# Thread: Related rates and implicit

1. ## Related rates and implicit

hi i had two questions first one

A spherical balloon is being filled with helium at a rate of 8cm^3/s at what rate is the radius increasing when the radius is 12cm, when the volume is 1435cm^3 and when it has bee filling for 33.5 sec

the angle between two intersecting curves is defined as the angle between their tangents at the point of intersection. If this angel is 90 degrees, the two curves are said to be orthogonal. prove that the curves defined by x^2-y^2=k and xy=p intersect orthogonally for all values of the constants k and p. Illustrate your proof with a sketch .

Thanks

2. hi i had two questions first one

A spherical balloon is being filled with helium at a rate of 8cm^3/s at what rate is the radius increasing when the radius is 12cm
The volume of a sphere is $\frac{4}{3}{\pi}r^{3}$

Differentiate: $\frac{dV}{dt}=4{\pi}r^{2}\cdot\frac{dr}{dt}$

Plug in your knowns and solve for dr/dt. dV/dt=8 and r=12.

When it has been filling for 33.5 seconds, its volume is 268 cm^3. 8(33.5)=268

Therefore, at that time its radius is $268=\frac{4}{3}{\pi}r^{3}$

Solve for r and plug all the info into the dV/dt formula and solve for dr/dt

3. ## Solutions

See attachment

For a complete discussion of Orthognal trajectories you can see the Differential Equations Page on my website Calculus Animations