Math Problem Statement
The intensity L(x) of light x feet beneath the surface of the ocean satisfies the differential equation StartFraction dL Over dx EndFraction equalsminuskL. From experience, a diver knows that diving to 19 ft in a sea cuts the intensity in half. He cannot work without artificial light when the intensity falls below one-fifth of the surface value. About how deep can he expect to work without artificial light? Question content area bottom Part 1 He can work approximately enter your response here ft deep without artificial light.
Solution
We are given that the intensity of light at a depth feet beneath the surface satisfies the differential equation:
where is a constant. The general solution to this differential equation is:
where is the intensity of light at the surface (i.e., ).
Step 1: Use the given information about intensity at 19 feet
We know that at feet, the intensity is half of the surface intensity. This gives the equation:
Substituting into the general solution:
We can cancel from both sides (since ):
Taking the natural logarithm of both sides:
This simplifies to:
Thus,
Step 2: Find the depth where intensity falls below one-fifth of the surface value
We want to find the depth at which the intensity falls to one-fifth of the surface intensity. That is, we need to solve for in:
Substitute this into the general solution:
Cancel from both sides:
Taking the natural logarithm of both sides:
This simplifies to:
Thus,
Step 3: Substitute the value of
We found earlier that , so substitute this into the equation:
Now, calculate this value:
Final Answer:
The diver can work approximately 44.2 feet deep without artificial light.
Would you like further details or have any questions?
Here are some related questions to expand on this topic:
- What happens to the intensity of light at greater depths?
- How can the constant be derived experimentally?
- What other factors might influence how deep a diver can go without artificial light?
- How does the rate of change of light intensity differ in other mediums, such as air or water with different properties?
- How would the solution change if the intensity decreased at a different rate?
Tip: Understanding exponential decay like this one can be useful in many fields, such as physics and biology, where processes like radioactive decay, population growth, or absorption follow similar patterns.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Exponential Decay
Logarithms
Formulas
General solution of first-order linear differential equation: L(x) = L_0 e^(-kx)
Natural logarithm to solve for k: k = ln(2)/19
Depth calculation using logarithmic equation: x = ln(5)/k
Theorems
Exponential Decay Law
Properties of Logarithms
Suitable Grade Level
College-Level Calculus
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