Find the rate of change of the distance between the origin and a moving point on the graph y = x^2 + 8 if dx/dt = 8 centimeters per second.

Answer 1

y = x^2 + 8
dy/dx = 2 x
dy/dt = dy/dx * dx/dt
dy/dt = 2x * 8 = 16 x

s = distance from origin
s^2 = x^2+y^2
2 s ds/dt = 2 x dx/dt +2 y dy/dt
so
s ds/dt = x dx/dt + y dy/dt
so
s ds/dt = [8 x + (x^2+8)(16 x)]

ds/dt = [8 x + (x^2+8)(16 x)]/[x^2 +(x^2+8)^2]^.5

= [16 x^3 + 136 x] / [x^2 +x^4 +16x^2 + 64]^.5

= 8(2x^3 + 17x) / (x^4 + 17 x^2 +64)^.5

Now if we knew x and if my arithmetic is right, we would know the speed component away from the origin at x.