$A$ particle is moving along the curve $x = at^2 + bt + c$. If $ac = b^2$,then the particle would be moving with uniform

If a metallic circular plate of radius $50 \, cm$ is heated so that its radius increases at the rate of $1 \, mm$ per hour,then the rate at which the area of the plate increases (in $cm^2/hour$) is (in $\pi$)

If the area of a circle increases at a uniform rate,then prove that the rate of change of its perimeter varies inversely as the radius.

$A$ spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute,then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is:

Two ships $A$ and $B$ are sailing straight away from a fixed point $O$ along routes such that $\angle AOB$ is always $120^o$. At a certain instance,$OA = 8 \ km$,$OB = 6 \ km$ and the ship $A$ is sailing at the rate of $20 \ km/hr$ while the ship $B$ is sailing at the rate of $30 \ km/hr$. Then the distance between $A$ and $B$ is changing at the rate (in $km/hr$):

The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = x^2 + 6x + 5$. The marginal revenue,when $x = 20$,is . . . . . . .