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The way that the intensity of light changes as it goes through a material is given by the Beer-Lambert law (below). In the equation, [tex] J [/tex] and [tex] I [/tex] are the intensity before and after going through the material, [tex] x [/tex] is the distance (in feet) traveled through the material, and [tex] k [/tex] is a constant depending on the material. For Lake Heron, [tex] k = 0.035 [/tex] and the light intensity at the surface is [tex] J = 10 [/tex] lumens. Find the light intensity, [tex] I [/tex], at a depth of 30 feet in Lake Heron.

[tex] \frac{-1}{k} \ln \left(\frac{I}{J}\right) = x [/tex] (Beer-Lambert Law)

A. [tex] I = 10 e^{(-0.035 \cdot 30)} [/tex]
B. [tex] I = 30 e^{(-0.035 \cdot 10)} [/tex]
C. [tex] I = 0.035 e^{(-10 \cdot 30)} [/tex]
D. [tex] I = 10 e^{(0.035 \cdot 30)} [/tex]



Answer :

GinnyAnswer
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.

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