I've got some calculus questions...Please help, A.S.A.P...?

Question

1. Suppose that the radius and Volume (V=4/3 pi.r^3) of a sphere are differentiable functions of t. Write an equation that relates dV/dt to dr/dt.

2. If a and be are legs of a right triangle, and c is the length of the hypotenuse. How is dc/dt related to da/dt and db/dt?

3. The range of a projectile is related to the initial velocity V and projection angle (theta) by R=((v^2)sin2(theta)/g), (g is cons) How is dR/dt related to dv/dt and d(theta)/dt if neither v nor theta is cons.

Answer 1

1)

V = (4/3)π*r^3

dV/dt = 4πr² * dr/dt


2)

a² + b² = c²

d/dt c² = d/dt (a² + b²)

2c*dc/dt = 2a*da/dt + 2b*db/dt

2(c*dc/dt) = 2(a*da/dt + b*db/dt)

dc/dt = (a * da/dt + b * db/dt) / c


3)

R = v²sin(2θ)/g

(1/g)*dR/dt = (d/dt v² * sin(2θ)) + (v² * cos(2θ)*2*dθ/dt)

(1/g)*dR/dt = (2v*sin(2θ)*dv/dt) + (2*v²cos(2θ)*dθ/dt)

dR/dt = 2v[sin(2θ) * dv/dt + vcos(2θ) * dθ/dt] / g

Answer 2

These are all straight forward exercises.
My random number generator says 2, so I'll only answer that one.

good ol' pythagorean theorem gives you the relationship among a,b,c

a^2 + b^2 = c^2
differentiate with respect to t

dc/dt = [a*da/dt + b*db/dt ]/c