Question

light intensity ($i$) is proportional ($\propto$) to the inverse square of distance ($d$) of a subject from a light source. the relationship in intensities for subjects at different distances from the same source can be likewise seen as a ratio or proportional relationship.

$i\propto\frac{1}{d^{2}}$

a proportional relationship can also be described as a percentage. you will now find the proportional relationship between two subjects, or subject a at $d_1$ and subject b at $d_2$. find the percentage of intensity experienced by subject b relative to subject a. in other words:

$\frac{\frac{1}{d_1^{2}}}{\frac{1}{d_2^{2}}}=\frac{a}{b}=$

solve the equation and convert the fraction to a percentage. use the value of 2 for $d_1$ and the value of 10 for $d_2$. your answer will essentially be the intensity of light for subject a as a rate of the intensity of light for subject b, so do not forget to convert to a percentage.

round the percentage, however, to the nearest hundredth if necessary. you do not need to include a sign for percentage. only the number, rounded to the hundredth, will be necessary.

Explanation:

Step1: Set up the proportion

Since $I\propto\frac{1}{d^{2}}$, the proportion of intensities $\frac{I_A}{I_B}=\frac{\frac{1}{d_1^{2}}}{\frac{1}{d_2^{2}}}=\frac{d_2^{2}}{d_1^{2}}$. Given $d_1 = 2$ and $d_2=10$, we substitute these values into the formula: $\frac{I_A}{I_B}=\frac{10^{2}}{2^{2}}$.

Step2: Calculate the fraction

$\frac{10^{2}}{2^{2}}=\frac{100}{4}=25$.

Step3: Convert to percentage

To convert the ratio to a percentage, we multiply by 100. So the percentage of intensity of light for subject A relative to subject B is $25\times100 = 2500\%$.

Answer:

2500