Question

a satisfactory x - ray of the thoracic spine is made at 60 inches using automatic exposure control. the exposure time was 100 milliseconds. if the exam is repeated at 40 inches, what will the new exposure time be?
225 milliseconds
150 milliseconds
67 milliseconds
44 milliseconds

Explanation:

Step1: Recall the inverse - square law

The intensity of the x - ray is inversely proportional to the square of the distance. Let $I_1$ and $I_2$ be the intensities at distances $d_1$ and $d_2$ respectively, and $t_1$ and $t_2$ be the exposure times. Since the amount of radiation reaching the detector should be the same in both cases (for a satisfactory x - ray), $I_1t_1=I_2t_2$. And $I\propto\frac{1}{d^{2}}$, so $\frac{d_1^{2}}{d_2^{2}}=\frac{t_2}{t_1}$.

Step2: Substitute the given values

We have $d_1 = 60$ inches, $d_2=40$ inches, and $t_1 = 100$ milliseconds. Substituting into the formula $\frac{d_1^{2}}{d_2^{2}}=\frac{t_2}{t_1}$, we get $\frac{60^{2}}{40^{2}}=\frac{t_2}{100}$. First, calculate $\frac{60^{2}}{40^{2}}=\frac{3600}{1600}=\frac{9}{4}$. Then, $\frac{9}{4}=\frac{t_2}{100}$. Cross - multiply: $4t_2=9\times100$.

Step3: Solve for $t_2$

$t_2=\frac{9\times100}{4}=225$ milliseconds.

Answer:

225 milliseconds