Question

1. in a body of water, each foot of water screens out 60% of the light above it. the equation (i = 1 - 0.40^{d}) represents this situation.

a. explain what the 1 and the 0.40 represent in this situation.
b. what percent of light is available after passing through 5 feet of water? show your thinking.
unit 1 lesson 5 9

Explanation:

a.

Step1: Analyze the meaning of 1

The 1 represents the initial or total amount of light before passing through any water. It is the light intensity at the water - surface level, considered as 100% or 1 in relative terms.

Step2: Analyze the meaning of 0.40

Since each foot of water screens out 60% of the light above it, the proportion of light that passes through each foot of water is \(1 - 0.60=0.40\). So 0.40 is the fraction of light that remains after passing through 1 foot of water.

b.

Step1: Identify the formula and the value of \(x\)

We are given the formula \(I = 0.40^{x}\), and we want to find the amount of light available after \(x = 5\) feet of water.

Step2: Substitute \(x = 5\) into the formula

\[I=0.40^{5}=0.40\times0.40\times0.40\times0.40\times0.40 = 0.01024\]

Step3: Convert the decimal to a percentage

To convert the decimal to a percentage, we multiply by 100. So \(I = 0.01024\times100 = 1.024\%\)

Answer:

a. 1 represents the initial amount of light (100% of the light at the water - surface), 0.40 represents the fraction of light that remains after passing through 1 foot of water.
b. 1.024%