Find pt. on 6x + y = 9 closest to (-3, 1) in terms of y

[gopher]

Find the point on the line 6x + y = 9 closest to the point (-3, 1) in terms of y

\(\displaystyle y=9-6x\)

Distance Formula:

\(\displaystyle d=sqrt{(x+3)^2 + (y-1)^2}\)

\(\displaystyle d(x)=sqrt{(x+3)^2 + (9-6x-1)^2}\)

\(\displaystyle d(x)=sqrt{37x^2-90x+73}\)

\(\displaystyle f(x)=d(x)^2=37x^2-90x+73\)

derivitive:

\(\displaystyle f'(x)=74x-90\)

Set the derivative equal to 0:

\(\displaystyle 0=74x-90\)

\(\displaystyle x=45/37\)

That is how far I can get by looking at the examples in the book. Then I get kinda lost... Any ideas? Thank you!

 

[skeeter]

6x + y = 9

slope of this line is m = -6

a perpendicular line has slope m = 1/6

the line passing through (-3,1) with slope m = 1/6 is

y - 1 = (1/6)(x + 3) edit mistake ... I wrote y + 1 instead of y - 1 on the left side of the linear equation
y = (1/6)x + (3/2)

the point of intersection of the line y = (1/6)x + (3/2) and 6x + y = 9 is ...

9 - 6x = (1/6)x + (3/2)
54 - 36x = x + 9
45 = 37x

x = 45/37, y = 9 - 6(45/37) = 63/37 now we should all agree

 

[gopher]

that was simple

thanks

 

[galactus]

I get the same thing as gopher.

When you are maximizing or minimizing a distance, there is a trick you can use to avoid radicals. The distance and the square of the distance have their max and min at the same point. Therefore, the minimum occurs at

\(\displaystyle \L\\S=L^{2}=(x+3)^{2}+(\underbrace{(9-6x)}_{\text{y}}-1)^{2}\)

\(\displaystyle \L\\S=37x^{2}-90x+73\)

\(\displaystyle \L\\S'=74-90x\)

\(\displaystyle \L\\x=\frac{45}{37}, \;\ y=\frac{63}{37}\)


Excuse me, skeet, but I think the y-intercept of that line should be 3/2, not 1/2.
Hence the discrepancy. \(\displaystyle y=\frac{1}{6}x+\frac{3}{2}\)
gives the same result as the calc way.

 

[gopher]

very good thank you both

 

[gopher]

ok im stuck again
Find the dimensions of a rectangle with area 1000m^2 whose perimeter is small as possible.


\(\displaystyle Area=x*y\)
\(\displaystyle 1000=x*y\)
\(\displaystyle y=1000x\)

\(\displaystyle Permiter=2x*2y\)

im stuck on how these releate to each other