Question

Find the point on the line
y = 3x + 5
that is closest to the origin

Answer 1

calculus?

Answer 2

algebra?

Answer 3

we can do either

Answer 4

calculus

Answer 5

i can only do calculus

Answer 6

I prefer the calculus way

Answer 7

First step find the distance from (x,3x+5) to (0,0)

Answer 8

You know the distance formula?

Answer 9

yeah its the distance= sqrt(X2-X1)^2(Y2-Y1)^2)

Answer 10

there should be a + between the x and y my bad

Answer 11

ok so input the points from our question
we are finding the distance from (x,y) to (0,0)
where it was given in your question y was 3x+5

Answer 12

so its distance= sqrt(X2-X)^2+(3x+5-y)^2)

Answer 13

You need to input the points (x,3x+5) and (0,0)

Answer 14

so sqrt(x^2-(3x+5)^2)

Answer 15

+?

Answer 16

yeah sorry i hate typing equations

Answer 17

\[d^2=x^2+(3x+5)^2 \\ \text{ let } f(x)=x^2+(3x+5)^2 \]
find f' then find the critical numbers

Answer 18

so f'= 2x +6(3x+5) then i would set that to 0 to find the critical points right?

Answer 19

yes

Answer 20

so the critical point is X= -20/30

Answer 21

you can reduce that

Answer 22

right to -2/3

Answer 23

and are you sure that is right?

Answer 24

Did you solve 20x+30=0 for x correctly?

Answer 25

Is not that if ax+b=0 then ax=-b then x=-b/a

Answer 26

duh i need to pay attention its -3/2

Answer 27

ok so you find the point by replacing x in y=3x+5 to get y

Answer 28

so y=3(-3/2)+5=?

Answer 29

y=1/2

Answer 30

so you have the point (-3/2,1/2) on y=3x+5 is closest to (0,0)

we could have use algebra...
y=3x+5
the perpendicular line will have slope -1/3
we want this perpendicular line to go through (0,0)
so we have the perpendicular line going through (0,0) is y=-1/3 *x
so find the intersection of y=3x+5 and y=-x/3 and we can find the point on y=3x+5 which has the shortest distance from (0,0)
so 3x+5=-x/3
multiply 3 on both sides
9x+15=-x
10x+15=0
10x=-15
x=-15/10=-3/2
so y=3x+5=3(-3/2)+5=-9/2+5=-9/2+10/2=1/2
so the point again is (-3/2,1/2)

Answer 31

thanks for all you help I'm getting most of the problems but everyone once and i while i get tripped in easy algebra thanks again and nice profile pic

Answer 32

lol thanks