Optimization with solids and distance. Finding closest point

Question

Find the point on the line  which is closest to the point 

Answer 1

The Algebra 2 way to solve this problem would be to define the line containing the point(-5, -4) that is orthogonal to the given line, then find where these 2 lines intersect. If you wish to use this method, we will have

3x+7y = 3 (the given line)

7x-3y = 7(-5) -3(-4) = -23 (the orthogonal line)

Solving

21x + 49y = 21

21x - 9y = -69

58y = 90

y = 45/29

3x + 7(45/29) = 3

3x = -228/29

x =-76/29

The Calculus way: get an equation for the distance of the given point from all points on the line and minimize the distance squared (equivalent to minimizing distance)

3x + 7y - 3 = 0

7y = -3x + 3

y = -3/7x + 3/7

Find the distance between (x, -3/7x + 3/7) and (-5, -4)

(x - -5)^2 + ((-3/7x +3/7) - -4)^2 =

(x + 5)^2 + (-3/7x +31/7)^2 =

x^2 + 10x + 25 + 9/49x^2 - 186/49x + 961/49 =

58/49 x^2 + 304/49x + 2186/49

f'(x) = 116/49 x + 304/49

116/49 x = -304/49

x = -76/29

y = -3/7x + 3/7 = 228/203 + 87/203 = 315/203 = 45/29

Calculus and Algebra 2 agree!