Find the point on the line y = 3x +

vadulgattp 2021-11-11 Answered
Find the point on the line y = 3x + 4 that is closest to the origin.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Susan Yang
Answered 2021-11-12 Author has 20 answers

Let's consider an arbitrary point p(x,y)
The distance from P to the origin is:
d=x2+y2
we notice that this the objective function. since p is on the line y=3x+4 the coordinates of P satisties this equation. This is the constrain for the objective function in (1).
Therefore, the objective function can be written as a function of one variable:
d=d(x)=x2+(3x+4)2
From the first devirative tes, we have that
d(x)=0
d2d(x)ddx(x2+(3x+4)2)=0
2x+2(3x+4)(3)=0
2x+18x+24=0
20x=24
x=2420=65
Since d(x)<0 for x<65 and d(x)>0 for x65, the minimum distance from p to the origin is achieved when x=65
Then, y=3x+4=3(65)+y
y=18+205=25
we found that, P(65,25) is the point on the line y=3x+4 closest to the origin.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions