Question

Solve each related rate problem.6) Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. Theradius of the spill increases at a rate of 4 m/min. How fast is the area of the spillincreasing when the radius is 7 m?A) 63 m²/min B) 54rt m®/min C) 5211 m²/min D) 560 m²/min7) A spherical balloon is inflated so that its radius increases at a rate of 2 cm/sec. How fast

Answer 1

6.

We know area of circle is

$\pi r^2$

Radius increases at a rate of 4.

So, we can write:

$\frac{dr}{dt}=4$

We want rate of change of area. Thus,

$\begin{gathered} A=\pi r^2 \\ dA=2\pi\text{rdr} \\ \frac{dA}{dt}=2\pi r\frac{dr}{dt} \end{gathered}$

We want dA/dt at r = 7, so substituting all that we know:

$\begin{gathered} \frac{dA}{dt}=2\pi r\frac{dr}{dt} \\ \frac{dA}{\text{dt}}=2\pi(7)(4) \\ =56\pi \end{gathered}$

The correct answer: 56*pi m^2/min