Questions: QUESTION 5 / 10 An x-ray exposure at 50 inches (127 cm) results in a beam intensity of 200 microgray ( μGy ). What is the new intensity for an x-ray exposure at 40 inches (102 cm)? 128 μGy 160 μGy 250 μGy 312 μGy

QUESTION 5 / 10 An x-ray exposure at 50 inches (127 cm) results in a beam intensity of 200 microgray ( μGy ). What is the new intensity for an x-ray exposure at 40 inches (102 cm)? 128 μGy 160 μGy 250 μGy 312 μGy
Transcript text: QUESTION $5 / 10$ An x-ray exposure at 50 inches $(127 \mathrm{~cm})$ results in a beam intensity of 200 microgray ( $\mu G y$ ). What is the new intensity for an $x$-ray exposure at 40 inches (102 cm )? $128 \mu \mathrm{~Gy}$ $160 \mu \mathrm{~Gy}$ $250 \mu \mathrm{~Gy}$ $312 \mu \mathrm{~Gy}$
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Solution

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Solution Steps

Step 1: Understand the Inverse Square Law

The intensity of x-ray exposure is inversely proportional to the square of the distance from the source. This relationship is described by the inverse square law:

\[ I_1 \cdot d_1^2 = I_2 \cdot d_2^2 \]

where \( I_1 \) and \( I_2 \) are the intensities at distances \( d_1 \) and \( d_2 \), respectively.

Step 2: Identify Known Values

From the problem, we know:

  • \( I_1 = 200 \, \mu\text{Gy} \)
  • \( d_1 = 50 \, \text{inches} \)
  • \( d_2 = 40 \, \text{inches} \)
Step 3: Apply the Inverse Square Law

Substitute the known values into the inverse square law equation:

\[ 200 \cdot 50^2 = I_2 \cdot 40^2 \]

Step 4: Solve for the New Intensity \( I_2 \)

Rearrange the equation to solve for \( I_2 \):

\[ I_2 = \frac{200 \cdot 50^2}{40^2} \]

Calculate \( I_2 \):

\[ I_2 = \frac{200 \cdot 2500}{1600} = \frac{500000}{1600} = 312.5 \, \mu\text{Gy} \]

Final Answer

The new intensity for an x-ray exposure at 40 inches is approximately \(\boxed{312 \, \mu\text{Gy}}\).

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