Question

it has been estimated that 1000 curies of a radioactive substance introduced at a point on the surface of the open sea would spread over an area of 30,000 km² in 30 days. assuming that the area covered by the radioactive substance is a linear function of time t and is always circular in shape, express the radius r of the contamination as a function of t. r(t) =

Explanation:

Step1: Find the area - time relationship

Since the area covered is a linear function of time $t$. In 30 days, the area $A = 30000$ $km^{2}$. Let the area - time relationship be $A(t)=kt$. Substituting $t = 30$ and $A=30000$, we get $30000 = k\times30$, so $k = 1000$. Thus, $A(t)=1000t$.

Step2: Relate area and radius

The area of a circle is $A=\pi r^{2}$. Since $A(t)=1000t$, we have $\pi r^{2}=1000t$.

Step3: Solve for $r$ as a function of $t$

First, isolate $r^{2}$: $r^{2}=\frac{1000t}{\pi}$. Then take the square - root of both sides. Since $r\geq0$, we get $r(t)=\sqrt{\frac{1000t}{\pi}}$.

Answer:

$r(t)=\sqrt{\frac{1000t}{\pi}}$