In calm waters oil spilling from the ruptured hull of a grounded tanker spreads in all directions. If the area polluted is a circle and its radius is increasing at a rate of 5 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft.

Question

In calm waters oil spilling from the ruptured hull of a grounded tanker spreads in all directions. If the area polluted is a circle and its radius is increasing at a rate of 5 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft. ​

Quizplus · Accepted Answer

The area of a circle is given by $A = \pi r^2$, where $r$ is the radius. The rate of change of the area with respect to time ($\frac{dA}{dt}$) can be found by differentiating both sides with respect to $t$, giving $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. Given $\frac{dr}{dt} = 5$ ft/sec and $r = 40$ ft, substituting these values gives $\frac{dA}{dt} = 2\pi(40)(5) = 400\pi$ ft²/sec.

In Calm Waters Oil Spilling from the Ruptured Hull of a Grounded