To solve the problem of finding the light intensity [tex]\( I \)[/tex] at a depth of 30 feet in Lake Heron using the Beer-Lambert law, let's break it down step by step.
### Given:
- [tex]\( k = 0.035 \)[/tex] (attenuation coefficient for Lake Heron)
- [tex]\( J = 10 \)[/tex] lumens (initial light intensity at the surface)
- [tex]\( x = 30 \)[/tex] feet (depth at which we want to find the light intensity)
The Beer-Lambert law is provided by the equation:
[tex]\[ \frac{-1}{k} \ln \left(\frac{I}{J}\right)=x \][/tex]
### Step-by-Step Solution:
1. Rewrite the Beer-Lambert Law in a usable form:
[tex]\[ \frac{-1}{k} \ln \left(\frac{I}{J}\right) = x \][/tex]
Multiply both sides by [tex]\( -k \)[/tex]:
[tex]\[ \ln \left( \frac{I}{J} \right) = -kx \][/tex]
2. Substitute the given values:
[tex]\[ \ln \left( \frac{I}{J} \right) = -0.035 \cdot 30 \][/tex]
Calculate the product on the right-hand side:
[tex]\[ \ln \left( \frac{I}{J} \right) = -1.05 \][/tex]
3. Solve for [tex]\( \frac{I}{J} \)[/tex]:
Take the exponential of both sides to remove the natural logarithm:
[tex]\[ \frac{I}{J} = e^{-1.05} \][/tex]
4. Solve for [tex]\( I \)[/tex]:
Multiply both sides by [tex]\( J \)[/tex]:
[tex]\[ I = J \cdot e^{-1.05} \][/tex]
Substitute [tex]\( J = 10 \)[/tex]:
[tex]\[ I = 10 \cdot e^{-1.05} \][/tex]
5. At this point, we already know from our calculations that the resulting intensity [tex]\( I \)[/tex] at a depth of 30 feet is approximately 3.499 lumens.
### Final Answer:
Therefore, the light intensity [tex]\( I \)[/tex] at a depth of 30 feet in Lake Heron is approximately 3.499 lumens.
