Question

A small footpath is shaped like the parabola y = x^2 − 9 on the domain [−3, 3]. There is a statue located at the point P = (0, −4). Use calculus methods to find the coordinates of the points on the path that are closest to the statue and the coordinates of the points on the path that are farthest away from the statue. Make sure to carefully explain your reasoning.

Answer 1

distance is maximum at coordinates (−3, 0) , (3, 0) and minimum at distance (0,-9)

Explanation:

since the distance to the statue is

D² = (x-x₀)²+ (y-y₀)²

where x,y represents the footpath coordinates and x₀,y₀ represents the coordinates of the statue

and

y= x²-9 , for x [−3, 3]

x² = y+9

thus

D² = x²+ y²

D² = y+9 +y²

since D² is minimised when d is minimised, then

the change in distance with y is

d (D²)/dy = 2*D*d(D)/dy =2*D*( 1+2*y)

d (D²)/dy =2*D*( 1+2*y)

since D>0 , d (D²)/dy >0 for y> -1/2

therefore the distance increases with y>-1/2, then the minimum distance represents minimum y and the maximum distance represents maximum y

since

y= x²-9 for [−3, 3]

y is maximum at x=−3 and x=3 → y=0

and minimum for x=0 → y=-9

then

distance is maximum at coordinates (−3, 0) , (3, 0) and minimum at distance (0,-9)