys it during the sale?
SALE
25% off
original price

Answers

15x.25=3.75
15-3.75=11.25.

The answer is $11.25

COULD SOMEONE HELP ME PLEASE I WILL MARK BRAINLIEST I BEG U PLEAASE PLEASEEEEEE

Answers

Answer:

I think it's B

Step-by-step explanation:

A triangle and a square are drawn on dotty paper with dots 1 cm apart. What is the area of the region coloured red? ​

Answers

Answer:

≈ 0.9 cm²

Step-by-step explanation:

I don't know if there is any easier way to solve this, I use 8th grade slope and line method to solve. suppose A is on origin, then the coordinates of triangle ABC are (0,0) (1,3) (4,1) and polygon DEFGH intersect line BC at F and G.

F is the intersect of y=2 and line BC, while G is intersect of x=3 and line BC.

Line BC: pass (1,3) and slope (m) = (1-3)/(4-1) = - 2/3,    

F(x,2)

(2-3) / (x-1) = -2/3            

2x-2 = 3       x = 2.5          

*F (2.5 , 2)

G (3,y)

(y-3) / (3-1) = -2/3     y = 5/3      

*G (3 , 5/3)

*point I (3 , 2)

segment FI = 3-2.5 = 0.5 = 1/2

segment GI = 2 - 5/3 = 1/3

area ΔFIG = (1/2 * 1/3) / 2 = 1/12 = 0.083

area of polygon  DEFGH (region colored RED) = 1 - 0.083 = 0.917 ≈ 0.9 cm²

Which function does f(-1/2) ≠ -1

Answers

Answer:

anything but 2

Step-by-step explanation:

f(-1/2)

If you plug in 2, you'd get -1. But if you plug in other values, it won't be equal to -1.

What is the 24th term in the sequence 1, 3, 5, 7

Answers

Answer:

47

Step-by-step explanation:

Each new term is the previous term +2.

1st term = 1

5th term = 9

10th term = 19

15th term = 29

20th term = 39

25th term = 49 - 2 = 47 = 24th term

You have been assigned to determine whether more people prefer Coke or Pepsi. Assume that roughly half the population prefers Coke and half prefers Pepsi. How large a sample do you need to take to ensure that you can estimate, with 95% confidence, the proportion of people preferring Coke within 3% of the actual value?

Answers

You have to add.

Step-by-step explanation:

I'm trash at math.

Please help me
I don’t know what to do for this and it’s really confusing

Answers

Answer:

Midpoint: (0, 5/2) or (0, 2.5)

Step-by-step explanation:

The diagonal BD is made of the points B(4,7) and D(-4,-2)

to find the midpoint of these points you add the two x coordinates and divide by two --> (4+-4)/2=0/2=0

Then you add the y coordinates and divide by 2 --> (7+-2)/2=5/2 or 2.5

Then you've got your midpoint.

hopefully I've understood the question correctly - hope this helps!

Jane needs to memorize words on a vocabulary list for German class. She has memorized 8 of the words, which is one-fourth of the list. How many words
are on the list?

Answers

Answer:

x = 32 words

Step-by-step explanation:

Okay...let's get solving! :D Firstly, we

let the word list be x and convert the long paragraph into an equation that we can use.

one-fourth = 1/4

Memroized = 8 words

[tex]x * \frac{1}{4} = 8[/tex]

Thus, we solve the equation.

x = 32 words

The store is 7⁄8 of a mile away from your house. You walked 1⁄5 of a mile towards the store before getting on the bus. If the bus went directly to the store, how many miles long was the bus ride?

Answers

7/8 = 35/40
1/5 = 8/40

35 8 27
—- ➖ —- = —-
40 40 40


27/40 miles

The ratio of the measures of the sides of a triangle is 5:7:8. Find the measures of each angle in the triangle.

Answers

Answer:

45°, 63°, 72°

Step-by-step explanation:

The sum of ratio units is 5+7+8=20, so each stands for 180°/20 = 9° of angle. Multiplying the ratio values by 9° gives the angle values:

  45° : 63° : 72°

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