Final answer:
To calculate the needed panel thickness to absorb 99.0% of the rays, one must use the exponential attenuation formula related to the initial and final intensities of the rays and the thickness of the initial material. The desired thickness is directly proportional to the ratio of natural logarithms of the initial and target intensity fractions. By plugging in the known values, one can find the required thickness in centimeters.
Step-by-step explanation:
To find out how thick of a panel of the same material is needed to absorb 99.0% of the rays, we use the principles of exponential attenuation of radiation. According to these principles, the thickness required to absorb a certain percentage of radiation is directly proportional to the natural logarithm of the inverse of the remaining intensity fraction. We can use the initial data provided, where a 1.48-cm panel absorbs 90.3% of the rays, to calculate the thickness needed for 99.0% absorption.
Let's define:
- I_0 as the initial intensity of the radioactive rays
- I as the final intensity of the radioactive rays
- t_1 as the initial thickness (1.48 cm)
- t_2 as the thickness required for the desired absorption
The formula for the thickness based on exponential attenuation is:
t_2 = t_1 * (ln(I_0/I)/ln(I_0/I_1))
Where:
- I_0/I is the fraction of rays we want to transmit through the second panel (0.01 for 99% absorption)
- I_0/I_1 is the fraction of rays transmitted through the first panel (0.097 for 90.3% absorption)
By substituting the known values, we can solve for t_2:
t_2 = 1.48 * (ln(1/0.01)/ln(1/0.097))
Finally, after calculating the natural logarithms and the multiplication, we get the value of t_2, which is the necessary thickness to achieve 99.0% absorption, in centimeters.