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

Chesley 2021-05-19 Answered

Find the point on the line \(y=5x+3\) that is closet to the origin.

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

mhalmantus
Answered 2021-05-20 Author has 18380 answers

Let (x,y) be the point on the line that is closet to the origine. Since, \(y=5x+3,\) so the point (x,y) on line becomes, \((x,5x+3)\) so now we use the distance dormula and minimize it.
Distance between (0,0) and \((x,5x+3)\) is given by
\(D=\sqrt{(x-0)^2+(5x+3-0)^2}\)
\(=\sqrt{x^2+(25x^2+30x+9)}\)
\(\Rightarrow D=\sqrt{26x^2+30x+9}\)
Now to minimize the distance, we find critical point by setting \(D'=0\). Here using chain rule, derivative is:
\(D'=\frac{1}{2}(26x^2+30x+9)^{-\frac{1}{2}}\cdot\frac{d}{dx}(26x^2+30x+9)\)
\(=\frac{1}{2\sqrt{26x^2+30x+9}}\cdot(26\cdot2x+30\cdot1+0)\)
\(\Rightarrow D'=\frac{52x+30}{2\sqrt{26x^2+30x+9}}\)
Now D'=0
\(\Rightarrow \frac{52x+30}{2\sqrt{26x^2+30x+9}}=0\)
\(\Rightarrow 52x+30=0\)
\(\Rightarrow x=-\frac{30}{52}=-\frac{15}{26}\)

Not exactly what you’re looking for?
Ask My Question
15
 

Expert Community at Your Service

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

Relevant Questions

asked 2021-05-14

Aidan knows that the observation deck on the Vancouver Lookout is 130 m above the ground. He measures the angle between the ground and his line of sight to the observation deck as \(77^{\circ}\). How far is Aidan from the base of the Lookout to the nearest metre?

asked 2021-05-08

The leg of a right triangle that lies on one ray of angle \(\theta\) is called the ? leg, and the leg that lies across triangle from \(\theta\) is called the ? leg.

asked 2021-06-13

Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to the top of the plateau is \(60^{\circ}\), how far are you from the base of the plateau?

asked 2020-12-27

Given theta is an acute angle such that sin theta = \(\displaystyle\frac{{5}}{{13}}\) find the value of \(\tan (\theta - \pi/4)\)

asked 2021-06-03
A classmate drew an acute triangle with sides 9 in. and 12 in. What is the greatest possible whole number that can be the length of the longest side of the triangle in inches? Provide evidence.
asked 2021-02-25

Can you draw two triangles each having two \(45^∘\) angles and one \(90^∘\) angle that are not similar? Justify your answer.

asked 2021-05-26
For a rope course and climbing wall, a guy wire R is attached 47 ft high on a vertical pole. Another guy wire S is attached 40 ft above the ground on the same pole. Find the angle α between the wires if they are attached to the ground 50 ft from the pole.
...