5. Applications of Derivatives

Given the equation below, find $dy/dt$ when $dx/dt=12$ and $x=\(\frac{15}{2}\)$.

$y=\(\sqrt{2x+1}\)$

Given the equation below, find $dz/dt$ when $dx/dt=3$, $dy/dt=2$, $x=2$, $y=4,$ and $z=1$.

$x^2+y^2-3z^2=125$

A sphere is growing at a rate of $50\(\frac{\operatorname{\mathrm{cm}\)}^3}{s}$. At what rate is the radius of the sphere increasing when the radius is $5\(\operatorname{\mathrm{cm}\)}$?

A right tringle has a base of $10\(\operatorname{\mathrm{cm}\)}$ and a height of $12\(\operatorname{\mathrm{cm}\)}$. The height of the right triangle is decreasing at a rate of $0.4\(\operatorname{\frac{\mathrm{cm}\)}{s}}$, at what rate is the area of the triangle decreasing?