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Illumination is the measure of the amount of light that hits a surface. It is often measured in candelas or lumens. Understanding how illumination works can help us make better decisions about lighting in different environments. For instance, whether we are installing streetlights or designing lighting for a living room, knowing how light intensity diminishes over distance is key. One important principle is the inverse square law: as you move away from a light source, the illumination decreases inversely with the square of the distance from the source. This means that if you double the distance from the light source, the illumination becomes one-fourth of the original value. This principle is essential in various fields like photography, astronomy, and even in everyday lighting setups.

Distance plays a crucial role in determining the level of illumination. In many physics and engineering problems, we use meters as the SI unit of distance.
In our exercise, we are given the illumination at \textbf{5 meters} to be 70 candela and need to find the illumination at \textbf{12 meters}.

The relationship between illumination and distance is captured by the inverse square law:
\[\text{I} = \frac{\text{k}}{\text{d}^2}\]
Here, we see that as we increase the distance, the illumination decreases exponentially.
For example:

• Illumination at 5 meters was calculated as 70 candelas.
• At 12 meters, the distance is more than doubled.
• By applying the formula, we use the inverse square relationship to find the new illumination.

This distance-illumination relationship helps us in planning lighting for large areas efficiently, ensuring sufficient lighting even at greater distances from the light source.

The constant \textbf{k} is an important value that characterizes the light source in our illumination formula.
To find \textbf{k}, we use initial conditions provided in the problem:

The illumination at 5 meters is 70 candela:

\[70 = \frac{k}{5^2}\]
Solve for \textbf{k} by rearranging and multiplying:

\[70 = \frac{k}{25}\] \[k = 70 \times 25\] \[k = 1750\]
This \textbf{k} value (1750 in this scenario) helps us predict the illumination at any other distance.

We use it to calculate illumination at 12 meters:

\[I = \frac{1750}{12^2} \] \[I = \frac{1750}{144}\]
\[I \approx 12.15 \text{ candela}\]
Understanding and calculating \textbf{k} is fundamental to translating the inverse square law into practical applications. It helps in scenarios from designing optimal lighting to setting up camera exposure settings for perfect photographs.