1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
DerKrebs [107]
3 years ago
13

Consider the three points ( 1 , 3 ) , ( 2 , 3 ) and ( 3 , 6 ) . Let ¯ x be the average x-coordinate of these points, and let ¯ y

be the mean y-coordinate of these points.
Then ( ¯ x , ¯ y ) =

Now consider a line through the point ( ¯ x , ¯ y ) that has slope m.

Add up the squares of the vertical distances between the line and these points (your result will be a quadratic function in terms of the variable m ).

What is the value of m that makes this total as small as possible?

Mathematics
1 answer:
loris [4]3 years ago
5 0

Answer:

m=\dfrac{3}{2}

Step-by-step explanation:

Given points are: ( 1 , 3 ) , ( 2 , 3 ) and ( 3 , 6 )

The average of x-coordinate will be:

\overline{x} = \dfrac{x_1+x_2+x_3}{\text{number of points}}

<u>1) Finding (\overline{x},\overline{y})</u>

  • Average of the x coordinates:

\overline{x} = \dfrac{1+2+3}{3}

\overline{x} = 2

  • Average of the y coordinates:

similarly for y

\overline{y} = \dfrac{3+3+6}{3}

\overline{y} = 4

<u>2) Finding the line through (\overline{x},\overline{y}) with slope m.</u>

Given a point and a slope, the equation of a line can be found using:

(y-y_1)=m(x-x_1)

in our case this will be

(y-\overline{y})=m(x-\overline{x})

(y-4)=m(x-2)

y=mx-2m+4

this is our equation of the line!

<u>3) Find the squared vertical distances between this line and the three points.</u>

So what we up till now is a line, and three points. We need to find how much further away (only in the y direction) each point is from the line.  

  • Distance from point (1,3)

We know that when x=1, y=3 for the point. But we need to find what does y equal when x=1 for the line?

we'll go back to our equation of the line and use x=1.

y=m(1)-2m+4

y=-m+4

now we know the two points at x=1: (1,3) and (1,-m+4)

to find the vertical distance we'll subtract the y-coordinates of each point.

d_1=3-(-m+4)

d_1=m-1

finally, as asked, we'll square the distance

(d_1)^2=(m-1)^2

  • Distance from point (2,3)

we'll do the same as above here:

y=m(2)-2m+4

y=4

vertical distance between the two points: (2,3) and (2,4)

d_2=3-4

d_2=-1

squaring:

(d_2)^2=1

  • Distance from point (3,6)

y=m(3)-2m+4

y=m+4

vertical distance between the two points: (3,6) and (3,m+4)

d_3=6-(m+4)

d_3=2-m

squaring:

(d_3)^2=(2-m)^2

3) Add up all the squared distances, we'll call this value R.

R=(d_1)^2+(d_2)^2+(d_3)^2

R=(m-1)^2+4+(2-m)^2

<u>4) Find the value of m that makes R minimum.</u>

Looking at the equation above, we can tell that R is a function of m:

R(m)=(m-1)^2+4+(2-m)^2

you can simplify this if you want to. What we're most concerned with is to find the minimum value of R at some value of m. To do that we'll need to derivate R with respect to m. (this is similar to finding the stationary point of a curve)

\dfrac{d}{dm}\left(R(m)\right)=\dfrac{d}{dm}\left((m-1)^2+4+(2-m)^2\right)

\dfrac{dR}{dm}=2(m-1)+0+2(2-m)(-1)

now to find the minimum value we'll just use a condition that \dfrac{dR}{dm}=0

0=2(m-1)+2(2-m)(-1)

now solve for m:

0=2m-2-4+2m

m=\dfrac{3}{2}

This is the value of m for which the sum of the squared vertical distances from the points and the line is small as possible!

You might be interested in
Solve the equation.<br> 9x=25
lidiya [134]

Answer:

x = \frac{25}{9}

Step-by-step explanation:

Divide both sides by 9 to isolate x:

\frac{9x}{9} = \frac{25}{9}  \\x = \frac{25}{9}\\

Hope this helps!

8 0
3 years ago
3x + 2 &gt;= 11 I need to understand how to do it on a number line.
alex41 [277]
3x+2\geq11\\&#10;3x\geq9\\&#10;x\geq3

5 0
3 years ago
How can you find the surface area of a composite solid made up of prisms ? (Help)
sammy [17]
Oh, this is easy. For any prism, you find the surface area by finding the are of all the sides. I will post all surface area formulas here, for reference
Cube:



Surface area = 6 × a2




Right circular cylinder:



Surface area = 2 × pi × r2   +  2 × pi × r × h

pi = 3.14
h is the height
r is the radius


Rectangular prism:



Surface area = 2 × l × w  +  2 × l × h  +  2 × w × h 




l is the length
w is the width
h is the height


Sphere:



Surface area = 4 × pi × r2 

pi = 3.14
r is the radius


Right circular cone:



Surface area = pi × r2  +  pi × r ×( √(h2 + r2)) 

pi = 3.14
r is the radius
h is the height
l is the slant height 


Right square pyramid:



Surface area = s2 + 2 × s × l

s is the length of the base
h is the height
l is the slant height 
8 0
3 years ago
pablo will rent a car for the weekend. he can choose one of two plans. the first plan has an initial fee of $55.96 with addition
kow [346]

Answer:

200 miles

Step-by-step explanation:

First, set up the equations.

plan 1: initial fee of $55.96, $0.12 per mile

  • y = 0.12x + 55.96

plan 2: initial fee of $63.96, $0.08 per mile

  • y = 0.08x + 63.96

We want to know at what distance will the cost be the same. So, set the equations equal to each other.

0.12x + 55.96 = 0.08x + 63.96

Combine the variables.

0.04x + 55.96 = 63.96

Combine the constants.

0.04x = 8

Divide by 0.04 to isolate x.

x = 200 miles

Check by plugging x back into each equation.

y = 0.12(200) + 55.96

y = 24 + 55.96

y = $79.96

y = 0.08(200) + 63.96

y = 16 + 63.96

y = $79.96

You are correct!

8 0
3 years ago
Billy is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How far is he
nikklg [1K]

He is  2√19 miles far way from starting point.

Suppose Billy starts at point A, turns at point B, and ends at point D, as shown below.

If Billy turns 60◦ northward and walks six miles, then we can draw a 30 − 60 − 90 triangle whose hypotenuse is 6 miles

It follows that Billy traveled 6/2 = 3 miles eastward during these 6 miles, and that he traveled 3√3 miles northward during these 6 miles. In total, Billy traveled 4 + 3 = 7 miles  eastward and 3√ 3 miles northward.

By the Pythagorean Theorem, the distance from his starting point is q:

(7)2 + (3√3)2

=√49 + 27

= √76

= 2√19 .

Learn more about Pythagorean Theorem on:

brainly.com/question/20545047

#SPJ4

5 0
2 years ago
Other questions:
  • B+11/3=-2 what number does b represent
    14·1 answer
  • In a multiple regression equation, two independent variables are considered, and the sample size is 30. The regression coefficie
    11·1 answer
  • Triangle abc is a right traingle if side ac=7 and side bc= 10 what is the measure of ab?
    11·1 answer
  • Why is 4 + (−3) equal to 1?
    9·1 answer
  • What’s the answer for IJG
    9·1 answer
  • Find the remainder when you divide <br><br> (−7) by 90.
    14·1 answer
  • Three students were asked to draw a diagram of the following situation and their responses are recorded below:
    14·1 answer
  • Television screens are usually described by the length of their diagonal measure. What would the diagonal measure be for a telev
    8·2 answers
  • The equation |p| = 2 represents the total number of points that can be earned or lost during one turn ofa game. Which
    11·1 answer
  • HELP!!!
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!