Question

Light penetrates" to a depth of about 15 feet in a typical Long Island lake. Assume that we are using a Secchi disk here that is barely visible when light intensity is reduced to about 0.10 (one tenth) of its value at the surface. If the decline in intensity is exponential, find the value of the decay constant in units of inverse feet. At what depth will the light intensity be reduced to 0.025 of the surface intensity? See notes and see the calculation we did for problem set One. (That problem had exponential increase, this one has exponential decrease.)

Answer 1

The decay constant for light intensity in water is found to be approximately 0.184 inverse feet, and the depth at which the light intensity will be 0.025 times the surface intensity is approximately 29.96 feet.

Step-by-step explanation:

The student's question involves determining the decay constant for the reduction of light intensity with depth in water, which is characterized by an exponential decay function. By definition, the intensity at a depth of 15 feet is 0.10 times the surface intensity. If we use the general form of the exponential decay I(d) = I0 * e^(-kd), where I(d) is the intensity at depth d, I0 is the surface intensity (which we can take as 1 for simplification), k is the decay constant, and e is the base of the natural logarithm, then we can substitute the known values to find k.

At 15 feet (d = 15), I(d) = 0.10, so 0.10 = e^(-15k). By taking the natural logarithm of both sides and solving for k, we get k =~ 0.184 feet-1. To find the depth at which the intensity is reduced to 0.025, we set I(d) = 0.025 and solve for d, which gives us a depth of approximately 29.96 feet.