Question

radiation machines, used to treat tumors, produce an intensity of radiation that varies inversely as the square of the distance from the machine. at 3 meters, the radiation intensity is 62.5 milliroentgens per hour. what is the intensity at a distance of 1.9 meters? the intensity is milliroentgens per hour. (round to the nearest tenth as needed.)

Explanation:

Step1: Establish inverse - square relationship formula

Let $I$ be the intensity of radiation and $d$ be the distance. The relationship is $I=\frac{k}{d^{2}}$, where $k$ is a constant. We know that when $d = 3$ meters, $I=62.5$ milliroentgens per hour. Substitute these values into the formula to find $k$: $62.5=\frac{k}{3^{2}}$.

Step2: Solve for the constant $k$

Multiply both sides of the equation $62.5=\frac{k}{9}$ by 9. So, $k = 62.5\times9=562.5$.

Step3: Find the intensity at a new distance

Now we want to find $I$ when $d = 1.9$ meters. Substitute $k = 562.5$ and $d = 1.9$ into the formula $I=\frac{k}{d^{2}}$. We get $I=\frac{562.5}{1.9^{2}}$.

Step4: Calculate the intensity value

First, calculate $1.9^{2}=3.61$. Then, $I=\frac{562.5}{3.61}\approx155.8$.

Answer:

$155.8$