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?

The perimeter of a rectangle is fixed at $30\operatorname{\mathrm{cm}}$. If the length $L$ is increasing at a rate of $2\operatorname{\frac{\mathrm{cm}}{s}}$, for what value of $L$ does the area start to decrease? Hint: the rectangle's area starts to decrease when the rate of change for the area is less than 0.

A 15-foot plank leans against a vertical pole. The top of the plank begins to slide down the pole at a steady speed of 2 inches per second. How fast is the bottom of the plank moving away from the pole when it is 8 feet away from the base of the pole (in inches per second)?

Two cars leave the same intersection and drive in perpendicular directions. Car A travels east at a speed of $60\frac{mi}{hr}$, Car B travels north at a speed of $40\frac{mi}{hr}$. Car A leaves the intersection at $2pm$, while Car B leaves at $2:30pm$. Determine the rate at which the distance between the two cars is changing at $3pm$.