Question

Scuba divers know that the deeper they dive, the more light is absorbed by the water above them. On a dive, Petra's light meter shows that the amount of light available decreases by \(10 \%\) for every \(10 \mathrm{m}\) that she descends. a) Write the exponential function that relates the amount, \(L,\) as a percent expressed as a decimal, of light available to the depth, \(d,\) in \(10-\mathrm{m}\) increments. b) Graph the function. c) What are the domain and range of the function for this situation? d) What percent of light will reach Petra if she dives to a depth of \(25 \mathrm{m} ?\)

Short answer

The exponential function is \( L = (0.9)^d \). The domain is \( [0, \infty ) \) and the range is \( (0, 1] \). At 25 meters, about 70.7% of the light reaches Petra.

Step-by-step solution

Understand the Problem

Petra needs to find how light decreases with depth. We know that light decreases by 10% for every 10 meters descended.

Identify the Exponential Function

Since the decrease in light is exponential, start with the general form of an exponential function: \[L = L_0 \times (r)^d\]where: - \(L_0\) is the initial amount of light (1 or 100%)- \(r\) is the rate of decay per 10 meters (0.9 since it’s 90% of the light after each 10 meters)- \(d\) is the depth in 10-meter increments.

Write the Exponential Function

Based on the above, if the initial light is 100%, the function becomes:\[L = (0.9)^d\]

Graph the Function

Graph the function \(L = (0.9)^d\). Plot points for depth increments (e.g., 0, 10, 20,...) and their corresponding lights. For example:- At \( d = 0 \): \( L = (0.9)^0 = 1\)- At \( d = 1 \): \( L = (0.9)^1 = 0.9 \)- At \( d = 2 \): \( L = (0.9)^2 = 0.81 \)

Determine the Domain and Range

The domain represents all feasible depths Petra can dive into. Since depths can go from 0 meters onwards, the domain is: \( [0, \infty ) \)The range is the amount of light available. Since the function is continuously decreasing, the range is: \( (0, 1] \)

Calculate Light at 25 meters

Convert the depth into the corresponding increment (\( d = 25/10 = 2.5 \)), and substitute into the function:\[L = (0.9)^{2.5}\]Evaluate to find the amount of light available.

Solve for the Amount of Light at 25 meters

Calculate the value using a calculator:\[L \approx 0.9^{2.5} = 0.707\]Therefore, 70.7% of the light will reach Petra at a depth of 25 meters.

Key concepts

Exponential Functions

Exponential functions describe how quantities grow or shrink over time. They are defined by the equation \( f(x) = a \times (b)^x \), where \( a \) is the initial value, \( b \) is the base or growth/decay factor, and \( x \) is the exponent. In the context of light absorption, the exponential function helps determine how the amount of light decreases with increasing depth underwater. For Petra's dive, the function is given by \( L = 1 \times (0.9)^d \), where:

• 1 represents the initial light amount (100%).
• 0.9 is the decay factor, indicating that 90% of the light from the previous depth level remains.
• d is the depth in 10-meter increments.

This function models the rapid decrease in light as Petra dives deeper.

Rate of Decay

The rate of decay specifies how quickly a quantity decreases over time or space. In the case of exponential decay, a fixed percentage of the quantity is lost in each time step or depth increment. For light absorption during Petra's dive, the rate of decay is 10% per 10 meters. This means that every 10 meters, the available light decreases by 10%. The formula for the remaining light is: \[L = (0.9)^d\]
This shows that as the depth (d) increases, the value of \( L \) (light) decreases, reflecting the exponential decay. Understanding the rate of decay is crucial for predicting how much light will be available at various depths.

Function Graphing

Graphing the function helps visualize the relationship between depth and light. In this case, the function \( L = (0.9)^d \) can be graphed with depth \( d \) on the x-axis and light \( L \) on the y-axis.
To graph it:

• Start at depth \( d = 0 \), where \( L = (0.9)^0 = 1 \) (100% light).
• For \( d = 1 \), \( L = (0.9) = 0.9 \) (90% light).
• For \( d = 2 \), \( L = (0.9)^2 = 0.81 \) (81% light).
• Continue plotting for more depths.

The graph will show a decreasing curve, indicating that less light penetrates as the depth increases. This visual representation provides a clear understanding of the exponential relationship.

Domain and Range

The domain and range of a function describe the set of possible inputs and outputs, respectively.
For Petra's light absorption function:

• The domain is all possible depths Petra can dive, which starts at 0 meters and extends indefinitely. This is mathematically represented as \([ 0, \infty )\).
• The range is the set of possible light values. Since the light amount cannot exceed the initial value (100%) and approaches zero but never quite reaches it, the range is \(( 0, 1 ]\).

Understanding the domain and range helps in interpreting the function's limits and the extent of light penetration underwater. It clearly communicates that as depth increases, the light theoretically approaches zero but never becomes negative.