Calculus-Applied Optimization Problem:

posted by .

Find the point on the line 6x + 3y-3 =0 which is closest to the point (3,1).

Note: Your answer should be a point in the xy-plane, and as such will be of the form (x-coordinate,y-coordinate)

  • Calculus-Applied Optimization Problem: -

    what a stupid note. Of course it will be a point with (x,y) coordinates! This is a calculus class - you know all that already.

    Given a point (x,y) on the line, y=1-2x

    The distance from (x,1-2x) to (3,1) is

    d^2 = (x-3)^2 + (1-(1-2x))^2 = 5x^2-6x+9

    so, we want minimum d, when dd/dx = 0:

    2d dd/dx = 10x-6
    dd/dx = 10x-6/2sqrt(blah blah)

    dd/dx=0 when x = 3/5.

    So, minimum d^2 is

    5(3/5)^2 - 6(3/5) + 9 = 36/5
    minimum d is 6/√5

    Or, as we all know, the distance from a point (x,y) to a line ax+by+c=0 is

    |ax+by+c|/√(a^2+b^2) = (6(3)+3(1)-3)/√(36+9) = 18/√45 = 6/√5

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. AP Calculus

    Consider the curve given by x^2+4y^2=7+3xy a) Show that dy/dx=(3y-2x)/(8y-3x) b) Show that there is a point P with x-coordinate 3 at which the line tangent to the curve at P is horizontal. Find the y-coordinate of P. c) Find the value …
  2. CALC

    point on the hyperbola 9x^2−6y^2=10 closest to the point (0, 7). y coordinate of each point= ?
  3. Calculus

    Verify please and thank you~ Does the line (x-7)/3=(y+8)/6=(z+4)/5 and the point P(10,-2,1) determine a plane?
  4. Calculus and vectors

    Verify please and thank you~ 1. Does the line (x-7)/3=(y+8)/6=(z+4)/5 and the point P(10,-2,1) determine a plane?
  5. Math

    19. Point A(4, 2) is translated according to the rule (x, y) (x + 1, y – 5) and then reflected across the y-axis. a) In which quadrant of the coordinate plane is point A located?
  6. math:quick help

    Let Fn be the nth number in the Fibonacci Sequence. Consider the 3 points (F30,F31),(F32,F33),(F34,F35) in the Cartesian plane. You are allowed to repeatedly apply the following operation: Let P be any one of the three points in the …
  7. math geometry

    Please help so hard. My problem... In the coordinate plane point C lies on line segment AB. The coordinates for MN are (2,4) and (16,12). If the ratio of the length of AC to the length of CB is 3:4, what is the x-coordinate of point …
  8. Calculus Reposted

    A point is moving along the curve xy=12. When the point is at (4,3), the x-coordinate decreases at the rate of 2cm/sec. How fast is the y-coordinate changing at that point?
  9. Math

    1. How can you represent a loss of 24 dollars as an integer?
  10. Math

    1. How can you represent a loss of 24 dollars as an integer?

More Similar Questions