Final answer:
The point (2,4) lies on the given curve as the substitution into the equation results in 0. The tangent and normal can be found by taking the derivative of the curve. The instantaneous rate of change of volume for a cone when r=3 is 12π cubic feet per foot.
Step-by-step explanation:
To show that the point (2,4) lies on the curve x³+y³-9xy=0, we substitute x=2 and y=4 into the equation and verify if the equation holds true.
Substitution: 2³ + 4³ - 9(2)(4) = 8 + 64 - 72 = 0, which verifies that (2,4) is indeed on the curve.
For the tangent and normal, we would need to find the derivative of the curve with respect to x (y’) and then find the slope of the tangent (y’ evaluated at x=2) and the normal (the negative reciprocal of the tangent slope at x=2).
The instantaneous rate of change of the volume V with respect to the radius r for a right circular cone when r=3 is given by the derivative of the volume formula with respect to r. We differentiate V(r) = (1/3)πr²h w.r.t. r, getting dV/dr = (1/3)π(2rh), and then we substitute h=6 and r=3 to find the instantaneous rate of change.
Derivation and substitution: dV/dr = (1/3)π(2)(6)r = 4πr, so at r=3, dV/dr = 4π(3) = 12π cubic feet per foot.